A general formula for walk determinants of rooted products with applications to DGS-graph constructions
Wei Wang, Jie Shen, Lihuan Mao
TL;DR
This work addresses the problem of determining the walk-determinant of rooted product graphs $G\circ H^{(v)}$ for general rooted graphs $H^{(v)}$. It introduces a general determinant formula for $\det W_M(G\circ H^{(v)})$ that factors through a resultant between $M(H)$-polynomials and a term depending on $G$ and $H$, together with a Kronecker-product-based eigenstructure analysis. The authors define $\mathcal{F}$-preservers and provide a three-condition sufficiency test (i)-(iii) ensuring rooted graphs preserve the $\mathcal{F}$-property under rooted products, enabling construction of infinite DGS-graph families. They also give concrete examples, computational listings of small-order preservers, and conjectures about unified determinant behavior, extending known path-based results to general rooted graphs and offering practical tools for building large DGS-graphs from small seeds.
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.