"""simple docstring"""





from collections import defaultdict



def      lowerCamelCase_(__SCREAMING_SNAKE_CASE  )->    int:
							_SCREAMING_SNAKE_CASE		:						int			  =    1
							_SCREAMING_SNAKE_CASE		:						Any			  =    True
							for v in tree[start]:
														if v not in visited:
																					ret += dfs(__SCREAMING_SNAKE_CASE  )
							if ret % 2 == 0:
														cuts.append(__SCREAMING_SNAKE_CASE  )
							return ret



def      lowerCamelCase_()->    Dict:
							dfs(1  )


if __name__ == "__main__":
			lowerCAmelCase_ ,					lowerCAmelCase_     =	10, 9
			lowerCAmelCase_     =	defaultdict(list)
			lowerCAmelCase_     =	{}
			lowerCAmelCase_     =	[]
			lowerCAmelCase_     =	0
			lowerCAmelCase_     =	[(2, 1), (3, 1), (4, 3), (5, 2), (6, 1), (7, 2), (8, 6), (9, 8), (10, 8)]
			for u, v in edges:
						tree[u].append(v)
						tree[v].append(u)
			even_tree()
			print(len(cuts) - 1)



"""simple docstring"""





from typing import Any

import numpy as np



def      lowerCamelCase_(__SCREAMING_SNAKE_CASE  )->    bool:
							return np.array_equal(__SCREAMING_SNAKE_CASE     ,	matrix.conjugate().T  )



def      lowerCamelCase_(__SCREAMING_SNAKE_CASE     ,	__SCREAMING_SNAKE_CASE  )->    Any:
							_SCREAMING_SNAKE_CASE		:						Optional[int]			  =    v.conjugate().T
							_SCREAMING_SNAKE_CASE		:						Optional[int]			  =    v_star.dot(__SCREAMING_SNAKE_CASE  )
							assert isinstance(__SCREAMING_SNAKE_CASE     ,	np.ndarray  )
							return (v_star_dot.dot(__SCREAMING_SNAKE_CASE  )) / (v_star.dot(__SCREAMING_SNAKE_CASE  ))



def      lowerCamelCase_()->    None:
							_SCREAMING_SNAKE_CASE		:						Optional[Any]			  =    np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]]  )
							_SCREAMING_SNAKE_CASE		:						int			  =    np.array([[1], [2], [3]]  )
							assert is_hermitian(__SCREAMING_SNAKE_CASE  ), F"""{a} is not hermitian."""
							print(rayleigh_quotient(__SCREAMING_SNAKE_CASE     ,	__SCREAMING_SNAKE_CASE  )  )

							_SCREAMING_SNAKE_CASE		:						int			  =    np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]  )
							assert is_hermitian(__SCREAMING_SNAKE_CASE  ), F"""{a} is not hermitian."""
							assert rayleigh_quotient(__SCREAMING_SNAKE_CASE     ,	__SCREAMING_SNAKE_CASE  ) == float(3  )


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			tests()


