On the levels of rational regular orthogonal matrices for generalized cospectral graphs

TL;DR

This paper investigates the divisibility properties of rational orthogonal matrices associated with generalized cospectral graphs, providing improved bounds on their levels relative to the walk matrix's determinant over finite fields.

Contribution

It establishes a tighter upper bound on the level of rational regular orthogonal matrices for generalized cospectral graphs, refining previous results.

Findings

Proves that v_p(ell(Q)) is at most half of v_p(det W(G)).

Improves the bound from previous work by Qiu et al.

Enhances understanding of the algebraic structure of cospectral graphs.

Abstract

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 $v_p(\ell(Q))\le v_p (\det W(G))-1.$