A general formula for walk determinants of rooted products with applications to DGS-graph constructions

TL;DR

This paper derives a general formula for walk determinants of rooted product graphs and explores their applications in constructing new families of graphs determined by their spectra, advancing graph controllability and spectral graph theory.

Contribution

It provides an explicit formula for walk matrix determinants of rooted product graphs and introduces the concept of -preservers to generate new DGS-graphs.

Findings

Explicit formula for walk matrix determinants of rooted product graphs.

Introduction of -preservers and their role in DGS-graph construction.

Many new infinite families of DGS-graphs derived from rooted products.

Abstract

For an $n$-vertex graph $G$, and a rooted graph $H^{(v)}$ with $v$ as the root, the rooted product graph $G\circ H^{(v)}$ is obtained from $G$ and $n$ copies of $H$ by identifying the root of the $i$th copy of $H$ with the $i$th vertex of $G$ for each $i$. As a refinement of the controllability criterion of $G\circ H^{(v)}$ obtained recently by Shan and Liu (2025), we obtain an explicit formula for the determinant of the walk matrix of $G\circ H^{(v)}$. Furthermore, for an important family of graphs $\mathcal{F}$ that are determined by their generalized spectrum (DGS), we introduce the concept of $\mathcal{F}$-preservers and provide a sufficient condition for a rooted graph to be an $\mathcal{F}$-preserver. A list of $\mathcal{F}$-preservers of small order is provided, which leads to many new infinite families of DGS-graphs using rooted products.