On the determinant of the walk matrix of the rooted product with a path
Zhidan Yan, Wei Wang
TL;DR
The paper extends the determinant formula for the walk matrix from the special case $\ell=1$ to general rooted-path attachments $G\circ P_m^{(\ell)}$, proving $\det W(G\circ P_m^{(\ell)})=\pm (\det A(G))^{\lfloor m/2\rfloor}(\det W(G))^{m}$ whenever $\gcd(\ell,m+1)=1$, and $0$ otherwise. The approach combines a spectral analysis of $G\circ P_m^{(\ell)}$ via $A(G\circ P_m^{(\ell)})=A(P_m)\otimes I_n+D_{\ell}\otimes A(G)$, the roots of Chebyshev-related polynomials, and resultant computations to evaluate critical constants $\Delta_1$ and $\Delta_2$. This yields a tool for constructing large families of graphs determined by their generalized spectrum (DGS) from smaller ones, by iterated rooted-path expansions while preserving determinant properties. The results have implications for spectral characterization and graph reconstruction, offering explicit determinations of walk-matrix determinants in a broad class of rooted products and enabling DGS-based graph constructions.
Abstract
For an $n$-vertex graph $G$, the walk matrix of $G$, denoted by $W(G)$, is the matrix $[e,A(G)e,\ldots,(A(G))^{n-1}e]$, where $A(G)$ is the adjacency matrix of $G$ and $e$ is the all-ones vector. For two integers $m$ and $\ell$ with $1\le \ell\le (m+1)/2$, let $G\circ P_m^{(\ell)}$ be the rooted product of $G$ and the path $P_m$ taking the $\ell$-th vertex of $P_m$ as the root, i.e., $G\circ P_m^{(\ell)}$ is a graph obtained from $G$ and $n$ copies of the path $P_m$ by identifying the $i$-th vertex of $G$ with the $\ell$-th vertex (the root vertex) of the $i$-th copy of $P_m$ for each $i$. We prove that, $\det W(G\circ P_m^{(\ell)})$ equals $\pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m$ if $\gcd(\ell,m+1)=1$, and equals 0 otherwise. This extends a recent result established in [Wang et al. Linear Multilinear Algebra 72 (2024): 828--840] which corresponds to the special case $\ell=1$. As a direct application, we prove that if $G$ satisfies $\det A(G)=\pm 1$ and $\det W(G)=\pm 2^{\lfloor n/2\rfloor}$, then for any sequence of integer pairs $(m_i,\ell_i)$ with $\gcd(\ell_i,m_i+1)=1$ for each $i$, all the graphs in the family \begin{equation*} G\circ P_{m_1}^{(\ell_1)}, (G\circ P_{m_1}^{(\ell_1)})\circ P_{m_2}^{(\ell_2)}, ((G\circ P_{m_1}^{(\ell_1)})\circ P_{m_2}^{(\ell_2)})\circ P_{m_3}^{(\ell_3)},\ldots \end{equation*} are determined by their generalized spectrum.