7 problems

Let $G$ be a controllable graph, let $p$ be an odd prime factor of $\det W(G)$, and let $d_{n-1}$ and $d_n$ be the last two invariant factors of the walk matrix $W(G)$. Write $v_p$…

Let $H^{(v)}$ be an $\mathcal{F}$-preserver of order $m$, and let $G$ be any graph. Write $A(G)$ for the adjacency matrix of $G$, $W_A(G)$ for its adjacency walk matrix, and…

Let $H_{4t+1}$ be a graph of order $4t+1$, let $H_{4t+1}^{(\ell)}$ denote the rooted graph obtained by choosing root $\ell$, and let $G\circ H_{4t+1}^{(\ell)}$ be their rooted prod…

Mao–Wang's conjecture. For any positive integer $m\ge 2$,

Let $D_n$ ($n\ge 4$) be the Dynkin graph, let $W(D_n)$ be its walk matrix, and let $\hat{W}(D_n)$ be the $(n-1)\times(n-1)$ matrix obtained from $W(D_n)$ by removing the first row…

Let $G$ be a self-converse mixed graph of order $n$, with walk-matrix … Here $\mathcal{G}_n^{sc}$ denotes the self-converse mixed graphs of order $n$, and $G$ is DGS when it is det…

Rank-$n-2$ walk-matrix isomorphism conjecture. The two graphs in part (ii) are always isomorphic.