On the determinant of the walk matrix of the rooted product with a path

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.

• The paper presents a closed-form determinant formula for the walk matrix of rooted product graphs with paths, using a coprimality condition to ensure nondegeneracy.
• It employs spectral analysis with Chebyshev polynomials and Kronecker product representations to express the graph's eigenvalues in relation to those of its components.
• The results facilitate the construction of DGS graph families by detailing when the walk matrix is non-singular based on root-position constraints.

Determinant of the Walk Matrix of Rooted Products with Paths

Introduction and Context

The paper investigates the determinant of the walk matrix for graphs constructed as rooted products with paths, notably generalizing previous results from the case where the root is the first vertex of the path to arbitrary roots. This extension is significant in advancing spectral graph theory, with direct implications for understanding generalized spectral characterizations and controllability of composite graphs.

Let $G$ be an $n$-vertex simple graph with adjacency matrix $A(G)$ and walk matrix $W(G) = [e, Ae, \ldots, A^{n-1}e]$, where $e$ is the all-ones vector. The central focus is the graph $G\circ P_m^{(\ell)}$, the rooted product of $G$ and the path $P_m$, where each vertex of $G$ is identified with the $\ell$-th vertex of its copy of $P_m$. The determinant of the walk matrix of such constructed graphs, especially its closed form and its zero-pattern, is analyzed completely.

Main Theorem and Its Analysis

The main result is the exact evaluation of $\det W(G\circ P_m^{(\ell)})$:

$$\det W(G\circ P_m^{(\ell)}) = \begin{cases} \pm (\det A(G))^{\lfloor\frac{m}{2}\rfloor}(\det W(G))^m & \text{if } \gcd(\ell, m+1) = 1,\ 0 & \text{otherwise.} \end{cases}$$

This formula characterizes, for any $n$-vertex graph $G$, any $m \geq 2$, and any root position $1\leq\ell\leq (m+1)/2$, exactly when the determinant is nonzero and how it relates multiplicatively to the original graph $G$. The result resolves the impact of root position—specifically, the coprimality constraint $\gcd(\ell,m+1)=1$ is necessary and sufficient for non-degeneracy. When this condition fails, the walk matrix becomes singular.

Structural and Spectral Properties

The authors harness the algebraic structure of $G\circ P_m^{(\ell)}$ through the Kronecker product representation of its adjacency matrix and employ the Chebyshev polynomials to succinctly capture characteristic polynomials of both the path and the product graph. In particular,

$$A(G\circ P_m^{(\ell)})=A(P_m)\otimes I_n+D_\ell\otimes A(G),$$

where $D_\ell$ is an indicator for the root attachment. The spectral analysis reveals that the eigenvalues of the rooted product admit explicit expressions in terms of those of $G$, parameterized by the spectrum of $P_m$ and Chebyshev polynomials, leading to the determinant formula.

A crucial intermediary is the fact that $G\circ P_m^{(\ell)}$ has only simple eigenvalues if and only if $\gcd(\ell, m+1) = 1$ and $G$ has simple eigenvalues. This interplay underpins both the non-vanishing and the precise algebraic form of the determinant.

Methodological Advances

Through deft use of resultants and combinatorial identities for Chebyshev polynomials, the determinant computation is reduced to evaluating certain products over roots of polynomial systems. The paper develops a framework in which the determinant of the walk matrix of a highly composite graph can be recursively or inductively expressed in terms of determinants from its constituent graphs.

Figure 1: Explicit depiction of $C_4\circ P_3^{(1)}$ and $C_4\circ P_3^{(2)}$, illustrating the distinct graph topologies induced by different root positions.

Implications: Generalized Spectra and DGS Constructions

A notable application is the construction of large families of graphs determined by their generalized spectrum (DGS graphs). For any $G$ in the family $\mathcal{F}_n$ (defined by $\det A(G)=\pm 1$ and $\det W(G)=\pm 2^{\lfloor n/2\rfloor}$), and any sequence of admissible rooted products (with the coprimality constraint on each pair), all iterated rooted products remain DGS. This demonstrates that the DGS property is robust under iterated rooted products with paths at suitable roots and provides a generative method for constructing infinite DGS-graph families.

This is formalized as follows: for any sequence $\{(m_i, \ell_i)\}$ with $\gcd(\ell_i, m_i+1)=1$, the family

$$G\circ P_{m_1}^{(\ell_1)}, (G\circ P_{m_1}^{(\ell_1)})\circ P_{m_2}^{(\ell_2)}, ((G\circ P_{m_1}^{(\ell_1)})\circ P_{m_2}^{(\ell_2)})\circ P_{m_3}^{(\ell_3)},\ldots$$

consists exclusively of DGS graphs whenever $G\in \mathcal{F}_n$. This result extends prior work which handled only the case $\ell=1$.

Technical Subtleties: Vanishing, Simplicity, and Combinatorics

A striking claim is the dichotomy in the determinant vanishing: if $\gcd(\ell, m+1)>1$ or if $G$ has repeated eigenvalues, then $\det W(G\circ P_m^{(\ell)})=0$ automatically. This follows from the effect of root placement on the multiplicity and location of eigenvalues. The paper also identifies connections to Wronskian vertices and the properties of totally unimodular matrices in combinatorial matrix theory.

Moreover, the proof strategy blends spectral graph theory, properties of walk matrices, symmetric functions over eigenvalues, and determinant identities. This multi-pronged approach enables the reduction of an intricate combinatorial-analytic object to a manageable algebraic form.

Conclusion

The paper delivers a comprehensive analysis and closed formula for the determinant of the walk matrix of rooted products of graphs with paths at general root locations, unified via the coprimality condition on the root parameter. The results provide new algebraic and spectral tools for constructing and classifying DGS graphs, cementing the bridge between adjacency matrix invariants, walk matrices, and spectral characterization theory. The explicit formulas and reduction machinery introduced open avenues for future research into walk-based invariants and further generalizations to rooted products with other graphs or under more relaxed spectral constraints.