# On certain edge-transitive bicirculants of twice odd order

István Kovács 1,2 I. Kovács
UP IAM, University of Primorska, Muzejski trg 2, SI-6000 Koper, Slovenia
UP FAMNIT, University of Primorska, Glagolǰaska ulica 8, SI-6000 Koper,
Slovenia<EMAIL_ADDRESS>and  János Ruff 2 J. Ruff
University of Pécs, Institute of Mathematics and Informatics, Ifjúság útja 6,
H-7624 Pécs, Hungary<EMAIL_ADDRESS>
###### Abstract.

A graph admitting an automorphism with two orbits of the same length is called
a bicirculant. Recently, Jajcay et al. initiated the investigation of the
edge-transitive bicirculants with the properties that one of the subgraphs
induced by the latter orbits is a cycle and the valence is at least $6$
(Electron. J. Combin., 2019). We show that the complement of the Petersen
graph is the only such graph whose order is twice an odd number.

###### Key words and phrases:

bicirculant, edge-transitive, primitive permutation group

###### 2010 Mathematics Subject Classification:

05C25, 20B25

1 Partially supported by the Slovenian Research Agency (research program
P1-0285, research projects N1-0062, J1-9108, J1-1695, J1-2451 and N1-0208).
2 Partially supported by the ARRS-NKFIH Slovenian-Hungarian Joint Research
Project, grant no. SNN 132625 (in Hungary) and N1-0140 (in Slovenia).

##  1\. Introduction

All groups and graphs in this paper will be finite. A graph admitting an
automorphism with two orbits of the same length is called a bicirculant. The
symmetry properties of bicirculants have attracted considerable attention
(see, e.g., [1, 4, 6, 15, 20, 21, 23, 28]). Recently, Jajcay et al. [10]
initiated the investigation of the edge-transitive bicirculants with the
properties that one of the subgraphs induced by the latter orbits is a cycle
and the valence is at least $6$. Motivated by this, we set the following
notation.

######  Notation.

For a positive integer $d\geq 3$, denote by $\mathcal{F}(d)$ the family of
regular graphs having valence $d$ and admitting an automorphism with two
orbits of the same length such that one of the subgraphs induces by these
orbits is a cycle.

The graphs in the family $\mathcal{F}(3)$ are the well studied generalised
Petersen graphs, which were introduced by Watkins [26] in 1969. The graphs in
$\mathcal{F}(4)$ were defined under the name Rose Window graphs by Wilson [27]
and those in $\mathcal{F}(5)$ under the name Tabačjn graphs by Arroyo et al.
[2]. The question which of these graphs are edge-transitive have been answered
in [9, 14, 2]. Moreover, the automorphism groups of all (not only the edge-
transitive) graphs in the families $\mathcal{F}(d)$, $d=3,4,5$, are also known
(see [9, 14, 8, 16]).

Jajcay et al. [10] focused primarily on the family $\mathcal{F}(6)$, they
called the members of this family Nest graphs (see also [25]). Their main
result was the classification of edge-transitive Nest graphs of girth $3$, the
task to classify all edge-transitive Nest graphs was posed as [10, Problem
1.2]. Regarding the families $\mathcal{F}(d)$ with $d>6$, the following
questions were raised (see [10, Question 1.1]):

  1. (1)

For which $d>6$ does the family $\mathcal{F}(d)$ contain at least one edge-
transitive graph?

  2. (2)

For which $d>6$ does the family $\mathcal{F}(d)$ contain infinitely many edge-
transitive graphs?

Jajcay et al. [10] also carried out an exhaustive computer search for edge-
transitive graphs of order at most $220$ and belonging to $\mathcal{F}(6)$,
and also for edge-transitive graphs of order at most $100$ and belonging to
the families $\mathcal{F}(d)$ with $7\leq d\leq 10$. By the order of a graph
we mean the number of its vertices. They obtained $66$ graphs in
$\mathcal{F}(6)$ (see [10, Table 1]) and none in the families
$\mathcal{F}(d)$, $7\leq d\leq 10$. Among the $66$ graphs, only one has twice
odd order, and this graph is the complement of the Petersen graph. Motivated
by these observations, in this paper we focus on the edge-transitive graphs in
the families $\mathcal{F}(d)$, $d\geq 6$, whose order is twice an odd number.
We would like to remark that the remaining edge-transitive graphs in
$\mathcal{F}(6)$ are dealt with in [13].

Our main result is the following theorem.

######  Theorem 1.1.

The family $\mathcal{F}(d)$ with $d>6$ contains no edge-transitive graph of
twice odd order. Furthermore, the complement of the Petersen graph is the only
edge-transitive graph in family $\mathcal{F}(6)$ of twice odd order.

The paper is organised as follows. Section 2 contains the needed results form
graph and group theory. The next two sections are devoted to the preparation
for the proof of our main theorem. The main result in Section 3 is Lemma 3.3,
which contains some necessary conditions for a graph in $\mathcal{F}(6)$ to be
edge-transitive. In Section 4 we analyse the blocks of imprimitivity for a
group of automorphisms acting transitively on the edges of a graph from
$\mathcal{F}(d)$. The core of our proof lies in this analysis, by which we
rely on the classification of primitive permutations groups containing a
semiregular cyclic subgroup with two orbits (see [22]), and the classification
of arc-transitive circulants (see [12, 17, 18]). The proof of Theorem 1.1 is
presented in Section 5.

##  2\. Preliminaries

###  2.1. Graph theory

For a graph $\Gamma$, let $V(\Gamma)$, $E(\Gamma)$, $A(\Gamma)$ and
$\operatorname{Aut}(\Gamma)$ denote its vertex set, edge set, arc set and
automorphism group, respectively. The set of vertices adjacent with a given
vertex $v$ is denoted by $\Gamma(v)$.

Let $G\leq\operatorname{Aut}(\Gamma)$ and let $v\in V(\Gamma)$. The stabiliser
of $v$ in $G$ is denoted by $G_{v}$ and the orbit of $v$ under $G$ by $v^{G}$.
For a subset $B\subseteq V(\Gamma)$, the set-wise stabiliser of $B$ in $G$ is
denoted by $G_{\\{B\\}}$. If $G$ is transitive on $V(\Gamma)$, then $\Gamma$
is said to be $G$-vertex-transitive, and $\Gamma$ is simply called vertex-
transitive when it is $\operatorname{Aut}(\Gamma)$-vertex-transitive. The
($G$-)edge- and ($G$-)arc-transitive graphs are defined correspondingly.

Let $\Gamma$ be a $G$-vertex-transitive graph. A subset $B\subseteq V(\Gamma)$
is called a block for $G$ (the term block of imprimitivity is also commonly
used) if $B^{g}\cap B=\emptyset$ or $B^{g}=B$ holds for every $g\in G$. The
block $B$ is non-trivial if $1<|B|<|V(\Gamma)|$, and it is minimal if it is
non-trivial and no non-trivial block is contained properly in $B$. The block
system induced by $B$ is the partition of $V(\Gamma)$ consisting of the images
$B^{g}$, where $g$ runs over $G$. A block system is called normal if it
consists of the orbits of a normal subgroup of $G$.

Let $\pi$ be an arbitrary partition of $V(\Gamma)$. For a vertex $v\in
V(\Gamma)$, let $\pi(v)$ denote the class containing $v$. The quotient graph
of $\Gamma$ with respect to $\pi$, denoted by $\Gamma/\pi$, is defined to have
vertex set $\pi$, and edges $\\{\pi(u),\pi(v)\\}$, where $\\{u,v\\}\in
E(\Gamma)$ such that $\pi(u)\neq\pi(v)$. If there exists a constant $r$ such
that

$\forall\\{u,v\\}\in
E(\Gamma):~{}\pi(u)\neq\pi(v)~{}\text{and}~{}|\Gamma(u)\cap\pi(v)|=r,$

then $\Gamma$ is called an $r$-cover of $\Gamma/\pi$ (our definition of an
$r$-cover generalises the definition given in [6], where $\pi$ is also assumed
to be a block system). The term cover will also be used instead of $1$-cover.
In the special case when $\pi$ is formed by the orbits of an intransitive
normal subgroup $N\lhd\operatorname{Aut}(\Gamma)$, $\Gamma/N$ will also be
written for $\Gamma/\pi$, and the term normal $r$-cover (normal cover,
respectively) will also be used instead of $r$-cover (cover, respectively).
The following properties are well-known.

######  Proposition 2.1.

Let $\Gamma$ be a connected $G$-vertex- and $G$-edge-transitive graph, let
$\mathcal{B}$ be a normal block system of $G$, and let $K$ be the kernel of
the action of $G$ on $\mathcal{B}$.

  1. (1)

$\Gamma$ is a normal $r$-cover of $\Gamma/\mathcal{B}$, where
$r=|\Gamma(v)\cap B|$, $v$ is any vertex and $B$ is any block in $\mathcal{B}$
containing a neighbour of $v$.

  2. (2)

If $\Gamma$ is a normal cover of $\Gamma/\mathcal{B}$, then $\Gamma$ and
$\Gamma/\mathcal{B}$ have the same valence, the kernel $K$ is regular on every
block in $\mathcal{B}$, and $\Gamma/\mathcal{B}$ is $G/K$-edge-transitive.

Let $S\subset H$ be a subset of a group $H$ such that $1_{H}\notin S$, where
$1_{H}$ denotes the identity element of $H$. The Cayley digraph
$\operatorname{Cay}(H,S)$ is defined to have vertex set $H$ and arcs $(h,sh)$,
where $h\in H$ and $s\in S$. In the case when $S$ is inverse-closed, we regard
$\operatorname{Cay}(H,S)$ as an undirected graph and use the term Cayley
graph. It is a well-known observation (see Sabidussi [24]) that, if $\Gamma$
is any graph, $v$ is any vertex, and $H\leq\operatorname{Aut}(\Gamma)$ is a
regular subgroup, then

$\Gamma\cong\operatorname{Cay}(H,S),~{}\text{where}~{}S=\\{x\in
H:v^{x}\in\Gamma(v)\\}.$ (1)

The Cayley digraphs of cyclic groups are shortly called circulants. A
recursive classification of finite arc-transitive circulants was obtained
independently by Kovács [12] and Li [17]. The paper [12] also provides an
explicit characterisation (see [12, Theorem 4]), which was rediscovered
recently by Li et a. [18]. The characterisation presented below follows from
the proof of [12, Theorem 4] or from [18, Theorem 1.1]. In order to state this
characterisation, some further definitions are introduced next.

For a finite group $H$, denote by $H^{\\#}$ the set of all non-identity
elements of $H$, and by $H_{R}$ the group of all right multiplications by the
elements of $H$. If $K\leq H$, then let $[H:K]$ denote the set of right
$K$-cosets in $H$. In the case when $K$ is a block for
$\operatorname{Aut}(\operatorname{Cay}(H,S))$, the block system induced by $K$
is equal to $[H:K]$. Now, if $K\lhd H$ also holds, then the image of the
action of $H_{R}$ on $[H:K]$ is regular, in particular, $\Gamma/[H:K]$ becomes
a Cayley graph of the group $H/K$.

A Cayley graph $\operatorname{Cay}(H,S)$ is called normal if
$H_{R}\lhd\operatorname{Aut}(\operatorname{Cay}(H,S))$. Note that, if
$\operatorname{Cay}(H,S)$ is a normal arc-transitive Cayley graph, then $S$ is
equal to an $A$-orbit for some subgroup $A\leq\operatorname{Aut}(H)$.

######  Theorem 2.2.

([12]) Let $\Gamma=\operatorname{Cay}(C,S)$ be a connected arc-transitive
graph, where $C$ is a cyclic group of order $n$. Then one of the following
holds.

  1. (a)

$\Gamma$ is the complete graph.

  2. (b)

$\Gamma$ is normal.

  3. (c)

There exists a subgroup $1<D<C$ such that $D$ is a block for
$\operatorname{Aut}(\Gamma)$ and $\Gamma/[C:D]$ is a connected arc-transitive
circulant. Furthermore, $S$ is a union of $D$-cosets.

  4. (d)

There exist subgroups $1<D,E<C$ such that both $D$ and $E$ are blocks for
$\operatorname{Aut}(\Gamma)$, $C=D\times E$, $|D|>3$ and $\gcd(|D|,|E|)=1$.
Furthermore, $S=D^{\\#}R$, where $R\subseteq E^{\\#}$, $R$ is inverse-closed,
$\operatorname{Cay}(E,R)$ is connected and arc-transitive.

Besides the Petersen graph, two further small arc-transitive graphs will
appear later. The Clebsch graph is obtained from the $4$ dimensional cube
graph $Q_{4}$ by adding the edges connecting antipodal points; the lattice
graph $L_{2}(4)$ is defined to have vertices the ordered pairs $(i,j)$, $1\leq
i,j\leq 4$, and two vertices are adjacent if and only if either their first or
second coordinates are the same. The graph $L_{2}(4)$ is depicted in Figure 1.
It can be easily checked that the mapping $\sigma:(i,j)\mapsto(j+1,i)$, where
the addition is computed modulo $4$, is an automorphism of $L_{2}(4)$,
$\sigma$ has two orbits of the same length, and one of the subgraphs induced
by these orbits is a cycle. All these show that $L_{2}(4)$ is an example of an
edge-transitive graph from the family $\mathcal{F}(6)$.

$(1,1)$$(2,1)$$(3,1)$$(4,1)$$(1,2)$$(2,2)$$(3,2)$$(4,2)$$(1,3)$$(2,3)$$(3,3)$$(4,3)$$(1,4)$$(2,4)$$(3,4)$$(4,4)$
Figure 1. The lattice graph $L_{2}(4)$ and its subgraph induced by the orbit
of the vertex $(1,1)$ under the automorphism $\sigma:(i,j)\mapsto(j+1,i)$,
which is shown with thick lines.

###  2.2. Group theory

Our terminology and notation are standard and we follow the books [7, 11]. The
socle of a group $G$, denoted by $\operatorname{soc}(G)$, is the subgroup
generated by the set of all minimal normal subgroups (see [7, page 111]). The
group $G$ is called almost simple if $\operatorname{soc}(G)=T$, where $T$ is a
non-abelian simple group. In this case $G$ is embedded in
$\operatorname{Aut}(T)$ so that its socle is embedded via the inner
automorphisms of $T$, and we also write $T\leq G\leq\operatorname{Aut}(T)$.

Our proof of Theorem 1.1 relies on the classification of primitive groups
containing a cyclic subgroup with two orbits due to Müller [22]. Here we need
only the special case when the cyclic subgroup is semiregular.

######  Theorem 2.3.

([22, Theorem 3.3]) Let $G$ be a primitive permutation group of degree $2n$
containing an element with two orbits of the same length. Then one of the
following holds.

  1. (1)

(Affine action) $G$ contains a regular normal subgroup isomorphic to
$\mathbb{Z}_{2}^{m}$, where $m\in\\{2,3,4\\}$.111The group $G$ is from a short
list, but as this possibility will not occur later, we omit the details.

  2. (2)

(Almost simple action) $G$ is an almost simple group and one of the following
holds.

    1. (a)

$n\geq 3$, $\operatorname{soc}(G)=A_{2n}$, and $A_{2n}\leq G\leq S_{2n}$ in
its natural action.

    2. (b)

$n=5$, $\operatorname{soc}(G)=A_{5}$, and $A_{5}\leq G\leq S_{5}$ in its
action on the set of $2$-subsets of $\\{1,2,3,4,5\\}$.

    3. (c)

$n=(q^{d}-1)/2(q-1)$, $\operatorname{soc}(G)=PSL_{d}(q)$, and $PSL_{d}(q)\leq
G\leq P\Gamma L_{d}(q)$ for some odd prime power $q$ and even number $d\geq 2$
such that $(d,q)\neq(2,3)$.

    4. (d)

$n=6$ and $\operatorname{soc}(G)=G=M_{12}$.

    5. (e)

$n=11$, $\operatorname{soc}(G)=M_{22}$, and $M_{22}\leq
G\leq\operatorname{Aut}(M_{22})$.

    6. (f)

$n=12$ and $\operatorname{soc}(G)=G=M_{24}$.

If $G$ is a group in one of the families (a)-(f) above, then it follows from
[7, Theorem 4.3B] that $\operatorname{soc}(G)$ is the unique minimal normal
subgroup of $G$. Therefore, we have the following corollary.

######  Corollary 2.4.

Let $G$ be a primitive permutation group in one of the families (a)-(f) in
part (2) of Theorem 2.3, and let $N\lhd G$, $N\neq 1$. Then $N$ is also
primitive.

For a transitive permutation group $G\leq\operatorname{Sym}(\Omega)$, the
subdegrees of $G$ are the lengths of the orbits of a point stabiliser
$G_{\omega}$, $\omega\in\Omega$. Since $G$ is transitive, it follows that the
subdegrees do not depend on the choice of $\omega$ (see [7, page 72]). The
number of orbits of $G_{\omega}$ is called the rank of $G$. The actions of a
group $G$ on sets $\Omega$ and $\Omega^{\prime}$ are said to be equivalent if
there is a bijection $\varphi:\Omega\to\Omega^{\prime}$ such that

$\forall\omega\in\Omega,~{}\forall g\in
G:~{}\varphi(\omega^{g})=(\varphi(\omega))^{g}.$

Now, suppose that $G$ is a group in one of the families (a)-(f) in part (2) of
Theorem 2.3. If $G$ is in family (a), then the action is unique up to
equivalence and $G$ is clearly $2$-transitive. If $G$ is in family (b), then
the action is unique up to equivalence and the subdegrees are $1,3$ and $6$.
Let $G$ be in family (c). The semiregular cyclic subgroup of $G$ with two
orbits is contained in a regular cyclic group, called the Singer subgroup of
$PGL_{d}(q)$ (see [11, Chapter 2, Theorem 7.3]). In this case the action is
unique up to equivalence if and only if $d=2$. If $d\geq 4$, then the action
of $G$ is equivalent to either its natural action on the set of points of the
projective geometry $PG_{d}(q)$, or to its natural action on the set of
hyperplanes of $PG_{d}(q)$. In both actions $G$ is $2$-transitive. Finally, if
$G$ is in the families (d)-(f), then the action is unique up to equivalence
and $G$ is $2$-transitive (this can also be read off from [5]). All this
information is summarised in the lemma below.

######  Lemma 2.5.

Let $G$ be a primitive permutation group in one of the families (a)-(f) in
part (2) of Theorem 2.3.

  1. (1)

$G$ is $2$-transitive, unless $G$ belongs to family (b). In the latter case
the subdegrees are $1,3$ and $6$.

  2. (2)

The action of $G$ is unique up to equivalence, unless $G$ is in family (c) and
$d\geq 4$. In the latter case $G$ admits two inequivalent faithful actions,
namely, the natural actions on the set of points and the set of hyperplanes,
respectively, of the projective geometry $PG_{d}(q)$.

The following result about $G$-arc-transitive bicirculants was proved by
Devillers et al. [6], but the proof works also for the edge-transitive
bicirculants as well. In fact, it is an easy consequence of Theorem 2.3.

######  Proposition 2.6.

([6, part (1) of Proposition 4.2]) Let $\Gamma$ be a $G$-edge-transitive
bicirculant such that $G$ is a primitive group. Then $\Gamma$ is one of the
following graphs:

  1. (1)

The complete graph, and $G$ is one of the $2$-transitive groups described in
part (2) of Theorem 2.3.

  2. (2)

The Petersen graph or its complement, and $A_{5}\leq G\leq S_{5}$.

  3. (3)

The lattice graph $L_{2}(4)$ or its complement, and $G$ is a rank $3$ subgroup
of $AGL(4,2)$.

  4. (4)

The Clebsch graph or its complement, and $G$ is a rank $3$ subgroup of
$AGL(4,2)$.

One can easily check which of the graphs in the families (1)–(4) above belongs
also to the family $\mathcal{F}(d)$ for some $d\geq 3$.

######  Corollary 2.7.

Let $\Gamma\in\mathcal{F}(d)$ be a $G$-edge-transitive graph for some $d\geq
3$. If $G$ is primitive on $V(\Gamma)$, then $\Gamma$ is isomorphic to
$K_{6}$, or the Petersen graph, or its complement, or the lattice graph
$L_{2}(4)$.

##  3\. A lemma on the graphs in the family $\mathcal{F}(6)$

In this section we derive some necessary conditions for a graph in
$\mathcal{F}(6)$ to be edge-transitive. Our main tool is the coset graph
construction defined next.

Let $G$ be a group, let $H$ be a core-free subgroup of $G$, and let $S$ be a
subset of $G$ such that $HSH=HS^{-1}H$. By core-free we mean that $H$ contains
no non-trivial normal subgroup of $G$. The coset graph
$\operatorname{Cos}(G,H,HSH)$ is defined to have vertex set $[G:H]$ (the set
of all right $H$-cosets in $G$), and edges $\\{Hx,Hy\\}$, where $x,y\in G$ and
$yx^{-1}\in HSH$. The lemma below is a folklore result. For the completeness
of the paper, we provide the reader with a proof.

######  Lemma 3.1.

Let $\Gamma$ be a both $G$-vertex- and $G$-edge-transitive graph. Write $H$
for the vertex stabiliser $\operatorname{Aut}(\Gamma)_{v}$, and let $g\in G$
such that $\\{v,v^{g}\\}$ is an edge. Then the mapping

$\varphi:[G:H]\to V(\Gamma),~{}Hx\mapsto v^{x}~{}\text{for}~{}x\in G$

is an isomorphism between $\operatorname{Cos}(G,H,H\\{g,g^{-1}\\}H)$ and
$\Gamma$.

###### Proof.

We check first that $\varphi$ is indeed a mapping. Suppose that $Hx=Hy$ for
some $x,y\in G$. Then $x=hy$ for some $h\in H$, and
$\varphi(Hx)=v^{x}=v^{hy}=v^{y}=\varphi(Hy)$. This shows that $\varphi$ is
well-defined. Since $\Gamma$ is $G$-vertex-transitive, it follows that
$\varphi$ is surjective. Using also that $|G|/|H|=|V(\Gamma)|$, we find that
$\varphi$ is a bijection.

It remains to prove that $\\{Hx,Hy\\}$ is an edge of
$\cos(G,H,H\\{g,g^{-1}\\}H)$ if and only if $\\{v^{x},v^{y}\\}$ is an edge of
$\Gamma$.

Assume first that $\\{Hx,Hy\\}$ is an edge. By definition,
$yx^{-1}=h_{1}gh_{2}$ or $h_{1}g^{-1}h_{2}$ for some $h_{1},h_{2}\in H$. We
deal only with the first case, the second one can be treated in the same way.
Then $v^{yx^{-1}}=v^{h_{1}gh_{2}}=v^{gh_{2}}$, and as $\\{v,v^{gh_{2}}\\}\in
E(\Gamma)$, we find $\\{v^{x},v^{y}\\}\in E(\Gamma)$.

Now suppose that $\\{v^{x},v^{y}\\}\in E(\Gamma)$ for some $x,y\in G$. Since
$\Gamma$ is edge-transitive, there is some $g^{\prime}\in G$, which maps the
edge $\\{v,v^{yx^{-1}}\\}$ to $\\{v,v^{g}\\}$. This means that

$v^{g^{\prime}}=v,~{}v^{yx^{-1}g^{\prime}}=v^{g}~{}\text{or}~{}v^{g^{\prime}}=v^{g},~{}v^{{yx^{-1}}g^{\prime}}=v.$

In the first case $g^{\prime}\in H$ and $yx^{-1}g^{\prime}=hg$ for some $h\in
H$, so $yx^{-1}=hg(g^{\prime})^{-1}\in HgH$. In the second case
$g^{\prime}=h_{1}g$ and $yx^{-1}g^{\prime}=h_{2}$ for some $h_{1},h_{2}\in H$,
so $yx^{-1}=h_{2}(g^{\prime})^{-1}=h_{2}g^{-1}h_{1}^{-1}\in Hg^{-1}H$. In
either case we obtain that $\\{Hx,Hy\\}$ is an edge of the coset graph. ∎

The valence of the graph $\operatorname{Cos}(G,H,H\\{g,g^{-1}\\}H)$ is given
below.

######  Lemma 3.2.

([19, Lemma 2.4]) The valence of $\operatorname{Cos}(G,H,H\\{g,g^{-1}\\}H)$ is
equal to $|H|/|H\cap H^{g}|$ if $HgH=Hg^{-1}H$, or $2|H|/|H\cap H^{g}|$
otherwise.

The main result of this section is the following lemma.

######  Lemma 3.3.

Let $\Gamma$ be a $G$-edge-transitive graph in $\mathcal{F}(6)$ of order $2n$,
and let $H$ be a vertex stabiliser of $G$. Then $G$ contains an element $g$ of
order $n$ satisfying one of the following sets of conditions:

  1. (1)

$HgH=Hg^{-1}H$ and $|H|=6|H\cap H^{g}|=\frac{1}{2}|H\langle g\rangle\cap
HgH|$.

  2. (2)

$HgH\neq Hg^{-1}H$ and $|H|=3|H\cap H^{g}|=|H\langle g\rangle\cap HgH|$.

###### Proof.

There is a subgroup $C\leq G$ and a vertex $v\in V(\Gamma)$ such that $C$ is
cyclic and semiregular with two orbits, and the subgraph of $\Gamma$ induced
by the orbit $v^{C}$ is a cycle. Choose $c\in C$ such that $\\{v,v^{c}\\}$ is
an edge. Note that $c$ has order $n$.

It is easy to see that $\Gamma$ is not only $G$-edge- , but also $G$-vertex-
transitive. Thus Lemma 3.1 can be applied to $G,K:=G_{v}$ and $c$, and this
yields

$\Gamma\cong\Gamma^{\prime}:=\operatorname{Cos}(G,K,K\\{c,c^{-1}\\}K).$

Note that, as both $H$ and $K$ are vertex stabilisers, $H=K^{g^{\prime}}$ for
some $g^{\prime}\in G$.

Assume first that $KcK=Kc^{-1}K$. Using the formula for the valence of
$\Gamma^{\prime}$ given in Lemma 3.2, we find

$|K|=6|K\cap K^{c}|.$ (2)

Let $\varphi$ be the isomorphism between $\Gamma^{\prime}$ and $\Gamma$
defined in Lemma 3.1. Since $KcK=Kc^{-1}K$, it follows that $\Gamma^{\prime}$
is $G$-arc-transitive. This means that $\Gamma$ is also $G$-arc-transitive,
hence the stabiliser $K$ is transitive on $\Gamma(v)$. It follows that
$v^{C}\cap(v^{c})^{K}=\\{v^{c},v^{c^{-1}}\\}$. Applying $\varphi^{-1}$, we get

$\\{Kc^{i}:0\leq i\leq n-1\\}\cap\\{Kck:k\in K\\}=\\{Kc,Kc^{-1}\\}.$

This shows that $K\langle c\rangle\cap KcK=Kc\cup Kc^{-1}$ holds in $G$, and
so

$|K\langle c\rangle\cap KcK|=2|K|.$ (3)

Using also that $H=K^{g^{\prime}}$, (2) and (3) show that choosing $g$ to be
$c^{g^{\prime}}$, part (1) of the lemma holds.

Now assume that $KcK\neq Kc^{-1}K$. Using again Lemma 3.2, we find

$|K|=3|K\cap K^{c}|.$ (4)

In this case $\Gamma^{\prime}$ is not arc-transitive. Thus neither is
$\Gamma$, and $\Gamma(v)$ splits into two $K$-orbits of the same size. We
claim that $v^{c}$ and $v^{c^{-1}}$ belong to different $K$-orbits. For
otherwise, $v^{c^{-1}}=v^{cg^{\prime\prime}}$ for some $g^{\prime\prime}\in
K$, and this would imply that the automorphism $cg^{\prime\prime}$ inverts the
arc $(v^{c^{-1}},v)$, contradicting that $\Gamma$ is not arc-transitive. Then
$v^{C}\cap(v^{c})^{K}=\\{v^{c}\\}$, and this yields $K\langle c\rangle\cap
KcK=Kc$, and so

$|K\langle c\rangle\cap KcK|=|K|.$ (5)

Then (4) and (5) show that part (2) of the lemma holds for $g=c^{g^{\prime}}$.
∎

##  4\. Blocks

Throughout this section we keep the following notation:

> $\Gamma\in\mathcal{F}(d)$ is a $G$-edge-transitive graph of order $2n$ for
> some $d\geq 6$.
>  $C\leq G$ is a cyclic semiregular subgroup with two orbits and one of the
> subgraphs induced by these orbits is a cycle.
>  $B$ is a non-trivial block for $G$ and $\mathcal{B}$ is the block system
> induced by $B$.

We say that $B$ is cyclic when it is contained in one of the $C$-orbits, and
non-cyclic otherwise.

Recall that, $C_{\\{B\\}}$ is the set-wise stabiliser of $B$ in $C$, and
$\mathcal{B}$ is said to be normal when there is a normal subgroup $N$ of $G$
such that $\mathcal{B}$ consists of the $N$-orbits.

######  Lemma 4.1.

Suppose that $B$ is cyclic such that $|B|<n/2$. Then the kernel of the action
of $G$ on $\mathcal{B}$ is equal to $C_{\\{B\\}}$. Furthermore, $\Gamma$ is a
normal cover $\Gamma/\mathcal{B}$ and $\Gamma/\mathcal{B}\in\mathcal{F}(d)$.

###### Proof.

Denote by $V_{i}$ the $C$-orbits, $i=1,2$, and by $K$ the kernel of the action
of $G$ on $\mathcal{B}$. It is clear that any block in $\mathcal{B}$ is
contained in either $V_{1}$ or $V_{2}$. Consider the blocks contained in
$V_{1}$. These form a block system for $C$, and as $C$ is regular on $V_{1}$,
it follows that these blocks are the $C_{\\{B_{1}\\}}$-orbits, where $B_{1}$
is any block contained in $V_{1}$. The group $C_{\\{B_{1}\\}}$ is regular on
$B_{1}$, hence $|C_{\\{B_{1}\\}}|=|B_{1}|=|B|$, from which it follows that
$C_{\\{B\\}}=C_{\\{B_{1}\\}}$. The same applies to $V_{2}$, and we conclude
that $\mathcal{B}$ consists of the $C_{\\{B\\}}$-orbits. Thus $K\geq
C_{\\{B\\}}$, in particular, $\mathcal{B}$ is normal.

It can be assumed w.l.o.g. that the subgraph of $\Gamma$ induced by $V_{1}$ is
a cycle. Now, fix an edge $\\{u,v\\}$ such that $u,v\in V_{1}$. Since
$|B|<n/2$, it follows that $\Gamma(u)\cap B^{\prime}=\\{v\\}$, where
$B^{\prime}$ is the block containing $v$. By Lemma 2.1(1), $\Gamma$ is a
normal cover of $\Gamma/\mathcal{B}$. Then part (2) of the same lemma shows
that $K$ is regular on every block, and so we have $K=C_{\\{B\\}}$.

In order to see that $\Gamma/\mathcal{B}$ belongs to $\mathcal{F}(d)$, one
only needs to observe that $\Gamma/\mathcal{B}$ has valence $d$,
$C/C_{\\{B\\}}$ is semiregular with two orbits, the induced cycle of $\Gamma$
on $V_{1}$ projects to an induced cycle of $\Gamma/\mathcal{B}$, and $V_{1}$
projects to a $C/C_{\\{B\\}}$-orbit. ∎

######  Lemma 4.2.

Suppose that $B$ is non-cyclic.

  1. (1)

$B$ is a union of two $C_{\\{B\\}}$-orbits. The group $C$ acts transitively on
$\mathcal{B}$ with kernel equal to $C_{\\{B\\}}$.

  2. (2)

If $|B|>2$ and $B$ is minimal, then $\mathcal{B}$ is normal.

###### Proof.

(1): Since there are two $C$-orbits on $V(\Gamma)$ of the same size and $B$
has a point in common with both, it follows that $B$ splits into two
$C_{\\{B\\}}$-orbits, hence $|B|=2|C_{\\{B\\}}|$.

Let $\bar{C}$ and $K$ be the image and the kernel, respectively, of the action
of $C$ on $\mathcal{B}$. It is clear that $C$ acts transitively on
$\mathcal{B}$. This shows that $\bar{C}$ is regular, hence $C_{\\{B\\}}\leq
K$, and we can write

$|C|/|K|=|\bar{C}|=|\mathcal{B}|=2n/|B|=|C|/|C_{\\{B\\}}|.$

This shows that $|K|=|C_{\\{B\\}}|$ also holds, and so $K=C_{\\{B\\}}$.

(2): For a subgroup $X\leq G_{\\{B\\}}$, denote by $X^{*}$ the image of the
action of $X$ on $B$. Let $M$ be the kernel of the action of $G$ on
$\mathcal{B}$. By the minimality of $B$, $(G_{\\{B\\}})^{*}$ is primitive. On
the other hand, as $1<(C_{\\{B\\}})^{*}\leq M^{*}\lhd(G_{\\{B\\}})^{*}$,
$M^{*}$ is transitive on $B$. This shows that $\mathcal{B}$ is normal. ∎

Form now on we focus on the case when $n$ is odd.

######  Lemma 4.3.

Suppose that $n$ is odd and $B$ is minimal non-cyclic block for $G$. Then
$|B|>2$.

###### Proof.

Assume on the contrary that $|B|=2$. Write $B=\\{u,v\\}$. Then $u^{C}\neq
v^{C}$, and we may assume that the subgraph of $\Gamma$ induced by $u^{C}$ is
a cycle. Let $c\in C$ such that $\\{u,u^{c}\\}$ is an edge. Clearly, $c$ has
order $n$.

There is a unique number $1\leq k\leq(n-1)/2$ such that $v^{c^{k}}$ and
$v^{c^{-k}}$ are the neighbours of $v$. Define the subset $S\subseteq C$ as

$S=\\{x\in C:\\{u,v^{x}\\}\in E(\Gamma)\\}.$

It is clear that $|S|=d-2$. Also, $1_{C}\notin S$, where $1_{C}$ is identity
element of $C$. For otherwise, $\\{u,v\\}$ is an edge, but as it is also a
block, $G_{u}=G_{v}$, and this contradicts that $\Gamma$ is edge-transitive.

We say that two blocks in $\mathcal{B}$ are adjacent when these are adjacent
as vertices of $\Gamma/\mathcal{B}$. It can be easily seen that any two
subgraphs of $\Gamma$ induced by the union of two adjacent blocks are
isomorphic to the same graph, say $\Delta$. We claim that

$\Delta\cong K_{2}~{}\text{or}~{}2K_{2}.$

Assume for the moment that there exists some $s\in S$ such that
$s\notin\\{c,c^{-1},c^{k},c^{-k}\\}$. Then the subgraph induced by
$\\{u,v,u^{s},v^{s}\\}\cong 2K_{2}$ or $K_{2}$ depending whether $s^{-1}\in S$
or not, and the claim follows. Now, as $|S|=d-2\geq 4$, we are left with the
case when $k\neq 1$ and $S=\\{c,c^{-1},c^{k},c^{-k}\\}$. In this case the
subgraph induced by $\\{u,u^{c},v,v^{c}\\}$ is the $3$-path
$(v,u^{c},u,v^{c})$. Since $\Gamma$ is $G$-edge-transitive, there is some
$g\in G$ mapping $\\{u,u^{c}\\}$ to $\\{u,v^{c}\\}$. This implies that $g$
maps the $3$-path to itself, hence it induces an automorphism of it. This is
clearly impossible, and so the claim is proved.

Moreover, the argument above also shows that we have the following options:

$(k=1~{}\text{and}~{}S=S^{-1})~{}\text{or}~{}(k\neq 1~{}\text{and}~{}S\cap
S^{-1}=\emptyset).$ (6)

Now, define the permutation $t$ of $V(\Gamma)$ as

$t=(u\,v)(u^{c}\,v^{c})\cdots(u^{c^{n-1}}\,v^{c^{n-1}}).$

Observe that $t$ commutes with any element of $G$. In particular,
$\hat{C}:=\langle c,t\rangle$ is a regular cyclic group.

Define next the graph $\Gamma^{\prime}$ by

$V(\Gamma^{\prime})=V(\Gamma)~{}\text{and}~{}E(\Gamma^{\prime})=\\{\\{u,u^{c}\\}^{g}:g\in\langle
G,t\rangle\\}.$

Using that $\langle G,t\rangle=G\times\langle t\rangle$, we find

$E(\Gamma^{\prime})=\\{\\{u,u^{c}\\}^{g}:g\in
G\\}~{}\cup~{}\\{\\{v,v^{c}\\}^{g}:g\in G\\}.$

This can be used to find the neighbourhood $\Gamma^{\prime}(u)$. If $k=1$,
then $E(\Gamma^{\prime})=E(\Gamma)$, and so $\Gamma^{\prime}(u)=\Gamma(u)$. If
$k\neq 1$, then $E(\Gamma^{\prime})$ splits into two edge-orbits under $G$,
and $t$ swaps these edge-orbits. This yields that
$\Gamma^{\prime}(u)=\Gamma(u)\cup\Gamma(u^{t})^{t}=\Gamma(u)\cup\Gamma(v)^{t}$.
Now, according to (1),
$\Gamma^{\prime}\cong\operatorname{Cay}(\hat{C},S^{\prime}\cup
S^{\prime\prime})$, where

$S^{\prime}=\\{c,c^{-1},c^{k},c^{-k}\\}~{}\text{and}~{}S^{\prime\prime}=tS\cup
tS^{-1}.$

It follows from (6) that the valence of $\Gamma^{\prime}$ is $2+|S|$ if $k=1$,
and $4+2|S|$ if $k\neq 1$.

By definition, $\Gamma^{\prime}$ is edge-transitive. It is well-known that it
must be then arc-transitive as well, and therefore, $\Gamma^{\prime}$ belongs
to one of the families (a)-(d) in Theorem 2.2. We consider below all
possibilities case by case.

Family (a): $\Gamma^{\prime}$ is the complete graph. This contradicts that
$\Gamma^{\prime}$ has even valence and order.

Family (b): $\Gamma^{\prime}$ is normal. Then $S^{\prime}\cup
S^{\prime\prime}=c^{A}$ for some subgroup $A\leq\operatorname{Aut}(\hat{C})$.
This contradicts that $c$ has order $n$, while $ts$ has even order for each
$s\in S$.

Family (c): There exists a subgroup $1<D<\hat{C}$ such that $S^{\prime}\cup
S^{\prime\prime}$ is a union of $D$-cosets. If $|D|$ is odd, then $D\leq C$,
and so $S^{\prime}$ would be a union of $D$-cosets. This is clearly
impossible. Let $|D|$ be even. Then $t\in D$, implying
$tS^{\prime}=S^{\prime\prime}$ and $tS^{\prime\prime}=S^{\prime}$, and so
$|S^{\prime}|=|S^{\prime\prime}|$. This contradicts that $|S^{\prime}|=2$ and
$|S^{\prime\prime}|=|S|\geq 4$ if $k=1$, and $|S^{\prime}|=4$ and
$|S^{\prime\prime}|=2|S|\geq 8$ otherwise.

Family (d): There exist subgroups $1<D,E<\hat{C}$ such that $\hat{C}=D\times
E$, $|D|>3$, $\gcd(|D|,|E|)=1$, and $S^{\prime}\cup S^{\prime\prime}=D^{\\#}R$
for some subset $R\subseteq E^{\\#}$.

Let $|D|$ be odd. Then $D\leq C$. For every $i\in\\{1,-1,k,-k\\}$,
$|Dc^{i}\cap S^{\prime}|=|D|-1\geq 3$. It follows that $|D|=5$, $k\neq 1$, and
$S^{\prime}\subset Dc$. This shows that $c^{2}\in D$, whence $D=C$. On the
other hand, $D$ is a block for $\operatorname{Aut}(\Gamma^{\prime})$, and so
$C$ is a block for $G$. This is impossible.

Let $|D|$ be even. Then $t\in D$ and $D$ can be written as $D=\langle
t\rangle\times D^{\prime}$. Also, $R\subset E\leq C$. As $S^{\prime}\cup
S^{\prime\prime}=D^{\\#}R$ is inverse-closed, so is $R$, in particular, $|R|$
is even. Also, $S^{\prime}=D^{\\#}R\cap C=(D^{\\#}\cap
C)R=(D^{\prime})^{\\#}R$. Thus $|S^{\prime}|=(|D^{\prime}|-1)|R|$, and these
imply in turn that $|R|=2$, $|D^{\prime}|=3$, $|S^{\prime}|=4$, and
$|S^{\prime\prime}|=|D^{\\#}R|-|S^{\prime}|=6$. We have seen above that this
is impossible. ∎

Our last lemma is one of the crucial steps towards Theorem 1.1.

######  Lemma 4.4.

If $n>5$ is odd, then $G$ admits a non-trivial cyclic block.

###### Proof.

Since $n>5$, it follows from Corollary 2.7 that $G$ is imprimitive. Choose a
minimal non-trivial block $B$ for $G$, denote by $\mathcal{B}$ the block
system induced by $B$, and let $K$ denote the kernel of the action of $G$ on
$\mathcal{B}$.

We are done if $B$ is cyclic, hence we assume that $B$ is non-cyclic. By Lemma
4.3, $|B|>2$. As before, for a subgroup $X\leq G_{\\{B\\}}$, $X^{*}$ denotes
the image of the action of $X$ on $B$.

Apply Lemma 4.2 to $B$. This shows that $\mathcal{B}$ is normal and
$(C_{\\{B\\}})^{*}$ is a cyclic semiregular subgroup of $(G_{\\{B\\}})^{*}$
with two orbits. As $B$ is minimal, $(G_{\\{B\\}})^{*}$ is also primitive, and
therefore described by Theorem 2.3. The fact that $n$ is odd shows that
$(G_{\\{B\\}})^{*}$ is one of the groups in the families (a)-(f) in part (2)
of Theorem 2.3. Then $K^{*}\lhd(G_{\\{B\\}})^{*}$. By Corollary 2.4, $K^{*}$
is also primitive. We derive the lemma in three steps.

Step 1. $K$ acts faithfully on every block in $\mathcal{B}$.

Assume on the contrary that $K$ acts unfaithfully on some block in
$\mathcal{B}$. Using the connectedness of $\Gamma$, it is easy to show that
there are adjacent blocks $B^{\prime},B^{\prime\prime}\in\mathcal{B}$ so that
the kernel of the action of $K$ on $B^{\prime}$ is non-trivial on
$B^{\prime\prime}$. Denote by $N$ the latter kernel. Now, as $N\lhd K$ and $K$
is primitive on $B^{\prime\prime}$, $N$ is transitive on $B^{\prime\prime}$.
This implies that any vertex in $B^{\prime}$ is adjacent with any vertex in
$B^{\prime\prime}$. This contradicts that the subgraph of $\Gamma$ induced by
one of the $C$-orbits is a cycle.

Fix a vertex $u\in B$.

Step 2. For each block $B^{\prime}\in\mathcal{B}$ there exists a unique vertex
$u^{\prime}\in B^{\prime}$ such that $K_{u}=K_{u^{\prime}}$.

Define the binary relation $\sim$ on $\mathcal{B}$ by letting $B^{\prime}\sim
B^{\prime\prime}$ if and only if the action of $K$ on $B^{\prime}$ and
$B^{\prime\prime}$, respectively, are equivalent. It is easy to show that
$\sim$ is an equivalence relation.

Let $B^{\prime},B^{\prime\prime}\in\mathcal{B}$ such that $B^{\prime}\sim
B^{\prime\prime}$ and let $g\in G$. We claim that
$(B^{\prime})^{g}\sim(B^{\prime\prime})^{g}$. There is a bijective mapping
$\varphi$ from $B^{\prime}$ to $B^{\prime\prime}$ such that

$\forall v\in B^{\prime},~{}\forall k\in
K:~{}\varphi(v^{k})=(\varphi(v))^{k}.$

Now, pick arbitrary $w\in(B^{\prime})^{g}$ and $k\in K$. Let $\gamma_{1}$ be
the bijection from $B^{\prime}$ to $(B^{\prime})^{g}$ defined by
$\gamma_{1}(x)=x^{g}$ for each $x\in B^{\prime}$, and let $\gamma_{2}$ be the
bijection from $B^{\prime\prime}$ to $(B^{\prime\prime})^{g}$ defined by
$\gamma_{2}(x)=x^{g}$ for each $x\in B^{\prime\prime}$. We finish the proof of
the claim by showing $\psi(w^{k})=(\psi(w))^{k}$, where $\psi$ is the
bijection defined as the composition
$\psi=\gamma_{2}\circ\varphi\circ\gamma_{1}^{-1}$. Then $w=v^{g}$ for some
$v\in B^{\prime}$ and $gk=k^{\prime}g$ for some $k^{\prime}\in K$ because
$K\lhd G$. Thus

$\psi(w^{k})=\psi(v^{k^{\prime}g})=(\varphi(v^{k^{\prime}}))^{g}=\varphi(v)^{k^{\prime}g}=(\varphi(v)^{g})^{k}=(\psi(w))^{k}.$

Thus $\sim$ is $G$-invariant, and therefore, it is a $G$-congruence. Due to
[7, Exercise 1.5.4], the $\sim$-classes form a block system for $G$ with
respect to its action on $\mathcal{B}$. Denote by $m$ the number of
$\sim$-classes. As $|\mathcal{B}|$ is odd, so is $m$. On the other hand, by
Lemma 2.5(2), $K$ has at most two inequivalent faithful actions, and we
conclude that $m=1$.

Let $B^{\prime}\in\mathcal{B}^{\prime}$ be an arbitrary block. Since $K$ acts
equivalently on $B$ and $B^{\prime}$, it follows by [7, Lemma 1.6B] that there
is an element $u^{\prime}\in B^{\prime}$ such that $K_{u}=K_{u^{\prime}}$. By
Lemma 2.5(1), $K$ is $2$-transitive on $B^{\prime}$, unless $|B^{\prime}|=10$,
$K=A_{5}$ or $S_{5}$, and it has subdegrees $1,3$ and $6$. This shows that
$K_{x}\neq K_{u^{\prime}}$ for any vertex $x\in B^{\prime}$ such that $x\neq
u^{\prime}$. On the other hand, $K_{u}=K_{u^{\prime}}$ and this finishes off
the proof of Step 2.

Step 3. The vertices $u^{\prime}$ defined in Step 2 form a cyclic block.

Denote by $\hat{B}$ the set of all vertices $u^{\prime}$ defined in Step 2.
The cardinality $|\hat{B}|=|\mathcal{B}|$, and as $|\mathcal{B}|$ is odd, we
are done if we show that $\hat{B}$ is a block. Equivalently,
$\hat{B}^{g}=\hat{B}$ or $B\cap\hat{B}=\emptyset$ holds for each $g\in G$.

Suppose that $v^{g}\in\hat{B}$ for some $v\in\hat{B}$ and $g\in G$. We have to
show that $\hat{B}^{g}=\hat{B}$. In fact, it is enough to show that
$\hat{B}^{g}\subseteq\hat{B}$. Choose an arbitrary element $w\in\hat{B}$. Then
$K_{u}=K_{v}=K_{v^{g}}=K_{w}$. Using also that $K\lhd G$, we can write

$K_{w^{g}}=K\cap G_{w^{g}}=K\cap(G_{w})^{g}=(K\cap G_{w})^{g}=(K_{w})^{g}.$

The same argument shows that $K_{v^{g}}=(K_{v})^{g}$, and thus
$K_{u}=K_{v^{g}}=(K_{v})^{g}=(K_{w})^{g}=K_{w^{g}}$. By the definition of the
set $\hat{B}$, $w^{g}\in\hat{B}$, and $\hat{B}^{g}\subseteq\hat{B}$ follows. ∎

##  5\. Proof of Theorem 1.1

Assume on the contrary that there is an edge-transitive graph
$\Gamma\in\mathcal{F}(d)$ of order $2n$ such that $d\geq 6$, $n$ is odd and
$n>5$. Choose $n$ to be the smallest possible, i.e., whenever
$\Gamma^{\prime}\in\mathcal{F}(d^{\prime})$ is edge-transitive of order
$2n^{\prime}$ such that $d^{\prime}\geq 6$, $n^{\prime}<n$ and $n^{\prime}$ is
odd, then $\Gamma^{\prime}$ is isomorphic to the complement of the Petersen
graph. In what follows, we denote the latter graph by
$\overline{\mathsf{Pet}}$.

For the sake of simplicity, write $G$ for $\operatorname{Aut}(\Gamma)$. Let
$C\leq G$ be a cyclic semiregular subgroup with two orbits such that one of
the subgraphs induced by these orbits is a cycle.

It follows from Corollary 2.7 that $G$ is imprimitive. Choose a minimal non-
trivial block $B$ for $G$, denote by $\mathcal{B}$ be the block system induced
by $B$, and let $K$ be the kernel of the action of $G$ on $\mathcal{B}$.

Due to Lemma 4.4 we may assume that $B$ is cyclic, i.e., any block in
$\mathcal{B}$ is contained in one of the two $C$-orbits.

As $n$ is odd, $|B|<n/2$. By Lemma 4.1, $C_{\\{B\\}}\lhd G$. Let $p$ be a
prime divisor of $|B|$, and let $P\leq C_{\\{B\\}}$ be the subgroup of order
$p$. The group $P$ is characteristic in $C_{\\{B\\}}$ and as $C_{\\{B\\}}\lhd
G$, it follows that $P\lhd G$. By Lemma 4.1, $\Gamma$ is a normal cover of
$\Gamma/P$ and $\Gamma/P\in\mathcal{F}(d)$. Due to Proposition 2.1(2),
$\Gamma/P$ is also $G/P$-edge-transitive. The order of $\Gamma/P$ is $2n/p$,
hence the minimality of $n$ yields

$n=5p,~{}\Gamma/P\cong\overline{\mathsf{Pet}},~{}\text{and}~{}G/P\cong
A_{5}~{}\text{or}~{}S_{5}.$

The last condition follows from the fact that $G/P$ acts transitively on the
edges of $\overline{\mathsf{Pet}}$.

If $p=3$ or $5$, then $\Gamma$ has order $30$ or $50$ and its valence is $6$.
It follows from [10, Table 1] that no graph in $\mathcal{F}(6)$ of order $30$
or $50$ is edge-transitive. Thus $p>5$, and the Zassenhaus theorem (see [11,
Chapter 1, Theorem 18.1]) shows that there exists a subgroup $L<G$ such that
$G=P\rtimes L$, $P\cong\mathbb{Z}_{p}$ and $L\cong G/P\cong A_{5}$ or $S_{5}$.

Fix $z$ to be a generator of $P$ and let $L$ be identified with $A_{5}$ or
$S_{5}$. Note that every element of $G$ can be expressed as a product

$z^{i}\lambda,~{}\text{where}~{}0\leq i\leq p-1~{}\text{and}~{}\lambda\in L.$

Let $N=C_{L}(P)$. Then $N\lhd L$ and $L/N$ is isomorphic to a subgroup of
$\operatorname{Aut}(P)$, in particular, it is a cyclic group. This implies
that either $N=L$, or $L=S_{5}$ and $N=A_{5}$. Consequently,

$G=\begin{cases}P\times L&\mbox{if}~{}N=L,\\\ P\rtimes
L&\mbox{if}~{}L=S_{5}~{}\text{and}~{}N=A_{5}.\end{cases}$ (7)

Furthermore, in the second case the action of $L$ on $P$ by conjugation is
defined by

$(z^{i})^{\lambda}=\begin{cases}z^{i}&\mbox{if}~{}\lambda~{}\text{is even},\\\
z^{-i}&\mbox{if}~{}\lambda~{}\text{is odd},\end{cases}$ (8)

where $0\leq i\leq p-1$ and $\lambda\in L=S_{5}$.

Let $H=N_{L}(\langle(1,2,3)\rangle)$. Then

$H=\begin{cases}\big{\langle}(1,2,3),(1,2)(4,5)\big{\rangle}\cong
S_{3}&\mbox{if}~{}L\cong A_{5},\\\
\big{\langle}(1,2,3),(1,2),(4,5)\big{\rangle}\cong
S_{3}\times\mathbb{Z}_{2}&\mbox{if}~{}L\cong S_{5}.\end{cases}$

We show next that $H$ is a vertex stabiliser in $G$. First, as $\Gamma$ is a
normal cover of $\overline{\mathsf{Pet}}$, the vertex stabilisers in $G$ are
isomorphic to the vertex stabilisers in $\bar{G}$, and therefore, they are
isomorphic to $H$. It follows immediately from (7) and (8) that all the
elements of order $3$ in $G$ are contained in $L$, and form a single conjugacy
class within $G$. In particular, there exists a vertex stabiliser in $G$
containing $(1,2,3)$, let this vertex stabiliser be denoted by $M$. Clearly,
$M\leq N_{G}(\langle(1,2,3)\rangle)$. Using (7) and (8), we obtain

$N_{G}(\langle(1,2,3)\rangle)=\begin{cases}P\times H&\mbox{if}~{}N=L,\\\
P\rtimes H&\mbox{if}~{}L=S_{5}~{}\text{and}~{}N=A_{5}.\end{cases}$

If $N_{G}(\langle(1,2,3)\rangle)=P\times H$, then it is clear that $H$ is its
only subgroup of $N_{G}(\langle(1,2,3)\rangle)$ isomorphic to $H$, so $M=H$,
i.e., $H$ is indeed a vertex stabiliser.

Let $N_{G}(\langle(1,2,3)\rangle)=P\rtimes H$ and suppose that $M\neq H$. Then
$z^{i}\lambda\in M$ for some $1\leq i\leq p-1$ and $\lambda\in H$. If
$\lambda$ is even, then $p$ divides the order of $z^{i}\lambda$, which is
impossible because $M\cong H$. Thus $\lambda$ must be odd. Using that
$|M|=|H|$, it is not hard to show that for any $\mu\in H$, there is an element
$z^{\prime}\in P$ such that $z^{\prime}\mu\in M$. It follows from this that
$(H\cap A_{5})\leq M$. If $i=2j$, then $(z^{i}\lambda)^{z^{j}}=\lambda$, and
we have

$M^{z^{j}}=\big{\langle}H\cap
A_{5},z^{i}\lambda\big{\rangle}^{z^{j}}=\big{\langle}(H\cap
A_{5})^{z_{j}},(z^{i}\lambda)^{z_{j}}\big{\rangle}=\big{\langle}H\cap
A_{5},\lambda\big{\rangle}=H.$

If $i=2j+1$, then one finds in the same way that $M^{\lambda
z^{(p-1)/2-j}}=H$. In either case we obtain that $H$ is a vertex stabiliser.

The desired contradiction will arise after applying Lemma 3.3 to $H$. Due to
this lemma, there is an element $g\in G$ of order $5p$ satisfying all
conditions in either part (1) or (2) of Lemma 3.3. W.l.o.g. we may write
$g=z\sigma$, where $\sigma$ is a $5$-cycle in $S_{5}$.

Case 1. $N=L$.

By (7), $G=P\times L$. Using that $z\lambda=\lambda z$ for every $\lambda\in
H$, it is easy to see that $g^{-1}\notin HgH$. Thus $HgH\neq Hg^{-1}H$, and so
we have $|H|=3|H\cap H^{g}|=|H\langle g\rangle\cap HgH|$. Since $H^{z}=H$, it
follows that $H^{g}=H^{\sigma}$. Also,

$H\langle
g\rangle=\bigcup_{i=0}^{p-1}z^{i}H\langle\sigma\rangle~{}\text{and}~{}HgH=zH\sigma
H.$

Thus the equalities $|H|=3|H\cap H^{g}|=|H\langle g\rangle\cap HgH|$ reduce to

$|H|=3|H\cap H^{\sigma}|=|H\langle\sigma\rangle\cap H\sigma H|.$

A computation with the computer package Magma [3] shows that no $5$-cycle
$\sigma$ satisfies these conditions.

Case 2. $L=S_{5}$ and $N=A_{5}$.

Let $H_{1}=H\cap A_{5}$. Note that $H=H_{1}\cup(4,5)H_{1}$. By (7),
$G=K\rtimes L$, and the action of $L$ on $P$ by conjugation is described in
(8).

Then $HgH=Hg^{-1}H$ if and only if $g^{-1}\in HgH$, and so
$g^{-1}=z^{-1}\sigma^{-1}=\lambda_{1}z\sigma\lambda_{2}$ for some
$\lambda_{1},\lambda_{2}\in H$. It can be seen that both $\lambda_{1}$ and
$\lambda_{2}$ must be odd. Then $\lambda_{i}=(4,5)\lambda_{i}^{\prime}$ for
some $\lambda_{i}^{\prime}\in H_{1}$, where $i=1,2$, and it holds
$\sigma^{-1}=(4,5)\lambda_{1}^{\prime}\sigma(4,5)\lambda_{2}^{\prime}$. A
computation with Magma [3] verifies that such $\lambda_{1}^{\prime}$ and
$\lambda_{2}^{\prime}$ exist for any $5$-cycle $\sigma$. Thus $HgH=Hg^{-1}H$,
and so we have $|H|=6|H\cap H^{g}|=\frac{1}{2}|H\langle g\rangle\cap HgH|$.
Then $H^{g}=H_{1}^{g}\cup((4,5)H_{1})^{g}=H_{1}^{\sigma}\cup
z^{-2}((4,5)H_{1})^{\sigma}$, hence the first equality reduces to

$|H|=6|H\cap H_{1}^{\sigma}|.$ (9)

In order to rewrite the second equality, observe first that

$H\langle
g\rangle=\bigcup_{i=0}^{p-1}Hz^{i}\langle\sigma\rangle=\bigcup_{i=0}^{p-1}\big{(}z^{i}H_{1}\langle\sigma\rangle\cup
z^{-i}(4,5)H_{1}\langle\sigma\rangle\big{)}=\bigcup_{i=0}^{p-1}z^{i}H\langle\sigma\rangle.$

On the other hand, $HgH=(H_{1}\cup(4,5)H_{1})z\sigma H=zH_{1}\sigma H\cup
z^{-1}(4,5)H_{1}\sigma H$. Thus $|H|=\frac{1}{2}|H\langle g\rangle\cap HgH|$
reduces to

$|H\langle\sigma\rangle\cap H_{1}\sigma
H|+|H\langle\sigma\rangle\cap(4,5)H_{1}\sigma H|=2|H|.$ (10)

A computation with Magma [3] shows that no $5$-cycle $\sigma$ satisfies both
(9) and (10). This completes the proof of Theorem 1.1.

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# Machine Learning Predictions of High-Curie-Temperature Materials

Joshua F. Belot  Brigham Young University, Provo, UT, USA, 84602  Valentin
Taufour  University of California, Davis, One Shields Avenue, Davis, CA, USA,
95616  Stefano Sanvito  School of Physics, AMBER and CRANN Institute, Trinity
College, Dublin 2, Ireland  Gus L. W. Hart<EMAIL_ADDRESS>Brigham Young
University, Provo, UT, USA, 84602

###### Abstract

Technologies that function at room temperature often require magnets with a
high Curie temperature, $T_{\mathrm{C}}$, and can be improved with better
materials. Discovering magnetic materials with a substantial $T_{\mathrm{C}}$
is challenging because of the large number of candidates and the cost of
fabricating and testing them. Using the two largest known data sets of
experimental Curie temperatures, we develop machine-learning models to make
rapid $T_{\mathrm{C}}$ predictions solely based on the chemical composition of
a material. We train a random forest model and a $k$-NN one and predict on an
initial dataset of over 2,500 materials and then validate the model on a new
dataset containing over 3,000 entries. The accuracy is compared for multiple
compounds’ representations (“descriptors”) and regression approaches. A random
forest model provides the most accurate predictions and is not improved by
dimensionality reduction or by using more complex descriptors based on atomic
properties. A random forest model trained on a combination of both datasets
shows that cobalt-rich and iron-rich materials have the highest Curie
temperatures for all binary and ternary compounds. An analysis of the model
reveals systematic error that causes the model to over-predict
low-$T_{\mathrm{C}}$ materials and under-predict high-$T_{\mathrm{C}}$
materials. For exhaustive searches to find new high-$T_{\mathrm{C}}$
materials, analysis of the learning rate suggests either that much more data
is needed or that more efficient descriptors are necessary.

††preprint: AIP/123-QED

Ferromagnetic materials are essential in modern technologies especially in
energy production and data storage. The threshold temperature where magnetism
disappears is called the Curie temperature, $T_{\mathrm{C}}$. Searches for
high-$T_{\mathrm{C}}$ materials typically look for magnets with a
$T_{\mathrm{C}}$ of at least 550–600 K, which is required for reliably running
an application at room temperature.Nelson and Sanvito (2019); Coey (2020)
High-$T_{\mathrm{C}}$ magnets are valuable but rare,Nelson and Sanvito (2019);
Curtarolo  _et al._ (2013); Merker  _et al._ (2022) in particular when other
properties are required on their electronic structure.Vedmedenko  _et al._
(2020) Although some empirical rules for the design of new magnets existCoey
(2010), rapid predictions of Curie temperatures could assist in identifying
candidate high-TC magnets in large-scale screening exercises.  Thousands of
known ferromagnetic materials exist Byland  _et al._ (2022), and while high-
throughput computation can help in identifying hundreds of thousands of
_potential_ magnets, experience suggests that only a fraction of them can
actually be realized.Curtarolo  _et al._ (2013); Kabiraj, Kumar, and Mahapatra
(2020); Graf, Felser, and Parkin (2011); Sanvito  _et al._ (2017a)

Measuring the Curie temperature of a compound is a relatively standard and
accurate procedure, but of course needs the material to be made first. In
contrast, in a computational design process, the $T_{\mathrm{C}}$ must be
predicted ahead of experiments, only using physical and chemical information.
This is a complex task prone to large errors. In fact, one needs to compute
the elementary magnetic excitations of a compound, most typically from density
functional theory (DFT), map these on a simple model, usually a Heisenberg-
type one, and then perform thermodynamic sampling with Monte Carlo
methods.Nolting, Vega, and Fauster (1995); Kübler (2006); Kübler, Fecher, and
Felser (2007); Halilov  _et al._ (1997); Körmann  _et al._ (2008); Rosengaard
and Johansson (1997); Ležaić, Mavropoulos, and Blügel (2007); Rusz  _et al._
(2006); Ruban  _et al._ (2007); Tanaka and Gohda (2020); Körmann  _et al._
(2009); Liechtenstein  _et al._ (1987); Pajda  _et al._ (2001); Turek, Rusz,
and Diviš (2005); Gong  _et al._ (2019); Takahashi, Ogura, and Akai (2007)
Then, the choice of DFT functional, the quality and appropriateness of the
mapping, and subtleties in the Monte Carlo algorithms, all contribute to a
large uncertainty on the predictions. Often this uncertainty is so severe that
blind predictions of $T_{\mathrm{C}}$ for unknown compounds are almost
impossible to make.

Machine-learning algorithms capture complex relationships in data that may be
difficult to recognize or understand.Song  _et al._ (2020) Even though
machine-learning models often have low interpretability, they can provide
cheap Curie temperature predictions. Rapid predictions can be a first step in
screening computer-suggested materials, so that experimental efforts can be
more selective.

Algorithms such as ridge regression, kernel ridge regression, neural networks,
and random forest have been trained on experimental data and used to predict
Curie temperatures.Sanvito  _et al._ (2017b); Nelson and Sanvito (2019);
Nguyen  _et al._ (2019); Zhai, Chen, and Lu (2018); Long  _et al._ (2021);
Kabiraj, Kumar, and Mahapatra (2020); Dam  _et al._ (2018); Hu  _et al._
(2020); Lu  _et al._ (2022). Most of these studies were limited to specific
structural or chemical families, and used a relatively small dataset ($<1800$)
for training which limited the ability to generalize the results to other less
known families. In this work, a random-forest model is trained on an initial
dataset of the chemical compositions of over 2,500 ferromagnetic materials
Nelson and Sanvito (2019). We then used a larger dataset of $\sim 3100$ Byland
_et al._ (2022) for a “blind test”, which shows our model to have good
generalization. Then we built a model using the combined dataset. This is, by
far, the largest dataset to be used in this kind of study.

[capbesideposition=right,center,capbesidewidth=6cm]figure[]

Figure 1: (a) Experimental data (DS1) projected from the 85-dimensional
feature space to two dimensions in a t-SNEPedregosa  _et al._ (2011) plot.
Colors were assigned based on the majority element in each compound. (b) DS1
represented as a two-dimensional t-SNE plot. Colors were assigned based on the
Curie temperature. Two compounds, Fe1Ni3 and Cr2Pt3, have anomalously high
Curie temperatures, relative to others in the same cluster.

[capbesideposition=right,center,capbesidewidth=6cm]figure[]

Figure 2: (a) Experimental data (DS2) projected from the 85-dimensional
feature space to two dimensions in a t-SNEPedregosa  _et al._ (2011) plot.
Colors were assigned based on the majority element in each compound. (b) DS2
represented as a two-dimensional t-SNE plot. Colors were assigned based on the
Curie temperature.

In general, the crystal structure affects the Curie temperature and there have
been attempts to incorporate structural information into different machine-
learning models with varying degrees of success. In some cases, Curie
temperature predictions that utilize known structural data are more accurate,
in particular when the structural diversity present in the dataset is
limited.Nguyen  _et al._ (2019); Long  _et al._ (2021); Balachandran (2019);
Balachandran, Xue, and Lookman (2016); Kabiraj, Kumar, and Mahapatra (2020);
Byland  _et al._ (2022); Chen  _et al._ (2022) However, the use of structural
information in a larger and more diverse set of data, may have an adverse
effect on the $T_{\mathrm{C}}$ predictions.Nelson and Sanvito (2019)
Furthermore, the lack of available structural information excludes a large
portion of experimental data which weakens the model’s predictive power. The
use of structural data in training a machine learning model also limits
predictions to materials with known structure. In this work, we avoid these
problems and exploit our large corpus of magnetic data in DS1 and DS2 by using
models and data that do not include structural information.

In reference [Nelson and Sanvito, 2019], a random forest model was used to
predict the Curie temperature based solely on chemical composition. In this
work, we first use the same data (DS1) and attempt to improve the predictions
by trying different machine-learning models and alternate descriptors. Using
the random forest model trained on DS1 to make predictions on a new set of
experimental data (DS2),Byland  _et al._ (2022) we find that the model
generalizes quite well but the prediction errors are 90 K (MAE) as compared to
69 K (MAE). Next, we combine DS1 and DS2 to build a new model and scan over
100s of thousands of potential new magnets. In the process, we find that the
errors between the experimental and predicted Curie temperatures reveal a
previously unnoticed systematic error.

Figure 3: $k$-nearest neighbor and random forest predictions. (Left)
Prediction with $k$-NN model with $k$ = 2. 44% of the $T_{\mathrm{C}}$
predictions were within 50 K of the actual values and 64% were within 100 K.
The mean absolute error was 109 K (A 2-NN model with randomly shuffled
$T_{\mathrm{C}}$ values gives a MAE of 270 K). (Right) Random forest
prediction. 62% of the $T_{\mathrm{C}}$ predictions were within 50 K of the
actual values and 78% were within 100 K. The mean absolute error was 69 K. (A
random forest model with randomly shuffled $T_{\mathrm{C}}$ values gives a MAE
of 240 K).  Figure 4: Random forest model trained on DS1 and validated with
DS2. 54% of the $T_{\mathrm{C}}$ predictions for DS2 were within 50 K of the
actual values and 70% were within 100 K. The mean absolute error was 90 K.

We cleaned both DS1 and DS2 according to the methods outlined in Ref. [Nelson
and Sanvito, 2019]. Both datasets had many duplicate compounds often with
different reported experimental Curie temperatures. The duplicates were
eliminated by selecting only the median Curie temperature in order to retain
an actual measured value of the $T_{\mathrm{C}}$. In DS1, we also added
entries for each of the non-magnetic elements found in the data and set the
Curie temperature to zero. After cleaning the raw data, DS1 contains the Curie
temperatures of 2,557 unique compounds and DS2 contains 3,194. There is an
overlap of 1,189 compounds between DS1 and DS2. Our feature vector has 85
features, each one describing a distinct element found in the data. Each
compound is characterized by placing the percentage that each element occupies
in the compound in the appropriate feature. For each machine-learning
prediction, a randomly selected third of the compounds in DS1 was used as the
test data and the other two thirds were used to train the models.

It is not possible to visualize the data in 85 dimensions. However, a
t-distributed stochastic neighbor embeddingPedregosa  _et al._ (2011) (t-SNE)
projection reduces the data to two dimensions and may reveal data clustering.
Figure 1(a) shows the t-SNE plot for DS1, where each point is colored by the
majority element in the compound. The red circles show areas where compounds
with the same majority element cluster together. Figure 1(b) shows the same
t-SNE plot but with the colors corresponding to the Curie temperatures of each
compound. Each point in these plots represents a magnetic compound in the
dataset. A comparison of the two plots reveal that most of the
high-$T_{\mathrm{C}}$ materials have a majority element of either cobalt or
iron. It also shows that there are occasional high-$T_{\mathrm{C}}$ spikes in
clusters where the $T_{\mathrm{C}}$ is typically very low (for example, FeNi3
and Cr2Pt3). Figure 2 shows similar plots for DS2.

In Ref. [Nelson and Sanvito, 2019], ridge regression, kernel ridge regression,
neural networks, and random-forest machine-learning methods were all tested
and the random-forest model made the most accurate predictions. We used DS1 in
a $k$-nearest neighbors model and compared its performance to the accuracy of
the random forest model. The $k$-NN algorithm was varied for 1 to 20 neighbors
to determine the optimal $k$. A $k$-NN model with 2 neighbors provided best
prediction accuracy but was not as good as the random forest (see Fig. 3).

To see how well the DS1 random forest model generalizes  on _unseen data_ , we
did a “blind” test on all the data in DS2 (see Fig. 4). 54% of the predicted
Curie temperatures were within 50 K and 70% within 100 K. The mean absolute
error was 90 K. This is a larger error than the CV score on DS1 data, however,
the model still makes relatively accurate predictions on around 2,000
previously unseen magnets. We also used k-fold cross validation to show how
the addition of DS2 data affects our predictive accuracy. 50 iterations of
3-fold cross validation shows that DS1 averages an MAE of 73 K and a standard
deviation of 3.2. A combination of DS1 and DS2 averages an MAE of 71 K and a
standard deviation of 2.3

Figure 5: Random forest prediction using features based on the composition and
20 of the features generated by the MAST-MLJacobs  _et al._ (2020) Python
library. 62% of the $T_{\mathrm{C}}$ predictions for DS1 were within 50 K of
the actual values and 78% were within 100 K The mean absolute error was 69 K.
57% of the $T_{\mathrm{C}}$ predictions for the MAST-ML data were within 50 K
of the actual values and 77% were within 100 K The mean absolute error was 70
K. Figure 6: Random forest prediction with a balanced dataset. The Curie
temperatures of the test data above 600 K were predicted with an accuracy of
37% of the data within 50 K and 49% within 100 K. The mean absolute error for
those points was 141 K. Figure 7: MAE vs the amount of training data in a
combined set of DS1 and DS2. For each iteration the test data was a new random
sample of 850 compounds while the training data was randomly sampled from the
remaining data.

While the random-forest results are encouraging, a few different strategies
were tried to improve the random-forest predictions. One of these strategies
is to reduce the number of dimensions in the DS1 design matrix. A random
forest was made for a range (5–85) of features using principal component
analysis (PCA). PCA is not helpful to our model because, although noisy, the
change in mean absolute error for each dataset shows that the accuracy
improves up to about 60 features and stops improving after that as already
observed in Ref. [Nelson and Sanvito, 2019].

Another improvement strategy is designing better features. The MAST-MLJacobs
_et al._ (2020) Python library can generate descriptors based on about one
hundred atomic properties (such as ionic radius, electronegativity, etc.).
Using MAST-ML, we generated 428 features for the compounds in DS1 and selected
the top 20 most meaningful ones identified through MAST-ML’s
`EnsembleModelFeatureSelector` method which uses a random forest model to rank
feature importance. Figure 5 shows these features used in a random forest and
the comparison between these predictions and our random forest model
predictions in Fig 3. The MAST-ML features yield practically the same accuracy
as with the original features. This is somewhat surprising because the MAST-ML
descriptors incorporate many atomic properties, not merely the composition.
However, reference [Murdock  _et al._ , 2020] shows that the domain knowledge
in the generated features is not expected to improve prediction accuracy over
simple fractional features when using large amounts of data. Our results
suggest that DS1 is large enough to make an accurate model using our original
fractional features. While the size of ML model using the MAST-ML descriptors
is significantly smaller than the size of the model using the composition-only
descriptors, both models have the same accuracy and both are efficient enough
to be used in the large searches below. Our final models used the composition-
only descriptors.

[capbesideposition=right,center,capbesidewidth=4cm]figure[]

Figure 8: Eight plots showing the Curie temperatures for eight different
binary compounds. Horizontal rows share the same scale on the y axis. The
black lines show the random forest predicted Curie temperatures across the
entire composition range (step size: +1 at - %). The red dots are experimental
data.

Figure 9: Prediction error of the random forest model (experimental minus
_prediction_). (Left) Error versus _experimental_ $T_{\mathrm{C}}$ values.
(Right) Error versus _predicted_ $T_{\mathrm{C}}$ values. The black line is
the line of best fit. The errors are correlated with the experimental
$T_{\mathrm{C}}$ but not with the predicted $T_{\mathrm{C}}$.

Getting more balanced training data may be another way to improve the model.
An analysis of DS1 shows a significant imbalance, 38% of the magnetic
materials have a $T_{\mathrm{C}}$ $0-100$ K, while only 15% have a
$T_{\mathrm{C}}$ $600-1400$ K. This could lead the model to reduce the
prediction error at low temperatures at the expense of making larger errors at
high temperatures. This is counter to the purpose of the model, which is that
to discover candidate high-$T_{\mathrm{C}}$ magnets. Thus, we downsampled the
low-$T_{\mathrm{C}}$ compounds to more evenly distribute DS1. The resulting
model is shown in Fig. 6. Even though there is a high density of
low-$T_{\mathrm{C}}$ compounds, apparently they are necessary for accurate
predictions of higher-$T_{\mathrm{C}}$ compounds. The balanced data
predictions for high $T_{\mathrm{C}}$ materials are not as good as with the
origin model, with an accuracy of 10% less within 50 K, 14% less within 100 K,
and the mean absolute error 14 K higher.

Let us make one final comment before using the ML model to search for new
high-$T_{\mathrm{C}}$. With this unprecedented large database of magnetic
compounds, a large range of training set sizes are possible so that the
learning rate of the model can be estimated. Figure 7 shows the MAE on a test
set of 850 compounds (approximately 1/3 the size of DS1, similar to the
predictions in Fig. 3) while the training set is increased, using the
remaining data. The process was repeated 100 times and averaged. A linear fit
reveals a slope of $\approx-0.22$. This learning rate is not particularly
fast, but a better learning rate is probably not possible with composition-
only descriptors. To reduce the MAE to 50 K in the current model class would
require a near tripling of the data.

The purpose of the ML model is to enable identification of candidate
high-$T_{\mathrm{C}}$ materials. Our first attempt was to sweep over
compositions of binary materials, but this revealed a systematic error. The
error persists independent of the training data (DS1 or DS2), the particular
ML method (random forest, $k$-NN), or the descriptors. Figure 8 compares the
predictions with the experimental $T_{\mathrm{C}}$ of eight different sets of
binary systems. The top four red subplots analyze four different
high-$T_{\mathrm{C}}$ materials and the four lower blue subplots analyze four
different low-$T_{\mathrm{C}}$ materials. Each subplot shows the actual
$T_{\mathrm{C}}$ value (red points) compared to the ML prediction (black
line). A clear pattern emerges: between experimental data points, the ML
model, instead of interpolating smoothly, drifts towards the average
$T_{\mathrm{C}}$ of the training data (293 K). In the case of
high-$T_{\mathrm{C}}$ materials, deep “valleys” occur in the absence of nearby
experimental data. And for low-$T_{\mathrm{C}}$ materials, the model tends to
overestimate between training points. In a pattern reminiscent of Gaussian
processes, the model tends to “regress toward the mean”.

This systematic drift toward the mean can be seen across all the test data in
Fig. 9(left), which shows the difference between the predicted and the
experimental $T_{\mathrm{C}}$ for the test data as a function of the
experimental $T_{\mathrm{C}}$ (the random forest model using DS1). Intuition
suggests that this systematic error could be mitigated simply by shifting and
re-scaling according to the black line in the figure. However, the Fig.
9(right) demonstrates that this does not work. The errors and the _predicted_
$T_{\mathrm{C}}$’s are not correlated. The reader should remember that
correlation is not transitive in general. Although the errors are correlated
with the experimental $T_{\mathrm{C}}$’s, and the experimental and predicted
$T_{\mathrm{C}}$’s are correlated, the errors and the _predicted_
$T_{\mathrm{C}}$’s are not necessarily correlated.

Recognizing the systematic error in predictions and being more interested in
the accuracy of high-$T_{\mathrm{C}}$ predictions, we trained a random forest
model on a combined set of DS1 and DS2 but only included the 967 compounds
with $T_{\mathrm{C}}>600$ K thus boosting the mean of the training data from
293 K to 825 K and mitigating the regression to the mean. Obviously this new
model cannot be used to predict low-$T_{\mathrm{C}}$ materials but as we are
primarily concerned with high-$T_{\mathrm{C}}$ materials, this new model with
a higher average protects the predictions from straying too far in data-
deficient areas.

Figure 10: Heat map showing the maximum predicted $T_{\mathrm{C}}$ of each
combination of cobalt, iron, and $X_{\mathrm{1}}$ where $X_{\mathrm{1}}$ can
be any element in materials with a $T_{\mathrm{C}}$ over 600 K found in DS1
and DS2 (excluding cobalt and iron). $T_{\mathrm{C}}$ is measured in Kelvins.
Plot created using Python-Ternary.Marc  _et al._ (2019) Figure 11: Heat map
showing the maximum predicted $T_{\mathrm{C}}$ of each combination of iron,
$X_{\mathrm{1}}$, and $X_{\mathrm{2}}$ where $X_{\mathrm{1}}$ and
$X_{\mathrm{2}}$ can be any element in materials with a $T_{\mathrm{C}}$ over
600 K found in DS1 and DS2 excluding cobalt and iron. Each compound is
composed of a minimum of 80% iron. $T_{\mathrm{C}}$ is measured in Kelvins.
Plot created using Python-Ternary.Marc  _et al._ (2019) (This plot should not
be directly compared with Fig. 10 because of the difference in color scales.)

This new high-$T_{\mathrm{C}}$-only model was used to search composition space
for all binary and ternary candidate compounds with exceptionally
high-$T_{\mathrm{C}}$. The high-$T_{\mathrm{C}}$-only model used only 71
features, restricted to elements that appear in compounds with
$T_{\mathrm{C}}>600$ K. Figs 1 and 2 show that most of the
highest-$T_{\mathrm{C}}$ compounds contain either cobalt, iron, or both. Using
these elements, we generated every possible ternary combination containing
iron, cobalt, and one of the other 69 elements ($X_{\mathrm{1}}$) in 1%
increments. The random forest model can rapidly predict $T_{\mathrm{C}}$ for
this entire set. Figure 10 displays the maximum predicted $T_{\mathrm{C}}$ of
each combination of cobalt, iron, and $X_{\mathrm{1}}$. While cobalt clearly
dominates the high-$T_{\mathrm{C}}$ compounds, materials with a high iron
concentration also are predicted to have a high $T_{\mathrm{C}}$ and are worth
exploring because iron is much cheaper than cobaltNational Minerals
Information Center (2023). Figure 11 shows similar predictions but for iron-
rich ternary and binary candidates, which are inexpensive. These predictions
suggest that, at least in general, for maximizing $T_{\mathrm{C}}$, it is
difficult to beat cobalt-rich materials, and that for minimizing materials
cost for a relatively high $T_{\mathrm{C}}$, iron-rich materials are difficult
to beat. Other searches excluding cobalt and iron were also performed but no
competitive materials emerged.

Unsuspected high-$T_{\mathrm{C}}$ materials sometimes appear. For example, see
FeNi3 and Cr2Pt3 in Fig. 1—these turn up in clusters with very low Curie
temperatures. But atomic structure, not merely composition, evidently plays an
essential role in such materials. Devoid of explicit structure in the training
data or the model, our predictions cannot be expected to include such
candidates. Our models can learn a major portion of the relationship between
chemical composition and Curie temperature and thus generally predict quite
accurately. But it seems that finding exceptional cases will require data that
includes structural information and a model that incorporates explicit
relationships between magnetism and structure. As discussed at the beginning,
the reliability of such models remains challenging, and corresponding data is
limited, so computational discovery of new high-$T_{\mathrm{C}}$ remains an
outstanding challenge.

## Supplementary Material

The supplementary material includes two error plots from the 2-NN predictions
shown in the left plot of Fig. 3. It also includes a plot of the number of
features reduced by PCA vs. the MAE of the predictions made by a random forest
model using that number of features.

## Acknowledgements

We would like to thank Mark Transtrum for useful discussions about the
systematic error in Fig. 9. J.F.B. was generously supported by the College of
Physical and Mathematical Sciences at Brigham Young University. J.F.B also
acknowledges the mentorship provided by Ben Afflerbach and the Informatics
Skunkworks Program. G.L.W.H. was supported by the U.S. National Science
Foundation under Award #DMR-1817321. V.T. acknowledges funding from the
Critical Materials Institute, an Energy Innovation Hub funded by the U.S.
Department of Energy Advanced Manufacturing Office. S.S. Acknowledges
financial support from the Irish Research Council (IRCLA/2019/127).

## Data Availability

The data and code to reproduce the findings of this study are available here:
https://github.com/msg-byu/ML-for-CurieTemp-Predictions.

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