For an $n$-vertex graph $G$ with adjacency matrix $A$, the walk matrix $W(G)$ of $G$ is the matrix $[e,Ae,\ldots,A^{n-1}e]$, where $e$ is the all-ones vector. Suppose that $W(G)$ is nonsingular and $p$ is an odd prime such that $W(G)$ has rank $n-1$ over the finite field $\mathbb{Z}/p\mathbb{Z}$. Let $H$ be a graph that is generalized cospectral with $G$, and $Q$ be the corresponding rational regular orthogonal matrix satisfying $Q^\mathsf{T} A(G) Q=A(H)$. We prove that
\begin{equation*}
v_p(\ell(Q))\le \frac{1}{2}v_p (\det W(G))
\end{equation*}
where $\ell(Q)$ is the minimum positive integer $k$ such that $kQ$ is an integral matrix, and $v_p(m)$ is the maximum nonnegative integer $s$ such that $p^s$ divides $m$. This significantly improves upon a recent result of Qiu et al. [Discrete Math. 346 (2023) 113177] stating that
$v_p(\ell(Q))\le v_p (\det W(G))-1.$
The smallest pair of cospectral cubic graphs with different chromatic indexes
通过对16阶三次图的穷举搜索，我们找到了一个唯一的具有不同色指数的共谱对。这个例子表明，正则图的色指数不是由它的谱来刻画的，这回答了最近在[O. Etesami，W. H. Haemers，关于由谱表征的NP-硬图性质，离散应用数学，285（2020）526-529]。我们证明了任何表示共谱对的两个邻接矩阵之间相似性的正交矩阵都不可能是有理数的。这意味着，不能使用原来的GM开关方法或其推广的基础上合理的正交矩阵的共谱对。
Switching methods of level 2 for the construction of cospectral graphs
Eindhoven University of Technology（埃因霍温理工大学）；Vrije Universiteit Brussels（布鲁塞尔自由大学）；Ghent University（根特大学）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
