$g\coloneqq g^{(M,P)}=g^{(M^{\prime},P^{\prime})}$. Then $M\cong M^{\prime}$
and $P\cong P^{\prime}$.

###### Proof.

We repeatedly reduce the problem to the largest basic direct summands and show
that they coincide. Write $M\cong B_{1}^{M}\oplus M_{1}$, where $B_{1}^{M}$ is
a basic direct summand of $M$ such that
$\operatorname{add}B_{1}^{M}=\operatorname{add}M$. Similarly, write
$M^{\prime}\cong B_{1}^{M^{\prime}}\oplus M_{1}^{\prime}$, such that
$B_{1}^{M^{\prime}}$ is basic and
$\operatorname{add}B_{1}^{M^{\prime}}=\operatorname{add}M^{\prime}$. In the
same way, write $P\cong Q_{1}^{P}\oplus P_{1}$ and $P^{\prime}\cong
Q_{1}^{P^{\prime}}\oplus P_{1}^{\prime}$ with $Q_{1}^{P}$ and
$Q_{1}^{P^{\prime}}$ largest basic direct summands. 6.12 implies
$\mathcal{C}_{(B_{1}^{M},Q_{1}^{P})}=\mathcal{C}_{(M,P)}$ and
$\mathcal{C}_{(B_{1}^{M^{\prime}},Q_{1}^{P^{\prime}})}=\mathcal{C}_{(M^{\prime},P^{\prime})}$.
Since $g$ is contained in the intersection of these cones, 6.11 implies that
$B_{1}^{M}\cong B_{1}^{M^{\prime}}$ and $Q_{1}^{P}\cong Q_{1}^{P^{\prime}}$.
We repeat the process with $g_{1}\coloneqq g-g^{(B_{1}^{M},Q_{1}^{P})}$ and
the $\tau$-rigid pairs $(M_{1},P_{1})$ and $(M_{1}^{\prime},P_{1}^{\prime})$.
Since the modules are finitely generated, eventually $M\cong M^{\prime}$ and
$P\cong P^{\prime}$. ∎

###  6.3. $\tau$-cluster morphism category

The definition of the category of a partitioned fan, 3.2, is inspired by the
geometric construction of the $\tau$-cluster morphism category from the
$g$-fan [46]. In the original constructions (cf. [16], [17], [30]) the objects
of the category are wide subcategories of the form
$\mathcal{W}_{E}\coloneqq\overline{\mathcal{T}}_{E}\cap\overline{\mathcal{F}}_{E}$
for some TF-equivalence class $E=\mathcal{C}_{(M,P)}$ coming from a
$\tau$-rigid pair $(M,P)$. The relationship between TF-equivalence classes and
the $g$-fan is described in 6.7. Let $\operatorname{TF}_{A}$ be the poset
category of TF-equivalence classes coming from $\tau$-rigid pairs with poset
relation $E_{1}\leq E_{2}$ whenever $E_{1}\subseteq\overline{E_{2}}$.

######  Definition 6.17.

[46, Sec. 2.1.4] Let $A$ be a finite-dimensional algebra. The $\tau$-cluster
morphism category $\mathfrak{C}(A)$ is the category

  * •

whose objects are objects of $\operatorname{TF}_{A}$ under the identification
$E_{1}\sim E_{2}$ whenever $\mathcal{W}_{E_{1}}=\mathcal{W}_{E_{2}}$;

  * •

whose morphisms $\operatorname{Hom}_{\mathfrak{C}(A)}([E],[F])$ are given by
the set of morphisms

$\bigcup_{E^{\prime}\in[E],F^{\prime}\in[F]}\operatorname{Hom}_{\operatorname{TF}_{A}}(E^{\prime},F^{\prime})$

under the identification $f_{EF}\sim f_{E^{\prime}F^{\prime}}$ whenever
$\pi_{E}(F)=\pi_{E^{\prime}}(F^{\prime})$.

For this category, the identification of cones implies the geometric
properties of possible identification at the start of Section 3, see [46, Sec.
3], which in turn imply that composition is well-defined. The identifications
of cones in the construction of the $\tau$-cluster morphism category are made
with respect to a wide subcategory coming from a brick [20], in such a way
that identified cones lie in the same hyperplane whose normal vector is the
dimension vector of the brick [15]. Since maximal cones correspond to torsion
classes, we may label an arrow of $\operatorname{Hasse}(\operatorname{tors}A)$
with the brick whose hyperplane is crossed [20]. Originally, the picture group
of an algebra was defined in [32] for representation-finite hereditary
algebras, and has since been extended to all $\tau$-tilting finite algebras in
[27] as follows.

######  Definition 6.18.

[27, Prop. 4.4a] Let $A$ be $\tau$-tilting finite. The picture group $G(A)$ is
generated by $\\{X_{S}:S\in\operatorname{brick}A\\}$ with a relation

$X_{S_{1}}\dots X_{S_{k}}=X_{S_{1}^{\prime}}\dots X_{S_{l}^{\prime}}$

if there exist torsion classes
$\mathcal{T},\mathcal{T}^{\prime}\in\operatorname{tors}A$ such that
$(S_{1},\dots,S_{k})$ and $(S_{1}^{\prime},\dots,S_{l}^{\prime})$ are
sequences of bricks labelling directed paths
$\mathcal{T}\to\mathcal{T}^{\prime}$ in
$\operatorname{Hasse}(\operatorname{tors}A)$, denote this element by
$X_{[\mathcal{T},\mathcal{T}^{\prime}]}$.

######  Proposition 6.19.

Let $A$ be $\tau$-tilting finite. Let
$\mathfrak{C}(A)=\mathfrak{C}(\Sigma,\mathfrak{P}_{\textnormal{WAC}})$, then
$G(\Sigma(A),\mathfrak{P}_{\textnormal{WAC}},\leq_{\operatorname{tors}A})$ is
isomorphic to $G(A)$, where $(\Sigma,\leq_{\operatorname{tors}A})$ is the fan
poset induced by $\operatorname{tors}A$.

###### Proof.

Each wide subcategory which implies identifications of codimension 1 cones is
given by exacty one brick [20, Thm. 3.3]. Hence equivalence classes of
codimension 1 cones are in bijection with $\operatorname{brick}A$ and thus the
generators of the two groups coincide. It follows that the set of relations of
$G(A)$ coincide with the set of type 1 relations for
$G(\Sigma(A),\mathfrak{P}_{\textnormal{WAC}},\leq_{\operatorname{tors}A})$ due
to the bijection between maximal cones and torsion classes in the
$\tau$-tilting finite case [19, Thm. 3.8]. The type 2 relations of
$G(\Sigma(A),\mathfrak{P}_{\textnormal{WAC}},\leq_{\operatorname{tors}A})$
follow from [20, Thm. 4.12, Prop. 4.13] which establishes a label-preserving
isomorphism between the dual graphs of the stars of cones whose corresponding
wide subcategories coincide and that of their shared projection. ∎

Therefore 4.7 is a generalisation of 6.18. We also remark that by [27, Thm.
4.10], the picture group $G(A)$ is isomorphic to the fundamental group of
$\mathcal{B}\mathfrak{C}(A)$. We conclude by applying the theory developed in
the previous sections to obtain a new algebra for which the classifying space
of the $\tau$-cluster morphism category is a $K(\pi,1)$ space for its picture
group.

######  Theorem 6.20.

Let $A=KQ/I$ be the Brauer cycle algebra of rank 3, which is given by

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\qquad I=\langle
ab,bc,ca,de,ef,fd,af-dc,be-fa,cd-eb\rangle.$

The $g$-fan is precisely the fan $\Sigma(A)=\Sigma_{\mathcal{H}}$ underlying
the hyperplane arrangement of 4.24. Moreover, the classifying space
$\mathcal{B}\mathfrak{C}(A)$ is a $K(\pi,1)$ space for the picture group
$G(A)$.

###### Proof.

This algebra is representation-infinite but $\tau$-tilting finite [1, Thm.
6.7][47, Thm. 6.1], hence it contains a finite number of bricks [19, Thm. 4.2]
which in this case are given by the representations:

${\tiny\begin{array}[]{cc}i\end{array}},\quad{\tiny\begin{array}[]{cc}i\\\
i+1\end{array}},\quad{\tiny\begin{array}[]{cc}i+1\\\
i\end{array}},\quad{\tiny\begin{array}[]{ccc}i-1\text{\phantom{xxx}}i+1\\\
i\end{array}},\quad{\tiny\begin{array}[]{ccc}i\\\
i-1\text{\phantom{xxx}}i+1\end{array}},$

for $i\in\mathbb{Z}_{3}$. It is easy to see that these bricks have exactly
dimension vectors of the normal vectors of $\mathcal{N}$ in 4.24 which define
the hyperplane arrangement $\mathcal{H}$. Now in our case it is easy to see
each set of bricks with the same dimension vector join together to form the
whole hyperplanes. In particular, [38, Sec. 4] connects the stability spaces
of bricks in the wall-and-chamber structure with the shards of the hyperplane
arrangement with respect to the base region given by the positive orthant. In
other words,
$\mathfrak{C}(\Sigma_{\mathcal{H}},\mathfrak{P}_{\operatorname{shard}})\cong\mathfrak{C}(A)$.
Hence by 5.5 there exists a faithful functor to the picture group of
$\mathfrak{C}(\Sigma_{\mathcal{H}},\mathfrak{P}_{\operatorname{flat}})$.
Moreover, [9, Thm. 1] implies that $\mathfrak{C}(A)$ has the pairwise
compatibility property for last factors since it has 3 simples. Thus, 4.15
implies that $\mathcal{B}\mathfrak{C}(A)$ is a $K(\pi,1)$ and [27, Thm. 4.10]
implies that $\pi$ is the picture group $G(A)$. ∎

Similar results were obtained for the $\tau$-cluster morphism category of
other special classes of algebras, see [26], [27], and [30]. However, the
algebra of 6.20 is the first example of an algebra of rank greater than 2
whose wall-and-chamber structure is a hyperplane arrangement satisfying the
$K(\pi,1)$ property. In dimension 4 and higher, the $\tau$-cluster morphism
categories of preprojective algebras, whose wall-and-chamber structure is also
a hyperplane arrangement by [37], no longer satisfy the pairwise compatibility
of last factors [9]. In general the question whether
$\mathcal{B}\mathfrak{C}(A)$ is a $K(\pi,1)$ for all $\tau$-tilting finite
algebras is still open.

### Acknowledgements

The author is thankful to Sibylle Schroll and Hipolito Treffinger for their
guidance and many meaningful conversations and suggestions. The author thanks
Eric J. Hanson whose comments helped greatly to improve this paper. The author
would also like to thank Edmund Heng, Lang Mou, Aran Tattar and José Vivero
for inspiring discussions. Finally, the author thanks Bethany Marsh whose
question was the starting point of this paper.

### Funding

MK is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) – Project-ID 281071066 – TRR 191.

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# The $\circ$ operation and $*$ operation of fan graphs

Guangjun Zhu∗, Yijun Cui, Yulong Yang and Yi yang  School of Mathematical
Sciences, Soochow University, Suzhou, Jiangsu, 215006, P. R. China
zhuguangjun@suda.edu.cn(Corresponding author:Guangjun Zhu),
237546805@qq.com(Yijun Cui), 1975992862@qq.com(Yulong Yang),
3201088194@qq.com(Yi Yang).

###### Abstract.

Let $G$ be a finite simple graph on the vertex set $V$ and let $I_{G}$ denote
its edge ideal in the polynomial ring $S={\mathbb{K}}[x_{V}]$. In this paper,
we compute the depth and the Castelnuovo–Mumford regularity of $S/I_{G}$ when
$G=F_{k}^{W}(K_{n})$ is a $k$-fan graph, or $G=G_{1}\circ G_{2}$ or
$G=G_{1}*G_{2}$ is the graph obtained from fan graphs $G_{1}$, $G_{2}$ by
$\circ$ operation or $*$ operation, respectively.

Keywords: Regularity, depth, fan graph, $\circ$ operation, $*$ operation

Mathematics Subject Classification. Primary 13C15, 13A15, 13D02; Secondary
05E40

∗ Corresponding author

## Introduction

For any two monomial ideals $I$ and $J$, it is known that
$\operatorname{reg}(I+J)\leq\operatorname{reg}(I)+\operatorname{reg}(J)-1$,
see [14] and [8]. Suppose we are given a finite simple graph $G=(V(G),E(G))$
with the vertex set $V(G)$ and the edge set $E(G)$ and its subgraphs
$G_{1}=(V(G_{1}),E(G_{1}))$ and $G_{2}=(V(G_{2}),E(G_{2}))$ such that
$E(G)=E(G_{1})\cup E(G_{2})$, then for the edge ideal the above inequalities
imply
$\operatorname{reg}(I_{G})\leq\operatorname{reg}(I_{G_{1}})+\operatorname{reg}(I_{G_{2}})-1$.
Herzog et al. in [10] introduced the concept of splitting graphs of a graph.
For two finite simple graphs $G=(V(G),E(G))$ and
$G^{\prime}=(V(G^{\prime}),E(G^{\prime}))$, if there exists a surjective map
$\alpha:V(G^{\prime})\rightarrow V(G)$ such that
$\alpha(e):=\\{\alpha(u),\alpha(v)\\}$ is an edge of $G$ for all edges
$e=\\{u,v\\}$ of $G^{\prime}$, and such that the map $E(G^{\prime})\rightarrow
E(G)$, $e\mapsto\alpha(e)$ is bijective. We call $G^{\prime}$ a splitting
graph of $G$ and $\alpha$ a splitting map of $G$. They proved in [10, Theorem
1.3 and Proposition 1.6] that
$\operatorname{reg}(I_{G})\leq\operatorname{reg}(I_{G^{\prime}})$ if
$\operatorname{reg}(I_{G})=\vartheta(G)+1$ where $\vartheta(G)$ is the induced
matching number of $G$, or the splitting map $\alpha:V(G^{\prime})\rightarrow
V(G)$ is special. They also proved that for path graphs and cycle graphs of
even length,
$\operatorname{depth}(S_{G^{\prime}}/I_{G^{\prime}})\geq\operatorname{depth}(S_{G}/I_{G})$,
where $S_{G}$ and $S_{G^{\prime}}$ are the polynomial rings over a field
${\mathbb{K}}$ in the variables corresponding to $V(G)$ and $V(G^{\prime})$,
respectively.

In this paper we are interested in whether formulas for the regularity and
depth of $S_{G}/I_{G}$ can be accurately determined in terms of this
information of $S_{G_{1}}/I_{G_{1}}$ and $S_{G_{2}}/I_{G_{2}}$ for a splitting
graph $G^{\prime}:=G_{1}\sqcup G_{2}$ of graph $G$ and subgraphs of
$G^{\prime}$. So far, there are no results, and it turns out to be a very hard
problem.

Bolognini et al. introduced in [2] a class of chordal graphs
$F_{k}^{W}(K_{n})$, called $k$-fan graphs, and a family of simple graphs
obtained from some graphs by the $*$ operation or the $\circ$ operation (see
Definition 3.1). In Section $2$, we give some formulas for the dimension,
depth and regularity of the quotient ring of the edge ideal of a $k$-fan graph
$F_{k}^{W}(K_{n})$. If $G=(G_{1},f_{1})*(G_{2},f_{2})$ where $f_{i}$ is a leaf
of $G_{i}$ for $i=1,2$. We set $G^{\prime}=G_{1}\sqcup G_{2}$ and define a map
$\alpha:V(G^{\prime})\rightarrow V(G)$ by $\alpha(f_{1})=\alpha(f_{2})=f$ and
$\alpha(u)=u$ with $u\in V(G^{\prime})\backslash\\{f_{1},f_{2}\\}$. It is
clear that $G^{\prime}$ is a splitting graph of $G$. If
$G=(G_{1},f_{1})\circ(G_{2},f_{2})$ where $f_{i}$ is a leaf of $G_{i}$ and
$v_{i}$ is its neighbor vertex in $G_{i}$ for $i=1,2$. We set
$G^{\prime}=(G_{1}\backslash f_{1})\sqcup(G_{2}\backslash f_{1})$ and define a
map $\alpha^{\prime}:V(G^{\prime})\rightarrow V(G)$ by
$\alpha^{\prime}(v_{1})=\alpha^{\prime}(v_{2})=v$ and $\alpha^{\prime}(u)=u$
with $u\in V(G^{\prime})\backslash\\{v_{1},v_{2}\\}$. It is also clear that
$G^{\prime}$ is a splitting graph of $G$ and $G^{\prime}$ is also an induced
subgraph of $G_{1}\sqcup G_{2}$. In Section $3$, we give some formulas for the
depth and regularity of some graphs obtained from the fan graphs by the two
special gluing operations mentioned earlier.

##  1\. Preliminary

In this section, we gather together the needed definitions and basic facts,
which will be used throughout this paper. However, for more details, we refer
the reader to [3, 9, 16].

###  1.1. Notions of simple graphs

For any finite simple graph $G=(V(G),E(G))$, with a set of vertices $V(G)$ and
a set of edges $E(G)$, we define some graph-theoretic notions as follows.

For a vertex $v\in V(G)$, its _neighborhood_ is defined as
$N_{G}(v)=\\{u\,|\,\\{u,v\\}\in E(G)\\}$ and its degree, denoted by
$\deg_{G}(v)$, is $|N_{G}(v)|$. Set $N_{G}[v]=N_{G}(v)\cup\\{v\\}$. For
$A\subset V(G)$, $G|_{A}$ denotes the _induced subgraph_ of $G$ on the set
$A$, i.e., for $i,j\in A$, $\\{i,j\\}\in E(G|_{A})$ if and only if
$\\{i,j\\}\in E(G)$. For $W\subseteq V(G)$, we denote by $G\backslash W$ the
induced subgraph of $G$ on $V(G)\setminus W$. For a vertex $v\in V(G)$, we
denote by $G\backslash v$ the induced subgraph of $G$ on the set
$V(G)\backslash\\{v\\}$ for simplicity.

A subset $M\subset E(G)$ is a _matching_ of $G$ if $e\cap
e^{\prime}=\emptyset$ for any pair of edges $e,e^{\prime}\in M$. A matching
$M$ of $G$ is called an _induced matching_ if the induced subgraph at the
vertices of $M$ contains no edges other than those already contained in M. The
_induced matching number_ of $G$, denoted by $\vartheta(G)$, is the maximum
size of an induced matching in $G$.

A _complete graph_ (or clique) on $n$ vertices is a graph where for any two
vertices there is an edge connecting them. It is denoted by $K_{n}$. A _cycle_
of length $n$ in a graph is a closed walk along its edges,
$\\{x_{1},x_{2}\\},\\{x_{2},x_{3}\\},\ldots,\\{x_{n-1},x_{n}\\},\\{x_{n},x_{1}\\}$,
such that $x_{i}\neq x_{j}$ for $i\neq j$. We denote the cycle on $n$ vertices
by $C_{n}$. A _chord_ in the cycle $C_{n}$ is an edge $\\{x_{i},x_{j}\\}$
where $x_{j}\neq x_{i-1},x_{i+1}$. A graph is said to be _chordal_ if it has a
chord for every cycle of length greater than or equal to $4$. It is clear that
a complete graph is a chordal graph. for every cycle of length greater than or
equal to $4$ there is a chord. It is clear that a complete graph is a chordal
graph.

For a simple graph $G=(V(G),E(G))$, the subset $C\subseteq V(G)$ is called a
_vertex cover_ of $G$ if $\\{x_{i},x_{j}\\}\in E(G)$ then $x_{i}\in C$ or
$x_{j}\in C$. A vertex cover $C$ of $G$ is called _minimal_ if every proper
subset of $C$ is not a vertex cover of $G$.

###  1.2. Algebraic preliminaries

Let $S={\mathbb{K}}[x_{1},\ldots,x_{n}]$ be a polynomial ring over a field
${\mathbb{K}}$. Let $M$ be a graded $S$-module with minimal free resolution

$0\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{p,j}}\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{p-1,j}}\rightarrow\cdots\rightarrow\bigoplus\limits_{j}S(-j)^{\beta_{0,j}}\rightarrow
M\rightarrow 0,$

where the maps are exact, $p\leq n$, and $S(-j)$ is the free module obtained
by shifting the degrees in $S$ by $j$. The numbers $\beta_{i,j}$’s are
positive integers and are called the $(i,j)$th graded Betti number of $M$. Two
very important homological invariants related to these numbers are the
Castelnuovo-Mumford regularity (or simply regularity) and the depth, denoted
by $\operatorname{reg}(M)$ and $\operatorname{depth}(M)$ respectively:

$\displaystyle\operatorname{reg}(M)$ $\displaystyle=\mbox{max}\,\\{j-i\ |\
\operatorname{Tor}_{i}(M,{\mathbb{K}})_{j}\neq 0\\}$
$\displaystyle\operatorname{depth}(M)$ $\displaystyle=\mbox{min}\,\\{i\ |\
\operatorname{Ext}^{i}({\mathbb{K}},M)\neq 0\\}.$

The following lemmas are often used to compute the depth and regularity of a
module or ideal. In particular, since the facts in Lemma 1.1 are well-known,
they are used implicitly in this paper.

######  Lemma 1.1.

Let $M,N$ be two finitely generated graded $S$-modules. Then,

  * (1)

$\operatorname{depth}(M\oplus
N)=\min\\{\operatorname{depth}(M),\operatorname{depth}(N)\\}$, and

  * (2)

$\operatorname{reg}(M\oplus
N)=\max\\{\operatorname{reg}(M),\operatorname{reg}(N)\\}$.

######  Lemma 1.2.

([11, Lemmas 2.1 and 3.1]) Let $0\longrightarrow M\longrightarrow
N\longrightarrow P\longrightarrow 0$ be an exact sequence of finitely
generated graded $S$-modules. Then we have

  1. (1)

$\operatorname{depth}\,(M)\geq\min\\{\operatorname{depth}\,(N),\operatorname{depth}\,(P)+1\\}$,
the equality holds if $\operatorname{depth}\,(N)\\\
\neq\operatorname{depth}\,(P)$.

  2. (2)

$\operatorname{reg}\,(M)\leq\max\\{\operatorname{reg}\,(N),\operatorname{reg}\,(P)+1\\}$,
the equality holds if $\operatorname{reg}\,(N)\neq\operatorname{reg}\,(P)$.

######  Lemma 1.3.

([11, Lemma 2.2, Lemma 3.2]) Let $S_{1}={\mathbb{K}}[x_{1},\dots,x_{m}]$ and
$S_{2}={\mathbb{K}}[x_{m+1},\dots,x_{n}]$ be two polynomial rings over
${\mathbb{K}}$, $I\subset S_{1}$ and $J\subset S_{2}$ be two non-zero
homogeneous ideals. Let $S=S_{1}\otimes_{\mathbb{K}}S_{2}$. Then we have

  1. (1)

$\operatorname{reg}\,(S/(I+J))=\operatorname{reg}\,(S_{1}/I)+\operatorname{reg}\,(S_{2}/J)$;

  2. (2)

$\operatorname{depth}\,(S/(I+J))=\operatorname{depth}\,(S_{1}/I)+\operatorname{depth}\,(S_{2}/J)$;

######  Lemma 1.4.

([7, Lemma 1.3]) Let $I\subset S$ be a proper non-zero homogeneous ideal. Then

$\operatorname{reg}\,(I)=\operatorname{reg}\,(S/I)+1.$

For a simple graph $G=(V(G),E(G))$, its edge ideal is defined as follows:

$I_{G}=(x_{i}x_{j}\mid\\{x_{i},x_{j}\\}\in E(G))\subset S],$

where $S={\mathbb{K}}[x_{V(G)}]$ is the polynomial ring over a field
${\mathbb{K}}$ and $x_{V(G)}=\\{x_{i}|i\in V(G)\\}$.

For a monomial ideal $I$, let $\mathcal{G}(I)$ be its unique minimal set of
monomial generators. For a subset $A\subset[n]$, we set $x_{A}:=\\{x_{i}|i\in
A\\}$ for simplicity. The following lemma is very important for the whole
paper.

######  Lemma 1.5.

Let $G=(V,E)$ be a simple connected graph. Let
$J=(x_{N_{G}(v)})+I_{G\backslash N_{G}[v]}$ and $K=(x_{v})+I_{G\backslash v}$,
where $v\in V$. Then,

  1. (1)

$J+K=(x_{N_{G}[v]})+I_{G\backslash N_{G}[v]}$;

  2. (2)

$I_{G}=J\cap K$;

  3. (3)

$\operatorname{depth}(S/J)=\operatorname{depth}(S/(J+K))+1$;

  4. (4)

$\operatorname{reg}(S/J)=\operatorname{reg}(S/(J+K))$.

###### Proof.

(1) Since $\mathcal{G}(I_{G\backslash
v})\subseteq(x_{N_{G}(v)})+\mathcal{G}(I_{G\backslash N_{G}[v]})$, we have

$\displaystyle J+K=(x_{N_{G}(v)})+I_{G\backslash
N_{G}[v]}+(x_{v})+I_{G\backslash v}=(x_{N_{G}[v]})+I_{G\backslash N_{G}[v]}.$

(2) It is clear that $I_{G}\subseteq J\cap K$. On the other hand, since
$(x_{N_{G}(v)})\cap(x_{v})\subseteq I_{G}$, both $G\backslash v$ and
$G\backslash N_{G}[v]$ are induced subgraphs of $G$. We have

$\displaystyle J\cap K$ $\displaystyle=[(x_{N_{G}(v)})+I_{G\backslash
N_{G}[v]}]\cap[(x_{v})+I_{G\backslash v}]$
$\displaystyle=(x_{N_{G}(v)})\cap(x_{v})+(x_{N_{G}(v)})\cap I_{G\backslash
v}+I_{G\backslash N_{G}[v]}\cap(x_{v})+I_{G\backslash N_{G}[v]}\cap
I_{G\backslash v}$ $\displaystyle\subseteq I_{G}.$

(3) and (4) follow from (1).

######  Lemma 1.6.

([5, Corollary 4.6]) Let $G$ be a complete graph on the set $[n]$, then

$\operatorname{depth}\,(S/I_{G})=\operatorname{reg}\,(S/I_{G})=1.$

######  Lemma 1.7.

([17, Theorem 3.3, Corollary 3.3]) Let $n\geq 2$ be an integer and $P_{n}$ be
a path graph on the set $[n]$, then

$\operatorname{depth}(S/I_{G})=\lceil\frac{n}{3}\rceil\text{\ and\
}\operatorname{reg}(S/I_{G})=\lfloor\frac{n+1}{3}\rfloor$

where $\lceil\frac{n}{3}\rceil$ is the smallest integer $\geq\frac{n}{3}$ and
$\lfloor\frac{n+1}{3}\rfloor$ is the largest integer $\leq\frac{n+1}{3}$.

######  Lemma 1.8.

([6, Corollary 6.9]) If $G$ is a chordal graph, then
$\operatorname{reg}(S_{G}/I_{G})=\vartheta(G)$, where $\vartheta(G)$ is the
induced matching number of $G$.

##  2\. Study of $F_{k}^{W}(K_{n})$

Bolognini et al. in [2] introduced the fan graphs of complete graphs, which
are a family of chordal graphs. In this section, we will study the dimension,
depth and regularity of the quotient ring of the edge ideal of these graphs.
We start with the definition reformulated in [12] and [15].

######  Definition 2.1 ([15, Definition 3.1]).

Let $K_{n}$ be a complete graph on the set $[n]$.

  1. (1)

Let $U=\\{u_{1},u_{2},\ldots,u_{r}\\}$ be a subset of $[n]$. Suppose that for
each $i\in[r]$, a new complete graph $K_{a_{i}}$ with $a_{i}>i$ is attached to
$K_{n}$ in such a way that $V(K_{n})\cap
V(K_{a_{i}})=\\{u_{1},u_{2},\dots,u_{i}\\}$. We say that the resulting graph
is obtained by _adding a fan_ to $K_{n}$ on the set $U$ and
$\\{K_{a_{1}},\ldots,K_{a_{r}}\\}$ is the _branch_ of that fan on $U$.

  2. (2)

Let $W$ be a subset of $[n]$ and suppose that $W=W_{1}\sqcup\cdots\sqcup
W_{k}$ is a partition of $W$. Let $F^{W}_{k}(K_{n})$ be a graph obtained from
$K_{n}$ by adding a fan on each $W_{i}$. The resulting graph
$F^{W}_{k}(K_{n})$ is called a _$k$ -fan graph_ of $K_{n}$ on the set $W$. For
future reference, for each $i\in[k]$, we assume that
$\\{K_{a_{i,1}},\ldots,K_{a_{i,r_{i}}}\\}$ is the branch of the fan on
$W_{i}=\\{w_{i,1},\ldots,w_{i,r_{i}}\\}$. For notational convenience, we also
set $h_{i,j}:=a_{i,j}-j$.

  3. (3)

A branch $\\{K_{a_{i,1}},\ldots,K_{a_{i,r_{i}}}\\}$ of the fan on $W_{i}$ is
called a _pure branch_ if $h_{i,j}=1$ for every $j\in[r_{i}]$. Furthermore, if
each branch is pure, then $F^{W}_{k}(K_{n})$ is said to be a _$k$ -pure fan
graph_ of $K_{n}$ on $W$.

  4. (4)

The complete graph $K_{n}$ can be considered as a degenerate fan without
branches, i.e., with $k=0$.

######  Example 2.2.

Below are examples of a $2$-fan graph and a $1$-fan graph.

$2-$$fan\,graph$$1-$$fan\,graph$

Figure $1$ Example of $2$-fan and $1$-fan graphs

For a positive integer $n\geq 2$ we set $[n]=\\{1,\ldots,n\\}$. We need the
following lemma.

######  Lemma 2.3.

([9, Lemma 9.1.4]) Let $G$ be a simple graph on the set $[n]$. A subset
$C=\\{i_{1},\ldots,i_{r}\\}\subset[n]$ is a vertex cover of $G$ if and only if
the prime ideal $P_{C}=(x_{i_{1}},\ldots,x_{i_{r}})$ contains $I_{G}$. In
particular, $C$ is a minimal vertex cover of $G$ if and only if $P_{C}$ is a
minimal prime ideal of $I_{G}$.

The following theorem provides an exact formula for the dimension of the
quotient ring of the edge ideal of a $k$-fan graph.

######  Theorem 2.4.

Let $G=F_{k}^{W}(K_{n})$ be a $k$-fan graph of the complete graph $K_{n}$ on
the set $W\subseteq[n]$ with $n\geq 2$ and $W=W_{1}\sqcup\cdots\sqcup W_{k}$
be a partition of $W$. Then

$\dim(S/I_{G})=\begin{cases}|W|+1&\text{if $W\subsetneq[n]$};\\\ n&\text{if
$W=[n]$}.\end{cases}$

###### Proof.

Let $I_{G}=P_{1}\cap\cdots\cap P_{m}$ be the irredundant primary decomposition
of $I_{G}$, where $P_{i}=(x_{i_{1}},x_{i_{2}},\ldots,x_{i_{r_{i}}})$ for
$i\in[m]$, then $\dim(S/I_{G})=|V(G)|-\min\\{r_{i}:i\in[m]\\}$ and each
$C_{i}=\\{i_{1},i_{2},\ldots,i_{r_{i}}\\}$ is a minimal vertex cover of $G$ by
Lemma 2.3. Therefore, to compute the dimension of $S/I_{G}$, we only need to
find the vertex cover of G with the smallest size. It is known that for a
complete graph $H$, the set $C\subsetneq V(H)$ is its minimal vertex cover if
and only if $|C|=|V(H)|-1$. For each $i\in[k]$, let
$W_{i}=\\{w_{i,1},\ldots,w_{i,r_{i}}\\}$ and
$\\{K_{a_{i,1}},\ldots,K_{a_{i,r_{i}}}\\}$ be the branch of the fan on
$W_{i}$. We distinguish between the following two cases:

(i) If $W\subsetneq[n]$, we choose $u_{i,j}\in V(K_{a_{i,j}})\backslash W_{i}$
for each $i\in[k],j\in[r_{i}]$ and $v\in[n]\backslash W$. Let
$C_{i,j}=V(K_{a_{i,j}})\backslash\\{u_{i,j}\\}$,
$C^{\prime}=V(K_{n})\backslash\\{v\\}$ and
$C=(\bigcup\limits_{i=1}^{k}\bigcup\limits_{j=1}^{r_{i}}C_{i,j})\cup
C^{\prime}$. Then it is clear that $u_{i,j},v\notin C$ and
$|C|=|V(G)|-\sum\limits_{i=1}^{k}|W_{i}|-1=|V(G)|-|W|-1$.

Claim: $C$ is a vertex cover of $G$ of minimum size.

Indeed, for any $e\in E(G)$, we have $e\in E(K_{n})$ or $e\in E(K_{a_{i,j}})$
for some $i\in[k]$, $j\in[r_{i}]$. It is clear that $e\cap C\neq\emptyset$ by
the choice of $C$, which implies that $C$ is a vertex cover of $G$. On the
other hand, if there exists some vertex cover $D$ of $G$ with
$|D|<|V(G)|-|W|-1$. Since there are exactly $(|W|+1)$ maximal cliques $K_{n}$
and $K_{a_{i,j}}$ for $i\in[k]$, $j\in[r_{i}]$ in $G$, there must be an edge
$\\{u,v\\}\in E(G)$ such that $u,v\notin D$, which contradicts with the fact
that $D$ is a vertex cover of $G$.

(ii) If $W=[n]$, we choose $u_{i,j}\in V(K_{a_{i,j}})\backslash W_{i}$ for
every $i\in[k],j\in[r_{i}]$, $C_{i,j}=V(K_{a_{i,j}})\backslash\\{u_{i,j}\\}$
and $C=\bigcup\limits_{i=1}^{k}\bigcup\limits_{j=1}^{r_{i}}C_{i,j}$. So
$|C|=|V(G)|-|W|$. In this case, we can obtain that $C$ is a vertex cover of
$G$ with the minimum size, using similar arguments as in (i). So in this case
we get $\dim(S/I_{G})=n$ in this case. And this completes the proof.

For a monomial $u\in S$, we set $\operatorname{supp}(u)=\\{x_{i}:x_{i}|u\\}$.
For a monomial ideal $I$ with $\mathcal{G}(I)=\\{u_{1},\ldots,u_{m}\\}$, we
set
$\operatorname{supp}(I)=\bigcup\limits_{i=1}^{m}\operatorname{supp}(u_{i})$.
We also denote by $S_{G}$ the polynomial rings in the variables corresponding
to $V(G)$ when more graphs are involved.

######  Proposition 2.5.

Let $G=F_{1}^{W}(K_{n})$ be a $1$-fan graph of the complete graph $K_{n}$ on
the set $W\subseteq[n]$ with $n\geq 2$. Then

$\operatorname{depth}\,(S/I_{G})=1.$

###### Proof.

Let $V(K_{n})\cap V(K_{a_{1}})=\\{w_{1},\ldots,w_{r_{1}}\\}$. We apply
induction on $|W|$. If $|W|=1$, then $V(K_{n})\cap V(K_{a_{1}})=\\{w_{1}\\}$.
In this case, let $J=(x_{N_{G}(w_{1})})$ and
$K=(x_{w_{1}})+I_{K_{n-1}}+I_{K_{a_{1}-1}}$, then
$\operatorname{depth}(S/J)=1$, $\operatorname{depth}(S/K)=2$ by Lemma 1.6,
since
$\\{x_{w_{1}}\\}\cap\operatorname{supp}(I_{K_{n-1}})\cap\operatorname{supp}(I_{K_{a_{1}-1}})=\emptyset$.
Meanwhile, we have $I_{G}=J\cap K$ and $J+K=(x_{V(G)})$. So
$\operatorname{depth}(S/(J+K))=0$. Hence the desired result holds by using
Lemma 1.1(1) and Lemma1.2(1) to the following exact sequence

(1) $0\longrightarrow\frac{S}{J\cap
K}\longrightarrow\frac{S}{J}\oplus\frac{S}{K}\longrightarrow\frac{S}{J+K}\longrightarrow
0,$

Now, we assume that $|W|\geq 2$. Let $J=(x_{N_{G}(w_{1})})$ and
$K=(x_{w_{1}})+I_{G\backslash w_{1}}$, then $J+K=(x_{V(G)})$. It follows that
$\operatorname{depth}(S/(J+K))=0$ and $\operatorname{depth}(S/J)=1$. Since
$G\backslash w_{1}$ is the disjoint union $K_{a_{1}-1}\sqcup
F_{1}^{W\backslash\\{w_{1}\\}}(K_{n-1})$, we have
$K=(x_{w_{1}})+I_{K_{a_{1}-1}}+I_{G\backslash V(K_{a_{1}})}$. Since
$|W\backslash\\{w_{1}\\}|=|W|-1$, we know by induction that

$\operatorname{depth}(S/K)=\operatorname{depth}(S_{K_{a_{1}-1}}/I_{K_{a_{1}-1}})+\operatorname{depth}(S_{G\backslash
V(K_{a_{1}})}/I_{G\backslash V(K_{a_{1}})})=1+1=2.$

Again applying Lemma 1.2(1) and Lemma 1.5(2) to the exact sequence (1), we get
the desired result.

######  Theorem 2.6.

Let $G=F_{k}^{W}(K_{n})$ be a $k$-fan graph of the complete graph $K_{n}$ on
the set $W\subseteq[n]$ with $n\geq 2$ and $W=W_{1}\sqcup\cdots\sqcup W_{k}$
be a partition of $W$. Then

$\operatorname{depth}(S/I_{G})=1+|W|-\max\\{|W_{1}|,|W_{2}|,\ldots,|W_{k}|\\}.$

###### Proof.

We prove the statement by induction on $k$ and $n$. The case for $k=1$ is
shown in Proposition 2.5. If $n=k=2$, then $|W_{1}|=|W_{2}|=1$. Let
$V(K_{n})\cap V(K_{a_{i,1}})=W_{i}$ with $W_{i}=\\{w_{i,1}\\}$ for $i=1,2$. In
this case, $G$ is a graph obtained by joining two complete graphs
$K_{a_{1,1}}$ and $K_{a_{2,1}}$ with a common edge $\\{w_{1,1},w_{2,1}\\}$. If
$K_{a_{1,1}}=K_{a_{2,1}}=K_{2}$, then in this case $G$ is a path with $4$
vertices. So we get $\operatorname{depth}(S/I_{G})=2$ by Lemma 1.7. Now, we
may assume that $K_{a_{2,1}}$ is a clique with at least three vertices. In
this case, we choose $J=(x_{N_{G}(w_{1})})+I_{G\backslash N_{G}[w_{1}]}$,
$K=(x_{w_{1}})+I_{G\backslash w_{1}}$ with $w_{1}=w_{1,1}$, then, by Lemma
1.5, we get that $I_{G}=J\cap K$,
$\operatorname{depth}\,(S/J)=\operatorname{depth}\,(S/(J+K))+1$ and
$J+K=(x_{N_{G}[w_{1}]})+I_{G\backslash N_{G}[w_{1}]}$, where $G\backslash
N_{G}[w_{1}]=K_{a_{2,1}-1}$ is a clique with at least two vertices. Hence
$\operatorname{depth}(S/(J+K))=1$. Note that $G\backslash w_{1}$ is the
disjoint union $K_{a_{1,1}-1}\sqcup K_{a_{2,1}}$, we have
$\operatorname{depth}(S/K)=2$. By Lemma 1.2(1) and the exact sequence (1), we
get $\operatorname{depth}(S/I_{G})=2$, which confirms the claim in this case.

Now we can assume that $n\geq 3$ and $k\geq 2$. Suppose that
$\max\\{|W_{1}|,\ldots,|W_{k}|\\}=T$ and $T=|W_{1}|$. Let $V(K_{n})\cap
V(K_{a_{1,r_{1}}})=\\{w_{1,1},\ldots,w_{1,r_{1}}\\}$. Let
$J=(x_{N_{G}(w_{1})})+I_{G\backslash N_{G}[w_{1}]}$,
$K=(x_{w_{1}})+I_{G\backslash w_{1}}$ where $w_{1}=w_{1,1}$, then $I_{G}=J\cap
K$, $\operatorname{depth}\,(S/J)=\operatorname{depth}\,(S/(J+K))+1$ and
$J+K=(x_{N_{G}[w_{1}]})+I_{G\backslash N_{G}[w_{1}]}$ by Lemma 1.5. We
distinguish between the following two cases:

(i) If $T=1$, then $|W_{i}|=1$ for each $i\in[k]$. In this case, $G\backslash
N_{G}[w_{1}]$ has $(k-1)$ connected components consisting of some cliques and
isolated vertices. Hence $\operatorname{depth}(S/(J+K))=k-1$ by Lemma 1.6. At
the same time, $G\backslash w_{1}$ is the disjoint union $K_{a_{1,1}-1}\sqcup
F_{k-1}^{W\backslash\\{w_{1}\\}}(K_{n-1})$, so we have
$\operatorname{depth}(S/K)=1+(k-1)=k$ by induction, Lemma 1.3(2) and Lemma
1.6. Applying Lemma 1.2(1) and Lemma 1.5(2) to the exact sequence (1), we get
$\operatorname{depth}(S/I_{G})=k=1+|W|-T$, as wished.

(ii) If $T\geq 2$, i.e., $|W_{1}|\geq 2$, then $G\backslash N_{G}[w_{1}]$ has
$\sum\limits_{i=2}^{k}|W_{i}|$ connected components, which in this case
consist of some cliques and isolated vertices in this case. Thus
$\operatorname{depth}(S/(J+K))=\sum\limits_{i=2}^{k}|W_{i}|=|W|-T$ by Lemma
1.6. Meanwhile, $G\backslash w_{1}$ is the disjoint union $K_{a_{1,1}-1}\sqcup
F_{k-1}^{W\backslash\\{w_{1}\\}}(K_{n-1})$, so we have
$K=(x_{w_{1}})+I_{K_{a_{1,1}-1}}+I_{G\backslash V(K_{a_{1,1}})}$. By
induction, Lemma 1.3(2) and Lemma 1.6, we obtain

$\displaystyle\operatorname{depth}(S/K)=$ $\displaystyle
1+[1+(T-1)+\sum\limits_{i=2}^{k}|W_{i}|-\max\\{T-1,|W_{2}|,\ldots,|W_{k}|\\}]$
$\displaystyle=$ $\displaystyle
1+\sum\limits_{i=1}^{k}|W_{i}|-\max\\{T-1,|W_{2}|,\ldots,|W_{k}|\\}$
$\displaystyle\geq$ $\displaystyle 1+|W|-T.$

Again applying Lemma 1.2(1) and Lemma 1.5(2) to the exact sequence (1), we can
obtain the desired result.

######  Lemma 2.7.

([1, Lemma 3.5]) Let $G$ be a simple graph and $H$ be its induced subgraph.
Then $\operatorname{reg}(I_{H})\leq\operatorname{reg}(I_{G})$.

######  Theorem 2.8.

Let $G=F_{k}^{W}(K_{n})$ be a $k$-fan graph of the complete graph $K_{n}$ on
the set $W\subseteq[n]$ with $n\geq 2$ and $W=W_{1}\sqcup\cdots\sqcup W_{k}$
be a partition of $W$. Suppose $p=|\\{h_{i,j}:h_{i,j}\geq 2\\}|$. If $|W|\geq
n-1$ and $h_{i,r_{i}}\geq 2$ for each $i\in[k]$, then
$\operatorname{reg}(S/I_{G})=p$. Otherwise, we have
$\operatorname{reg}(S/I_{G})=p+1$.

###### Proof.

Since $G$ is a chordal graph, we get
$\operatorname{reg}(S/I_{G})=\vartheta(G)$ by Lemma 1.8. Let
$M=\\{e_{1},\ldots,e_{\vartheta(G)}\\}$ be an induced matching of $G$. It
suffices to show that $\vartheta(G)=p$ if $|W|\geq n-1$ and $h_{i,r_{i}}\geq
2$ for each $i\in[k]$. Otherwise, $\vartheta(G)=p+1$. We consider the
following three cases:

(1) If $|W|\geq n-1$ and $h_{i,r_{i}}\geq 2$ for each $i\in[k]$, then
$|V(K_{n}\backslash W)|\leq 1$ and $G\backslash W$ is in this case the
disjoint union of $p$ cliques and some isolated vertices in this case. Let
$K_{b_{1}},\dots,K_{b_{p}}$ be all the cliques in $G\backslash W$. Since the
induced matching $M$ is maximal, thus, for each $K_{b_{i}}$, there exists some
$e\in M$ such that $e\in E(K_{b_{i}})$. Say $e_{i}\in E(K_{b_{i}})$ for
$i=1,\ldots,p$. If $\vartheta(G)\geq p+1$, we set $e_{p+1}=\\{u,v\\}$, then
$u,v\notin\cup_{i=1}^{p}V(K_{b_{i}})$ and $u,v$ cannot simultaneously belong
to $V(K_{n})$. Otherwise, $u$ (or $v$) and the two endpoints of some $e_{i}$
form a path with three vertices, which is contradictory. It follows that $u,v$
belong to the set of isolated vertices in $V(G\backslash W)$, which
contradicts with $e_{p+1}=\\{u,v\\}\in E(G)$.

(2) If $|W|\geq n-1$ and $h_{i,r_{i}}=1$ for some $i\in[k]$, let $V(K_{n})\cap
V(K_{a_{i,r_{i}}})=\\{w_{i,1},\ldots,w_{i,r_{i}}\\}$, then
$G\backslash(V(K_{n})\backslash w_{i,r_{i}})$ is the disjoint union of $(p+1)$
cliques and some isolated vertices.

(3) If $|W|\leq n-2$, then $|V(K_{n}\backslash W)|\geq 2$. In this case,
$G\backslash W$ is the disjoint union of $(p+1)$ cliques and some isolated
vertices.

For cases (2) and (3), by arguments similar to (1), we obtain that
$\vartheta(G)=p+1$ in these two cases, as claimed.

##  3\. Results involving $*$ operations and $\circ$ operations

In this section, we will study some simple graphs obtained from the fan graphs
by using the $*$ operation or the $\circ$ operation. The main task of this
section is to provide some formulas for the depth and regularity of some
graphs. We start by recalling from [2] the two special gluing operations
mentioned earlier.

######  Definition 3.1.

For $i=1,2$, let $G_{i}$ be a graph with at least one leaf $f_{i}$ and $v_{i}$
be its neighbor with $\deg_{G_{i}}(v_{i})\geq 2$.

  1. (1)

Let $G$ be the graph obtained from $G_{1}$ and $G_{2}$ by identifying the
vertices $f_{1}$ and $f_{2}$. In this case, we say that $G$ is obtained from
$G_{1}$ and $G_{2}$ by the _$*$ operation_ and write
$G=(G_{1},f_{1})*(G_{2},f_{2})$ or simply $G=G_{1}*G_{2}$. If we denote the
identified vertex in $G$ by $f$, then we also write $G=G_{1}*_{f}G_{2}$.

  2. (2)

Let $H$ be the graph obtained from $G_{1}$ and $G_{2}$ by first removing the
leaves $f_{1},f_{2}$, and then identifying the vertices $v_{1}$ and $v_{2}$.
In this case, we say that $H$ is obtained from $G_{1}$ and $G_{2}$ by the
_$\circ$ operation_ and write $H=(G_{1},f_{1})\circ(G_{2},f_{2})$ or simply
$H=G_{1}\circ G_{2}$. If $v_{1}$ and $v_{2}$ are identified as the vertex $v$
in $G$, then we also write $H=G_{1}\circ_{v}G_{2}$.

######  Remark 3.2.

From the above definition we can see that $G_{1}\sqcup G_{2}$ is a splitting
graph of $G$ and $(G_{1}\backslash f_{1})\sqcup(G_{2}\backslash f_{2})$ is a
splitting graph of $H$.

######  Example 3.3.

The following are examples of the graphs obtained from $G_{1}$ and $G_{2}$ by
$\circ$ and $*$ operations.

$G_{1}$$v_{1}$$f_{1}$$G_{2}$$v_{2}$$f_{2}$$G_{1}\circ
G_{2}$$v$$Figure\;2$$\circ$$operation$

$G_{1}$$v_{1}$$f_{1}$$G_{2}$$v_{2}$$f_{2}$$G_{1}*G_{2}$$v_{1}$$f$$v_{2}$$Figure\;3$$*$$operation$

###  3.1. Glue via the $\circ$ operation

First of all, we compute the depth and regularity of the quotient ring of the
edge ideal of a graph, which is obtained by a single $\circ$ operation.

######  Lemma 3.4.

Suppose that $G=(G_{1},f_{1})\circ(G_{2},f_{2})$, where
$G_{i}=F_{k_{i}}^{W_{i}}(K_{n_{i}})$ is a $k_{i}$-fan graph of $K_{n_{i}}$ on
the set $W_{i}$ with $n_{i}\geq 2$ and $k_{i}\geq 1$ for $i\in[2]$. Then we
have

$\operatorname{depth}(S_{G}/I_{G})\geq\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-2.$

###### Proof.

Let $v_{i}$ be the neighbor point of $f_{i}$ in $G_{i}$ and
$W_{i}=W_{i,1}\sqcup\cdots\sqcup W_{i,k_{i}}$ be a partition of $W_{i}$ with
$v_{i}\in W_{i,1}$ for $i\in[2]$. Suppose that $v_{1}$ and $v_{2}$ are
identified with $v$ in $G$ by the $\circ$ operation. Let
$J=(x_{N_{G}(v)})+I_{G\setminus N_{G}[v]}$ and $K=(x_{v})+I_{G\backslash v}$,
then $G\backslash N_{G}[v]$ has $\sum\limits_{i=1}^{2}(|W_{i}|-|W_{i,1}|)$
connected components consisting of cliques and isolated vertices. Let
$T_{i}=\max\\{|W_{i,1}|,|W_{i,2}|,\ldots,|W_{i,k_{i}}|\\}$ for $i\in[2]$, then
$|W_{i,1}|\leq T_{i}$. Thus
$\operatorname{depth}(S_{G}/J)=1+\sum\limits_{i=1}^{2}(|W_{i}|-|W_{i,1}|)\geq
1+\sum\limits_{i=1}^{2}(|W_{i}|-T_{i})$ by and Lemma 1.6. Meanwhile,
$G\backslash
v=(G_{1}\backslash\\{v,f_{1}\\})\sqcup(G_{2}\backslash\\{v,f_{2}\\})$ and
every $G_{i}\backslash\\{v,f_{i}\\}$ is a fan graph, so we have
$K=(x_{v})+I_{G_{1}\backslash\\{v,f_{1}\\}}+I_{G_{2}\backslash\\{v,f_{2}\\}}$.
Then, by Theorem 2.6, we get

$\displaystyle\operatorname{depth}(S_{G}/K)=$
$\displaystyle\operatorname{depth}(S_{G_{1}\backslash\\{v,f_{1}\\}}/I_{G_{1}\backslash\\{v,f_{1}\\}})+\operatorname{depth}(S_{G_{2}\backslash\\{v,f_{2}\\}}/I_{G_{2}\backslash\\{v,f_{2}\\}})$
$\displaystyle=$
$\displaystyle[1+(|W_{1}|-1)-T^{\prime}_{1}]+[1+(|W_{2}|-1)-T^{\prime}_{2}]$
$\displaystyle=$
$\displaystyle(1+|W_{1}|-T^{\prime}_{1})+(1+|W_{2}|-T^{\prime}_{2})-2$
$\displaystyle\geq$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-2.$

where $T^{\prime}_{i}=\max\\{|W_{i,1}|-1,|W_{i,2}|,\ldots,|W_{i,k_{i}}|\\}$
for $i\in[2]$, and the last inequality holds because of every
$T^{\prime}_{i}\leq T_{i}$ and Theorem 2.6. Again applying Lemma 1.2(1) and
Lemma 1.5 to the exact sequence (1), we obtain the desired result.

Jayanthan et al. in [13] introduced the notion of clique sum. Here, we make a
simple application of it. By a clique sum $G_{1}\cup_{v}G_{2}$, we mean a
union of graphs $G_{1}$ and $G_{2}$ such that $V(G_{1})\cap V(G_{2})=\\{v\\}$.

######  Lemma 3.5.

Let $G=F_{k}^{W}(K_{n})$ be a $k$-fan graph of $K_{n}$ on the set $W$ with
$n\geq 2$ and $k\geq 1$. Suppose $f$ is a leaf of $G$, $v$ is its neighbor
point, and $W=W_{1}\sqcup\cdots\sqcup W_{k}$ is a partition of $W$ with $v\in
W_{1}$. Let $T=\max\\{|W_{1}|,|W_{2}|,\ldots,|W_{k}|\\}$,
$T^{\prime}=\max\\{|W_{1}|-1,|W_{2}|,\ldots,|W_{k}|\\}$ with $T^{\prime}=T-1$.
Then $\operatorname{depth}(S_{G\backslash f}/I_{G\backslash
f})=\operatorname{depth}(S_{G}/I_{G})$.

###### Proof.

Let $G^{\prime}=G\backslash f$. Choose
$J=(x_{N_{G^{\prime}}(v)})+I_{{G^{\prime}}\backslash N_{G^{\prime}}[v]}$ and
$K=(x_{v})+I_{{G^{\prime}}\backslash v}$, then
$J+K=(x_{N_{G^{\prime}}[v]})+I_{G\backslash N_{G^{\prime}}[v]}$ by Lemma
1.5(1) and $G^{\prime}\backslash N_{G^{\prime}}[v]$ has $(|W|-|W_{1}|)$
connected components consisting of cliques and isolated vertices. Thus
$\operatorname{depth}(S_{G^{\prime}}/(J+K))=|W|-|W_{1}|=\operatorname{depth}(S_{G}/I_{G})-1$
by Lemma 1.6. Meanwhile, $G^{\prime}\backslash v$ is a fan graph of $K_{n-1}$
on the set $W\backslash v$, so
$\operatorname{depth}(S_{G^{\prime}}/K)=1+(|W|-1)-T^{\prime}=\operatorname{depth}(S_{G}/I_{G})$.
Applying Lemma 1.2(1) and Lemma 1.5 to the exact sequence (1) by substituting
$S_{G^{\prime}}$ for $S$, we get the desired result.

Now we are ready to prove the first major result of this subsection.

######  Theorem 3.6.

Suppose that $G=(G_{1},f_{1})\circ(G_{2},f_{2})$, where
$G_{i}=F_{k_{i}}^{W_{i}}(K_{n_{i}})$ is a $k_{i}$-fan graph of $K_{n_{i}}$ on
the set $W_{i}$ with $n_{i}\geq 2$ and $k_{i}\geq 1$ for $i\in[2]$. Let
$v_{i}$ be the neighbor of $f_{i}$ in $G_{i}$ and
$W_{i}=W_{i,1}\sqcup\cdots\sqcup W_{i,k_{i}}$ be a partition of $W_{i}$ with
$v_{i}\in W_{i,1}$ for $i\in[2]$. Let
$T_{i}=\max\\{|W_{i,1}|,|W_{i,2}|,\ldots,|W_{i,k_{i}}|\\}$ and
$T^{\prime}_{i}=\max\\{|W_{i,1}|-1,|W_{i,2}|,\ldots,|W_{i,k_{i}}|\\}$ for
$i\in[2]$, and $t=|\\{i:T^{\prime}_{i}\neq T_{i}-1\\}|$. Then

$\operatorname{depth}(S_{G}/I_{G})=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-s$

where $s=1$ if $t\leq 1$, otherwise, $s=2$.

###### Proof.

Let $v=v_{1}=v_{2}$. By the definition of $t$, we have $t\in\\{0,1,2\\}$. We
distinguish the following three cases:

(I) If $t=0$, then $T^{\prime}_{i}=T_{i}-1$ for $i\in[2]$. This implies that
each $T_{i}=|W_{i,1}|$ and $|W_{i,1}|>|W_{i,j}|$ for $j=2,\ldots,k_{i}$. In
this case, we set $J=(x_{N_{G}(v)})+I_{G\backslash N_{G}[v]}$ and
$K=(x_{v})+I_{G\backslash v}$, then $J+K=(x_{N_{G}[v]})+I_{G\backslash
N_{G}[v]}$ and $G\backslash N_{G}[v]$ has
$\sum\limits_{i=1}^{2}(|W_{i}|-|W_{i,1}|)$ connected components consisting of
some cliques and isolated vertices. Thus
$\operatorname{depth}(S_{G}/(J+K))=\sum\limits_{i=1}^{2}(|W_{i}|-|W_{i,1}|)=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-2$
by Lemma 1.6 and Theorem 2.6. Meanwhile, $G\backslash
v=(G_{1}\backslash\\{v,f_{1}\\})\sqcup(G_{2}\backslash\\{v,f_{2}\\})$ and
every $G_{i}\backslash\\{v,f_{i}\\}$ is a fan graph. So we have
$K=(x_{v})+I_{G_{1}\backslash\\{v,f_{1}\\}}+I_{G_{2}\backslash\\{v,f_{2}\\}}$.
By Theorem 2.6, we get

$\displaystyle\operatorname{depth}(S_{G}/K)=$
$\displaystyle\operatorname{depth}(S_{G_{1}\backslash\\{v,f_{1}\\}}/I_{G_{1}\backslash\\{v,f_{1}\\}})+\operatorname{depth}(S_{G_{2}\backslash\\{v,f_{2}\\}}/I_{G_{2}\backslash\\{v,f_{2}\\}})$
$\displaystyle=$
$\displaystyle[1+(|W_{1}|-1)-T^{\prime}_{1}]+[1+(|W_{2}|-1)-T^{\prime}_{2}]$
$\displaystyle=$
$\displaystyle[1+(|W_{1}|-1)-(T_{1}-1)]+[1+(|W_{2}|-1)-(T_{2}-1)]$
$\displaystyle=$ $\displaystyle(1+|W_{1}|-T_{1})+(1+|W_{2}|-T_{2})$
$\displaystyle=$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}}).$

By Lemma 1.2(1) and Lemma 1.5(2) and the exact sequence (1), we get the
desired result.

(II) If $t=1$, we assume that $T^{\prime}_{1}=T_{1}-1$ and $T^{\prime}_{2}\neq
T_{2}-1$. It follows that $T_{1}=|W_{1,1}|>|W_{1,j}|$ for $j=2,\ldots,k_{1}$
and $T_{2}=|W_{2,j}|\geq|W_{2,1}|$ for some $j\neq 1$. Say $T_{2}=|W_{2,2}|$.
In this case, we prove by induction on $n_{2}$ that

$\operatorname{depth}(S_{G}/I_{G})=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-1.$

If $n_{2}=2$, then $G_{2}$ can only be $G_{2}=P_{3}\cup_{u}K_{b}$, which is
the clique sum of $P_{3}$ and some $K_{b}$. So we have
$\operatorname{depth}(S_{G_{2}}/I_{G_{2}})=2$ by Theorem 2.6. In this case,
let $J=(x_{N_{G}(u)})+I_{G\backslash N_{G}[u]}$ and $K=(x_{u})+I_{G\backslash
u}$, then $J+K=(x_{N_{G}[u]})+I_{G\backslash N_{G}[u]}$, $G\backslash
N_{G}[u]=G_{1}\backslash\\{v_{1},f_{1}\\}$ is a fan graph and $G\backslash
u=(G_{1}\backslash f_{1})\sqcup K_{b-1}$. Thus

$\displaystyle\operatorname{depth}(S_{G}/(J+K))$
$\displaystyle=\operatorname{depth}(S_{G_{1}\backslash\\{v_{1},f_{1}\\}}/I_{G_{1}\backslash\\{v_{1},f_{1}\\}})$
$\displaystyle=1+(|W_{1}|-1)-T^{\prime}_{1}=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})$

and

$\operatorname{depth}(S_{G}/K)=1+\operatorname{depth}(S_{G_{1}\backslash
f_{1}}/I_{G_{1}\backslash f_{1}})=1+\operatorname{depth}(S_{G_{1}}/I_{G_{1}})$

by Lemma 3.5. The desired results follow from Lemma 1.2(1), Lemma 1.5(2) and
the exact sequence (1) by substituting $S_{G}$ for $S$.

Now suppose $n_{2}\geq 3$. In this case, let
$W_{2,2}=\\{w_{1},\ldots,w_{r_{2}}\\}$ and
$\\{K_{a_{2,1}},\ldots,K_{a_{2,r_{2}}}\\}$ be a branch of the fan on $W_{2,2}$
such that $V(K_{n_{2}})\cap V(K_{a_{2},j})=\\{w_{1},\ldots,w_{j}\\}$ for
$j\in[r_{2}]$. Choose $J=(x_{N_{G}(w_{1})})+I_{G\backslash N_{G}[w_{1}]}$ and
$K=(x_{w_{1}})+I_{G\backslash w_{1}}$, then
$J+K=(x_{N_{G}[w_{1}]})+I_{G\backslash N_{G}[w_{1}]}$, $G\backslash
N_{G}[w_{1}]$ is the disjoint union of the graph
$G_{1}\backslash\\{v_{1},f_{1}\\}$ and $(|W_{2}|-|W_{2,2}|-1)$ connected
components consisting of some cliques and isolated vertices. Then we obtain

$\displaystyle\operatorname{depth}(S_{G}/(J+K))$
$\displaystyle=\operatorname{depth}(S_{G_{1}\backslash\\{v_{1},f_{1}\\}}/I_{G_{1}\setminus\\{v_{1},f_{1}\\}})+(|W_{2}|-|W_{2,2}|-1)$
$\displaystyle=[1+(|W_{1}|-1)-T^{\prime}_{1}]+(1+|W_{2}|-|W_{2,2}|-2)$ (2)
$\displaystyle=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-2.$

Meanwhile, $G\backslash w_{1}=G_{1}\circ(G_{2}\setminus w_{1})$ and
$G_{2}\backslash w_{1}=F_{q}^{W_{2}\backslash\\{w_{1}\\}}(K_{n_{2}-1})$ for
some $q$. In this case, to compute $\operatorname{depth}(S_{G\backslash
w_{1}}/I_{G\backslash w_{1}})$, we set $T^{*}_{1}=T_{1}$,
$T^{\prime*}_{1}=T^{\prime}_{1}$,
$T^{*}_{2}=\max\\{|W_{2,1}|,|W_{2,2}|-1,|W_{2,3}|,\ldots,|W_{2,k_{2}}|\\}$ and
$T^{\prime*}_{2}=\max\\{|W_{2,1}|-1,|W_{2,2}|-1,|W_{2,3}|,\\\
\ldots,|W_{2,k_{2}}|\\}$. Let $t^{\prime}=|\\{i:T^{\prime*}_{i}\neq
T^{*}_{i}-1\\}|$, then $t^{\prime}\leq 1$. From the above case (I) and by
induction we have

$\operatorname{depth}(S_{G}/K)=1+\operatorname{depth}(S_{G\backslash
w_{1}}/I_{G\backslash
w_{1}})=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}\backslash
w_{1}}/I_{G_{2}\backslash w_{1}}).$

The desired result follows from Lemma 1.2(1), the equality (2), Lemma 1.5 and
the exact sequence (1).

(III) If $t=2$, then $T^{\prime}_{i}\neq T_{i}-1$ for $i\in[2]$. This implies
that $T_{i}=|W_{i,j_{i}}|\geq|W_{i,1}|$ for some $j_{i}\neq 1$. Suppose
$T_{i}=|W_{i,2}|$ for $i\in[2]$. In this case, let
$W_{2,2}=\\{w_{1},\ldots,w_{r_{2}}\\}$ and
$\\{K_{a_{2,1}},\ldots,K_{a_{2,r_{2}}}\\}$ be a branch of the fan on $W_{2,2}$
with $V(K_{n_{2}})\cap V(K_{a_{2},j})=\\{w_{1},\ldots,w_{j}\\}$ for
$j\in[r_{2}]$. Choose $J=(x_{N_{G}(w_{1})})+I_{G\backslash N_{G}[w_{1}]}$ and
$K=(x_{w_{1}})+I_{G\backslash w_{1}}$, then
$J+K=(x_{N_{G}[w_{1}]})+I_{G\backslash N_{G}[w_{1}]}$. Similar to the case
$t=1$, $G\backslash N_{G}[w_{1}]$ is the disjoint union of a graph
$G_{1}\backslash\\{v_{1},f_{1}\\}$ and $(|W_{2}|-|W_{2,2}|-1)$ connected
components consisting of some cliques and isolated vertices, and $G\backslash
w_{1}=G_{1}\circ(G_{2}\backslash w_{1})$ and $G_{2}\backslash
w_{1}=F_{q}^{W_{2}\backslash\\{w_{1}\\}}(K_{n_{2}-1})$ for some $q$. So we
have

$\displaystyle\operatorname{depth}(S_{G}/(J+K))$
$\displaystyle=\operatorname{depth}(S_{G_{1}\backslash\\{v_{1},f_{1}\\}}/I_{G_{1}\backslash\\{v_{1},f_{1}\\}})+(|W_{2}|-|W_{2,2}|-1)$
(3) $\displaystyle=[1+(|W_{1}|-1)-T^{\prime}_{1}]+(1+|W_{2}|-|W_{2,2}|-2)$
$\displaystyle=[1+(|W_{1}|-1)-T_{1}]+(1+|W_{2}|-|W_{2,2}|-2)$
$\displaystyle=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-3.$

and

$\displaystyle\operatorname{depth}(S/K)=$ $\displaystyle
1+\operatorname{depth}(S_{G\backslash w_{1}}/I_{G\backslash w_{1}})$
$\displaystyle\geq$ $\displaystyle
1+\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}\backslash
w_{1}}/I_{G_{2}\backslash w_{1}})-2$ $\displaystyle=$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+[1+(|W_{2}|-1)-T^{*}_{2}]-1$
$\displaystyle\geq$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+(1+|W_{2}|-T_{2})-2$
(4) $\displaystyle=$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-2.$

where the first inequality holds by Lemma 3.4 and the second inequality holds
by $T^{*}_{2}\leq T_{2}$, where
$T^{*}_{2}=\max\\{|W_{2,1}|,|W_{2,2}|-1,|W_{2,3}|,\ldots,|W_{2,k_{2}}|\\}$. By
Lemma 1.2(1), the relations (3), (4), Lemma 1.5(2) and the exact sequence (1),
we get the wished result.

######  Lemma 3.7.

Let $G_{1}=F_{k}^{W}(K_{n})$ be a $k$-fan graph of $K_{n}$ on the set
$W\subsetneq[n]$ with $n\geq 2$ and $P_{2}$ be a path with $2$ vertices.
Suppose that $G=G_{1}\cup_{v}P_{2}$ is the clique sum of graphs $G_{1}$ and
$P_{2}$ with $V(G_{1})\cap V(P_{2})=\\{v\\}$ and $v\in V(K_{n})\backslash W$.
Let $W=W_{1}\sqcup\cdots\sqcup W_{k}$ be a partition of $W$. Then

$\operatorname{reg}(S_{G\backslash v}/I_{G\backslash
v})=\operatorname{reg}(S_{G}/I_{G})-s$

where $s=1$ if $|W|\geq n-2$ and $h_{i,r_{i}}\geq 2$ for all $i\in[k]$,
otherwise, $s=0$.

###### Proof.

$G$ is actually a $(k+1)$-fan graph of $K_{n}$ on the set $W\cup\\{v\\}$ and
$G\backslash v$ is the disjoint union of a graph $F_{k}^{W}(K_{n-1})$ and an
isolated vertex. Suppose $p=|\\{h_{i,j}:h_{i,j}\geq 2\\}|$. By Lemma 1.8 and
Theorem 2.8, we getthat if $|W|\geq n-2$ and $h_{i,r_{i}}\geq 2$ for all
$i\in[k]$, then $\operatorname{reg}(S_{G}/I_{G})=p+1$ and
$\operatorname{reg}(S_{G\backslash v}/I_{G\backslash v})=p$. Otherwise,
$\operatorname{reg}(S_{G}/I_{G})=\vartheta(G\backslash v)=p+1$, as expected.

Next, we are ready to prove another major result of this subsection.

######  Theorem 3.8.

Suppose that $G=(G_{1},f_{1})\circ(G_{2},f_{2})$, where
$G_{i}=F_{k_{i}}^{W_{i}}(K_{n_{i}})$ is a $k_{i}$-fan graph of $K_{n_{i}}$ on
the set $W_{i}$ with $n_{i}\geq 2$ and $k_{i}\geq 1$ for $i\in[2]$. Let
$v_{i}$ be the neighbor point of $f_{i}$ in $G_{i}$ and
$t=|\\{i\in[2]:\operatorname{reg}(S_{G_{i}\backslash v_{i}}/I_{G_{i}\backslash
v_{i}})\neq\operatorname{reg}(S_{G_{i}}/I_{G_{i}})\\}|$. We have

  * (1)

If $t\leq 1$, then
$\operatorname{reg}(S_{G}/I_{G})=\operatorname{reg}(S_{G_{1}}/I_{G_{1}})+\operatorname{reg}(S_{G_{2}}/I_{G_{2}})-t$,

  * (2)

If $t=2$ and $|W_{i}|=n_{i}-1$ for some $i\in[2]$, then

$\operatorname{reg}(S_{G}/I_{G})=\operatorname{reg}(S_{G_{1}}/I_{G_{1}})+\operatorname{reg}(S_{G_{2}}/I_{G_{2}})-1,$

  * (3)

If $t=2$ and $|W_{i}|=n_{i}$ for all $i\in[2]$, then

$\operatorname{reg}(S_{G}/I_{G})=\operatorname{reg}(S_{G_{1}}/I_{G_{1}})+\operatorname{reg}(S_{G_{2}}/I_{G_{2}})-2.$

###### Proof.

From the definition of $t$, we get $t\in\\{0,1,2\\}$. Meanwhile, we have
$\operatorname{reg}(S_{G}/I_{G})=\vartheta(G)$ and
$\operatorname{reg}(S_{G_{i}}/I_{G_{i}})=\vartheta(G_{i})$ for $i\in[2]$ by
Lemma 1.8, where $\vartheta(G)$ is the induced matching number of $G$. We
divide into the following three cases:

(1) If $t=0$, then $\operatorname{reg}(S_{G_{i}\backslash
v_{i}}/I_{G_{i}\backslash v_{i}})=\operatorname{reg}(S_{G_{i}}/I_{G_{i}})$ for
$i\in[2]$, i.e., every $\vartheta(G_{i}\backslash v_{i})=\vartheta(G_{i})$. In
this case, it is clear that $M=M_{1}\sqcup M_{2}$ is an induced matching of
$G$, where every $M_{i}=\\{e_{i,1},\ldots,e_{i,\vartheta(G_{i}\backslash
v_{i})}\\}$ is an induced matching of $G_{i}\backslash v_{i}$. So
$\vartheta(G)\geq\vartheta(G_{1}\backslash v_{1})+\vartheta(G_{2}\backslash
v_{2})=\vartheta(G_{1})+\vartheta(G_{2})$. If
$\vartheta(G)\geq\vartheta(G_{1})+\vartheta(G_{2})+1$. Suppose
$M^{\prime}=\\{e_{1},\ldots,e_{\vartheta(G_{1})+\vartheta(G_{2})+1}\\}$ is an
induced matching of $G$, then there exist at least $(\vartheta(G_{1})+1)$
elements in $M^{\prime}$ belonging to $E(G_{1}\backslash f_{1})$ or
$(\vartheta(G_{2})+1)$ elements in $M^{\prime}$ belonging to
$E(G_{2}\backslash f_{2})$. This forces
$\vartheta(G_{1})+1\leq\vartheta(G_{1}\backslash f_{1})\leq\vartheta(G_{1})$
or $\vartheta(G_{2})+1\leq\vartheta(G_{2}\backslash
f_{2})\leq\vartheta(G_{2})$, a contradiction.

(2) If $t=1$. By symmetry, we assume that
$\operatorname{reg}(S_{G_{1}\backslash v_{1}}/I_{G_{1}\backslash
v_{1}})=\operatorname{reg}(S_{G_{1}}/I_{G_{1}})$ and
$\operatorname{reg}(S_{G_{2}\backslash v_{2}}/I_{G_{2}\backslash
v_{2}})\neq\operatorname{reg}(S_{G_{2}}/I_{G_{2}})$, i.e.,
$\vartheta(G_{1}\backslash v_{1})=\vartheta(G_{1})$ and
$\vartheta(G_{2}\backslash v_{2})\neq\vartheta(G_{2})$. In this case, let
$M_{i}=\\{e_{i,1},\ldots,e_{i,\vartheta(G_{i}\backslash v_{i})}\\}$ be an
induced matching of $G_{i}\backslash v_{i}$ for $i\in[2]$, then $M=M_{1}\sqcup
M_{2}$ is an induced matching of $G$. Thus
$\vartheta(G)\geq\vartheta(G_{1}\backslash v_{1})+\vartheta(G_{2}\backslash
v_{2})=\vartheta(G_{1})+\vartheta(G_{2})-1$, since $\vartheta(G_{2}\backslash
v_{2})=\vartheta(G_{2})-1$. If
$\vartheta(G)\geq\vartheta(G_{1})+\vartheta(G_{2})$ and
$M^{\prime\prime}=\\{e_{1},\ldots,e_{\vartheta(G_{1})+\vartheta(G_{2})}\\}$ is
an induced matching of $G$, then, by the assumption that
$\vartheta(G_{1}\backslash v_{1})=\vartheta(G_{1})$ and
$\vartheta(G_{2}\backslash v_{2})=\vartheta(G_{2})-1$, we obtain that there
exists at least $(\vartheta(G_{1})+1)$ elements in $M^{\prime\prime}$ belong
to $E(G_{1}\backslash f_{1})$ or $\vartheta(G_{2})$ elements in
$M^{\prime\prime}$ belong to $E(G_{2}\backslash f_{2})$, implying
$\vartheta(G_{1})+1\leq\vartheta(G_{1}\backslash f_{1})$ or
$\vartheta(G_{2})\leq\vartheta(G_{2}\backslash f_{2})$. We consider two
subcases:

(i) If $\vartheta(G_{1})+1\leq\vartheta(G_{1}\backslash f_{1})$, then
$\vartheta(G_{1})+1\leq\vartheta(G_{1})$ since $\vartheta(G_{1}\backslash
f_{1})\leq\vartheta(G_{1})$, a contradiction.

(ii) If $\vartheta(G_{2})\leq\vartheta(G_{2}\backslash f_{2})$, then
$\vartheta(G_{2}\backslash f_{2})=\vartheta(G_{2})$, since
$\vartheta(G_{2}\backslash f_{2})\leq\vartheta(G_{2})$. In this case, there
exist $\vartheta(G_{1})$ elements in $M^{\prime\prime}$, say
$e_{i_{1}},\ldots,e_{i_{\vartheta(G_{1})}}$, consisting of an induced matching
of $G_{1}\backslash f_{1}$, and the remaining elements consist of an induced
matching of $G_{2}\backslash f_{2}$. Note that
$\vartheta(G_{2})=\vartheta(G_{2}\backslash v_{2})+1$, there exists some $e$
with $v_{2}$ as its an endpoint belonging to
$M^{\prime\prime}\backslash\\{e_{i_{1}},\ldots,e_{i_{\vartheta(G_{1})}}\\}$.
Let $e^{\prime}=\\{v,f_{1}\\}\in E(G_{1})$ where $v=v_{1}=v_{2}$, then
$\\{e_{i_{1}},\ldots,e_{i_{\vartheta(G_{1})}},e^{\prime}\\}$ is an induced
matching of $G_{1}$, a contradiction.

(3) If $t=2$, then $\operatorname{reg}(S_{G_{i}\backslash
v_{i}}/I_{G_{i}\backslash v_{i}})\neq\operatorname{reg}(S_{G_{i}}/I_{G_{i}})$
for $i\in[2]$, i.e., every $\vartheta(G_{i}\backslash
v_{i})\neq\vartheta(G_{i})$. Let $W_{i}=W_{i_{1}}\sqcup\cdots\sqcup
W_{i_{k_{i}}}$ be a partition of $W_{i}$ with $v_{i}\in W_{i_{1}}$. From the
assumptions that for each $i$, $\operatorname{reg}(S_{G_{i}\backslash
v_{i}}/I_{G_{i}\backslash v_{i}})\neq\operatorname{reg}(S_{G_{i}}/I_{G_{i}})$
and $f_{i}$ is a leaf of $G_{i}$ with a unique neighbor point $v_{i}$, we have
$|W_{i_{1}}|=1$ from the proof of Theorem 2.8. Let
$W_{i_{\ell}}=\\{w_{i_{\ell},1},\ldots,w_{i_{\ell},r_{i_{\ell}}}\\}$ for
$\ell\in[k_{i}]$ and $i\in[2]$, then by Lemma 3.7 we get that $|W_{i}|\geq
n_{i}-1$ and $h_{i_{\ell},r_{i_{\ell}}}\geq 2$ for all $\ell=2,\ldots,k_{i}$
and $i\in[2]$. In this case, each $G_{i}\backslash W_{i}$ is the disjoint
union of $(\vartheta(G_{i})-1)$ cliques $K^{\prime}_{i,1}$,
…,$K^{\prime}_{i,\vartheta(G_{i})-1}$ and some isolated vertices. Let
$M_{i}=\\{e_{ij}|e_{ij}\in E(K^{\prime}_{i,j}),j\in[\vartheta(G_{i})-1]\\}$
for $i\in[2]$, then $M_{i}$ is an induced matching of $G_{i}\backslash v_{i}$.
We consider two subcases:

(i) If $|W_{i}|=n_{i}-1$ for some $i\in[2]$. Suppose that $|W_{1}|=n_{1}-1$.
In this case, let $M=M_{1}\sqcup M_{2}\sqcup\\{e\\}$, where edge
$e=\\{v_{1},u\\}$ with $u\in V(K_{n_{1}})\backslash W_{1}$. Then $M$ is an
induced matching of $G$ by the choice of $M_{1}$ and $M_{2}$. So
$\vartheta(G)\geq\vartheta(G_{1})+\vartheta(G_{2})-1$. If
$\vartheta(G)\geq\vartheta(G_{1})+\vartheta(G_{2})$ and
$M^{\prime\prime\prime}=\\{e_{1},\ldots,e_{\vartheta(G_{1})+\vartheta(G_{2})}\\}$
is an induced matching of $G$, then there exist at least $\vartheta(G_{1})$
elements in $M^{\prime\prime\prime}$ belonging to $E(G_{1}\backslash f_{1})$
and $\vartheta(G_{2})$ elements in $M^{\prime\prime\prime}$ belonging to
$E(G_{2}\backslash f_{2})$, implying
$\vartheta(G_{1})\leq\vartheta(G_{1}\backslash f_{1})$ and
$\vartheta(G_{2})\leq\vartheta(G_{2}\backslash f_{2})$. So
$\vartheta(G_{i}\backslash f_{i})=\vartheta(G_{i})$ for $i\in[2]$. Note that
each $\vartheta(G_{i})=\vartheta(G_{i}\backslash v_{i})+1$. Therefore, there
exist elements $e_{i_{1}}$ and $e_{i_{2}}$ in $M^{\prime\prime\prime}$ with
$v_{1}$ and $v_{2}$ as one of their endpoints respectively. This contradicts
the fact that $v_{1}=v_{2}=v$ in $G$ and $M^{\prime\prime\prime}$ is an
induced matching of $G$.

(ii) If $|W_{i}|=n_{i}$ for all $i\in[2]$, then $M_{1}\sqcup M_{2}$ is an
induced matching of $G$. Thus
$\vartheta(G)\geq\vartheta(G_{1})+\vartheta(G_{2})-2$. If
$\vartheta(G)\geq\vartheta(G_{1})+\vartheta(G_{2})-1$ and
$M^{*}=\\{e_{1},\ldots,e_{\vartheta(G_{1})+\vartheta(G_{2})-1}\\}$ is an
induced matching of $G$, then there exist at least $\vartheta(G_{1})$ elements
in $M^{*}$ belonging to $E(G_{1}\backslash f_{1})$ or $\vartheta(G_{2})$
elements in $M^{*}$ belonging to $E(G_{2}\backslash f_{2})$. This forces
$\vartheta(G_{1})=\vartheta(G_{1}\backslash f_{1})$ or
$\vartheta(G_{2})=\vartheta(G_{2}\backslash f_{2})$, since every
$\vartheta(G_{i}\backslash f_{i})\leq\vartheta(G_{i})$. Thus there exists an
edge $e=\\{u,v\\}\in M^{*}$ with $u,v\in V(K_{n_{i}})$. Since $|W_{i}|=n_{i}$
for all $i\in[2]$, we get that $u,v$ and the two endpoints of another edge
form a path with at least three vertices, a contradiction.

###  3.2. Glue via the $*$ operation

In this subsection, we study the depth and regularity of the quotient ring of
the edge ideal of a graph, which is obtained by $*$ operations. First, we show
the first major result of this subsection.

######  Theorem 3.9.

Suppose that $G=(G_{1},f_{1})*(G_{2},f_{2})$, where
$G_{i}=F_{k_{i}}^{W_{i}}(K_{n_{i}})$ is a $k_{i}$-fan graph of $K_{n_{i}}$ on
the set $W_{i}$ with $n_{i}\geq 2$ and $k_{i}\geq 1$ for $i\in[2]$. Let
$v_{i}$ be the neighbor of $f_{i}$ in $G_{i}$ and
$W_{i}=W_{i,1}\sqcup\cdots\sqcup W_{i,k_{i}}$ be a partition of $W_{i}$ with
$v_{i}\in W_{i,1}$ for $i\in[2]$. Let
$T_{i}=\max\\{|W_{i,1}|,|W_{i,2}|,\ldots,|W_{i,k_{i}}|\\}$ and
$T^{\prime}_{i}=\max\\{|W_{i,1}|-1,|W_{i,2}|,\ldots,|W_{i,k_{i}}|\\}$ for
$i\in[2]$, and $t=|\\{i:T^{\prime}_{i}\neq T_{i}-1\\}|$. Then

$\operatorname{depth}(S_{G}/I_{G})=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-s$

where $s=0$ if $t=0$, otherwise, $s=1$.

###### Proof.

First, we have $t\in\\{0,1,2\\}$ by the definition of $t$. We distinguish
between the following two cases:

(1) If $t=0$, then $T^{\prime}_{i}=T_{i}-1$ for $i\in[2]$, which forces each
$T_{i}=|W_{i,1}|$ and $|W_{i,1}|>|W_{i,j}|$ for $j=2,\ldots,k_{i}$. In this
case, we set $J=(x_{N_{G}(v_{2})})+I_{G\backslash N_{G}[v_{2}]}$ and
$K=(x_{v_{2}})+I_{G\backslash v_{2}}$, then
$J+K=(x_{N_{G}[v_{2}]})+I_{G\backslash N_{G}[v_{2}]}$, $G\backslash
v_{2}=G_{1}\sqcup(G_{2}\backslash\\{v_{2},f_{2}\\})$ and $G\backslash
N_{G}[v_{2}]$ has $(|W_{2}|-|W_{2,1}|+1)$ connected components consisting of
$G_{1}\backslash f_{1}$ and $(|W_{2}|-|W_{2,1}|)$ cliques or isolated
vertices. So we have

$\displaystyle\operatorname{depth}(S_{G}/K)=$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}\backslash\\{v_{2},f_{2}\\}}/I_{G_{2}\backslash\\{v_{2},f_{2}\\}})$
$\displaystyle=$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+(1+(|W_{2}|-1)-T^{\prime}_{2})$
$\displaystyle=$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+(1+|W_{2}|-T_{2})$
$\displaystyle=$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}}).$

and

$\displaystyle\operatorname{depth}(S_{G}/(J+K))=$
$\displaystyle\operatorname{depth}(S_{G_{1}\backslash
f_{1}}/I_{G_{1}\backslash f_{1}})+(|W_{2}|-|W_{2,1}|)$ $\displaystyle=$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-1.$

where the second equality holds by Lemma 3.5 and Theorem 2.6. Applying Lemma
1.2(1) and Lemma 1.5 to the exact sequence (1), we get the desired result.

(2) If $t\geq 1$, we assume that $T^{\prime}_{2}\neq T_{2}-1$. This implies
that $T_{2}=|W_{2,j}|\geq|W_{2,1}|$ for some $j\neq 1$. Suppose
$T_{2}=|W_{2,2}|$. In this case, we prove by induction on $n_{2}$ that

$\operatorname{depth}(S_{G}/I_{G})=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-1.$

If $n_{2}=2$, then $G_{2}=P_{3}\cup_{u}K_{b}$ is the clique sum of $P_{3}$ and
some $K_{b}$, we get $\operatorname{depth}(S_{G_{2}}/I_{G_{2}})=2$ by Theorem
2.6. Let $J=(x_{N_{G}(u)})+I_{G\backslash N_{G}[u]}$ and
$K=(x_{u})+I_{G\backslash u}$, then $J+K=(x_{N_{G}[u]})+I_{G\backslash
N_{G}[u]}$, $G\backslash N_{G}[u]=G_{1}$ and $G\backslash
u=K_{b-1}\sqcup(G_{1}\circ P_{4})$. So
$\operatorname{depth}(S_{G}/(J+K))=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})$
and

$\displaystyle\operatorname{depth}(S_{G}/K)$
$\displaystyle=\operatorname{depth}(S_{G\backslash u}/I_{G\backslash u})$
$\displaystyle=1+\operatorname{depth}(S_{G_{1}\circ P_{4}}/I_{G_{1}\circ
P_{4}})$ $\displaystyle\geq
1+(\operatorname{depth}(S_{P_{4}}/I_{P_{4}})+\operatorname{depth}(S_{G_{1}}/I_{G_{1}})-2)$
$\displaystyle=\operatorname{depth}(S_{P_{4}}/I_{P_{4}})+\operatorname{depth}(S_{G_{1}}/I_{G_{1}})-1$
$\displaystyle=2+\operatorname{depth}(S_{G_{1}}/I_{G_{1}})-1$
$\displaystyle=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+1$

where the inequality holds by Theorem 3.6. Applying Lemma 1.2(1) and Lemma 1.5
to the exact sequence (1), we get

$\operatorname{depth}(S_{G}/I_{G})=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+1=\operatorname{depth}(S_{1}/I_{G_{1}})+\operatorname{depth}(S_{2}/I_{G_{2}})-1.$

Now we assume that $n_{2}\geq 3$. Let $W_{2,2}=\\{w_{1},\ldots,w_{r_{2}}\\}$
and $\\{K_{a_{2,1}},\ldots,K_{a_{2,r_{2}}}\\}$ be a branch of the fan on
$W_{2,2}$ with $V(K_{n_{2}})\cap V(K_{a_{2,i}})=\\{w_{1},\ldots,w_{i}\\}$. Let
$J=(x_{N_{G}(w_{1})})+I_{G\backslash N_{G}[w_{1}]}$ and
$K=(x_{w_{1}})+I_{G\backslash w_{1}}$, then
$J+K=(x_{N_{G}[w_{1}]})+I_{G\backslash N_{G}[w_{1}]}$, $G\backslash
N_{G}[w_{1}]$ has $(|W_{2}|-|W_{2,2}|)$ connected components, which consisting
of $G_{1}$ and $(|W_{2}|-|W_{2,2}|-1)$ connected components consisting of some
cliques and isolated vertices. Then we have

$\displaystyle\operatorname{depth}(S_{G}/(J+K))=$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+(|W_{2}|-|W_{2,2}|-1)$
$\displaystyle=$
$\displaystyle\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}\,(S_{G_{2}}/I_{G_{2}})-2.$

where the last equality holds by Theorem 2.6. At the same time, $G\backslash
w_{1}=(G_{1}*(G_{2}\backslash w_{1}))\sqcup K_{b-1}$, then

$\operatorname{depth}(S_{G}/K)=\operatorname{depth}(S_{G\backslash
w_{1}}/I_{G\backslash w_{1}})=1+\operatorname{depth}(S_{G_{1}*(G_{2}\backslash
w_{1})}/I_{G_{1}*(G_{2}\backslash w_{1})}).$

To compute the $\operatorname{depth}(S_{G_{1}*(G_{2}\backslash
w_{1})}/I_{G_{1}*(G_{2}\backslash w_{1})})$, let $T^{*}_{1}=T_{1}$,
$T^{\prime*}_{1}=T^{\prime}_{1}$, $T^{*}_{2}=\max\\{|W_{2,1}|,\\\
|W_{2,2}|-1,|W_{2,3}|,\ldots,|W_{2,k_{2}}|\\}$,
$T^{\prime*}_{2}=\max\\{|W_{2,1}|-1,|W_{2,2}|-1,|W_{2,3}|,\ldots,|W_{2,k_{2}}|\\}$
and $t^{\prime}=|\\{i:T^{\prime*}_{i}\neq T^{*}_{i}-1\\}|$. We distinguish
into the following two subcases:

(i) If $t=1$, then $T^{\prime}_{1}=T_{1}-1$. In this case, for the graph
$G_{1}*(G_{2}\backslash w_{1})$, we have $t^{\prime}\leq 1$. Thus, by the
above case $(1)$ and by induction, we have

$\displaystyle\operatorname{depth}(S_{G\backslash w_{1}}/I_{G\backslash
w_{1}})$
$\displaystyle=\begin{cases}\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}\backslash
w_{1}}/I_{G_{2}\backslash w_{1}}),&\text{if $t^{\prime}=0$,}\\\
\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}\backslash
w_{1}}/I_{G_{2}\backslash w_{1}})-1,&\text{if $t^{\prime}=1$,}\end{cases}$
$\displaystyle\geq\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}\backslash
w_{1}}/I_{G_{2}\backslash w_{1}})-1.$

(ii) If $t=2$, then $T^{\prime}_{1}\neq T_{1}-1$. In this case, for the graph
$G_{1}*(G_{2}\backslash w_{1})$, we have $t^{\prime}\geq 1$. By induction, we
have

$\operatorname{depth}(S_{G_{1}*(G_{2}\backslash
w_{1})}/I_{G_{1}*(G_{2}\backslash
w_{1})})=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}\setminus
w_{1}}/I_{G_{2}\setminus w_{1}})-1.$

In both cases we obtain

$\displaystyle\operatorname{depth}(S_{G}/K)$
$\displaystyle=1+\operatorname{depth}(S_{G_{1}*(G_{2}\backslash
w_{1})}/I_{G_{1}*(G_{2}\backslash w_{1})})$ $\displaystyle\geq
1+(\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}\backslash
w_{1}}/I_{G_{2}\backslash w_{1}})-1)$
$\displaystyle=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+(1+(|W_{2}|-1)-T^{*}_{2})$
$\displaystyle\geq\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+(|W_{2}|-T_{2})$
$\displaystyle=\operatorname{depth}(S_{G_{1}}/I_{G_{1}})+\operatorname{depth}(S_{G_{2}}/I_{G_{2}})-1$

where the second inequality holds because of $T^{*}_{2}\leq T_{2}$. Applying
Lemma 1.2(1) and Lemma 1.5 to the exact sequence (1), we obtain the desired
results.

Next, we are ready to prove another major result of this subsection.

######  Theorem 3.10.

Suppose that $G=(G_{1},f_{1})*(G_{2},f_{2})$, where
$G_{i}=F_{k_{i}}^{W_{i}}(K_{n_{i}})$ is a $k_{i}$-fan graph of $K_{n_{i}}$ on
the set $W_{i}$ with $n_{i}\geq 2$ and $k_{i}\geq 1$ for $i\in[2]$. Let
$v_{i}$ be the neighbor of $f_{i}$ in $G_{i}$ and
$t=|\\{i\in[2]:\operatorname{reg}(S_{G_{i}\backslash v_{i}}/I_{G_{i}\backslash
v_{i}})\neq\operatorname{reg}(S_{G_{i}}/I_{G_{i}})\\}|$. We have

  * (1)

If $t\leq 1$, then
$\operatorname{reg}(S_{G}/I_{G})=\operatorname{reg}(S_{G_{1}}/I_{G_{1}})+\operatorname{reg}(S_{G_{2}}/I_{G_{2}})$;

  * (2)

If $t=2$ and $|W_{i}|=n_{i}-1$ for all $i\in[2]$, then

$\operatorname{reg}(S_{G}/I_{G})=\operatorname{reg}(S_{G_{1}}/I_{G_{1}})+\operatorname{reg}(S_{G_{2}}/I_{G_{2}}),$

  * (3)

If $t=2$ and $|W_{i}|=n_{i}$ for some $i\in[2]$, then

$\operatorname{reg}(S_{G}/I_{G})=\operatorname{reg}(S_{G_{1}}/I_{G_{1}})+\operatorname{reg}(S_{G_{2}}/I_{G_{2}})-1.$

###### Proof.

First we have $t\in\\{0,1,2\\}$ and
$\operatorname{reg}(S_{G}/I_{G})=\vartheta(G)$ and
$\operatorname{reg}(S_{G_{i}}/I_{G_{i}})=\vartheta(G_{i})$ for $i\in[2]$ by
the definition of $t$ and Lemma 1.8. We distinguish between the following two
cases:

(1) If $t\leq 1$, we assume that $\operatorname{reg}(S_{G_{1}\backslash
v_{1}}/I_{G_{1}\backslash v_{1}})=\operatorname{reg}(S_{G_{1}}/I_{G_{1}})$,
i.e., $\vartheta(G_{1}\backslash v_{1})=\vartheta(G_{1})$. In this case,
$M=M_{1}\sqcup M_{2}$ is an induced matching of $G$, where
$M_{1}=\\{e_{1,1},\ldots,e_{1,\vartheta(G_{1}\backslash v_{1})}\\}$ and
$M_{2}=\\{e_{2,1},\ldots,e_{2,\vartheta(G_{2})}\\}$ are induced matchings of
$G_{1}\backslash v_{1}$ and $G_{2}$, respectively. Thus
$\vartheta(G)\geq\vartheta(G_{1}\backslash
v_{1})+\vartheta(G_{2})=\vartheta(G_{1})+\vartheta(G_{2})$. If
$\vartheta(G)\geq\vartheta(G_{1})+\vartheta(G_{2})+1$, then there exists an
induced matching $M^{\prime}$ of $G$, which has at least
$\vartheta(G_{1})+\vartheta(G_{2})+1$ elements. So there exist at least
$(\vartheta(G_{1})+1)$ elements in $M^{\prime}$ belonging to $E(G_{1})$ or
$(\vartheta(G_{2})+1)$ elements in $M^{\prime}$ belonging to $E(G_{2})$. This
forces $\vartheta(G_{1})+1\leq\vartheta(G_{1})$ or
$\vartheta(G_{2})+1\leq\vartheta(G_{2})$, a contradiction.

(2) If $t=2$, then each $\operatorname{reg}(S_{G_{i}\backslash
v_{i}}/I_{G_{i}\backslash v_{i}})\neq\operatorname{reg}(S_{G_{i}}/I_{G_{i}})$,
i.e., each $\vartheta(G_{i}\backslash v_{i})<\vartheta(G_{i})$. Let
$W_{i}=W_{i_{1}}\sqcup\cdots\sqcup W_{i_{k_{i}}}$ be a partition of $W_{i}$
with $v_{i}\in W_{i_{1}}$. From the assumptions that for each $i$,
$\operatorname{reg}(S_{G_{i}\backslash v_{i}}/I_{G_{i}\backslash
v_{i}})\neq\operatorname{reg}(S_{G_{i}}/I_{G_{i}})$ and $f_{i}$ is a leaf of
$G_{i}$ with a unique neighbor point $v_{i}$, we have $|W_{i_{1}}|=1$ from the
proof of Theorem 2.8. Let
$W_{i_{\ell}}=\\{w_{i_{\ell},1},\ldots,w_{i_{\ell},r_{i_{\ell}}}\\}$ for
$\ell\in[k_{i}]$ and $i\in[2]$, then by Lemma 3.7 we get that $|W_{i}|\geq
n_{i}-1$ and $h_{i_{\ell},r_{i_{\ell}}}\geq 2$ for all $\ell=2,\ldots,k_{i}$
and $i\in[2]$. In this case, each $G_{i}\backslash W_{i}$ is the disjoint
union of $(\vartheta(G_{i})-1)$ cliques $K^{\prime}_{i,1}$,
…,$K^{\prime}_{i,\vartheta(G_{i})-1}$ and some isolated vertices. Let
$M_{i}=\\{e_{ij}|e_{ij}\in E(K^{\prime}_{i,j}),j\in[\vartheta(G_{i})-1]\\}$
for $i\in[2]$, then $M_{i}$ is an induced matching of $G_{i}\backslash v_{i}$.
We consider two subcases:

(i) If $|W_{i}|=n_{i}-1$ for all $i\in[2]$. In this case, let $M=M_{1}\sqcup
M_{2}\sqcup\\{e_{1},e_{2}\\}$, where the edge $e_{i}=\\{v_{i},u_{i}\\}$ with
$u_{i}\in V(K_{n_{i}})\backslash W_{i}$. Then $M$ is an induced matching of
$G$ by the choice of $M_{1}$ and $M_{2}$. Hence
$\vartheta(G)\geq\vartheta(G_{1}\backslash v_{1})+\vartheta(G_{2}\backslash
v_{2})+2=\vartheta(G_{1})+\vartheta(G_{2})$. If
$\vartheta(G)\geq\vartheta(G_{1})+\vartheta(G_{2})+1$, then there exists an
induced matching $M^{\prime\prime}$ of $G$, which has at least
$\vartheta(G_{1})+\vartheta(G_{2})+1$ elements. So there are at least
$(\vartheta(G_{1})+1)$ elements in $M^{\prime\prime}$ belonging to $E(G_{1})$
or $(\vartheta(G_{2})+1)$ elements in $M^{\prime\prime}$ belonging to
$E(G_{2})$. This forces $\vartheta(G_{1})+1\leq\vartheta(G_{1})$ or
$\vartheta(G_{2})+1\leq\vartheta(G_{2})$, a contradiction.

(ii) If $|W_{i}|=n_{i}$ for some $i\in[2]$, then we assume that
$|W_{1}|=n_{1}$. In this case, let $M=M_{1}\sqcup M_{2}\sqcup\\{e\\}$, where
the edge $e=\\{v_{1},f\\}$ with $f=f_{1}=f_{2}$ in $G$. Then $M$ is an induced
matching of $G$ by the choice of $M_{1}$ and $M_{2}$. So
$\vartheta(G)\geq\vartheta(G_{1}\backslash v_{1})+\vartheta(G_{2}\backslash
v_{2})+1=\vartheta(G_{1})+\vartheta(G_{2})-1$. If
$\vartheta(G)\geq\vartheta(G_{1})+\vartheta(G_{2})$, then there exists an
induced matching $M^{\prime\prime\prime}$ of $G$, which has at least
$\vartheta(G_{1})+\vartheta(G_{2})$ elements. So there are at least
$\vartheta(G_{1})$ elements in $M^{\prime\prime\prime}$ belonging to
$E(G_{1})$ or $\vartheta(G_{2})$ elements in $M^{\prime\prime\prime}$
belonging to $E(G_{2})$. Since each
$\vartheta(G_{i})=\vartheta(G_{i}\backslash v_{i})+1$ and $|W_{1}|=n_{1}$,
there are elements $e_{i_{1}}$ and $e_{i_{2}}$ in $M^{\prime\prime\prime}$
with $e_{i_{1}}=\\{v_{1},f\\}$ and $e_{i_{2}}=\\{v_{2},f\\}$. Thus $v_{1}$,
$v_{2}$ and $f$ form a path in $M^{\prime\prime\prime}$, which contradicts
that $M^{\prime\prime\prime}$ is an induced matching of $G$.

Acknowledgments

This research is supported by the Natural Science Foundation of Jiangsu
Province (No. BK20221353). The authors are grateful to the computer algebra
systems CoCoA [4] for providing us with a large number of examples.

Data availability statement

The data used to support the findings of this study are included within the
article.

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