Bounding the multiplicities of eigenvalues of graph matrices in terms of circuit rank using a new approach

TL;DR

This paper establishes new bounds on the eigenvalue multiplicities of various graph matrices based on the circuit rank, generalizing previous results and applying to a broader class of graphs and eigenvalues.

Contribution

It introduces bounds on eigenvalue multiplicities in terms of circuit rank that are tighter and more general than previous bounds, applicable to multiple graph matrices and eigenvalues.

Findings

Bounds depend only on circuit rank, not multiples of it.

Generalizes previous results to all even eigenvalues of various matrices.

Extends bounds to rational eigenvalues and generalized adjacency matrices.

Abstract

Let $G$ be a simple undirected graph, $\theta(G)$ be the circuit rank of $G$, $\eta_M(G)$ and $m_M(G,\lambda)$ be the nullity and the multiplicity of eigenvalue $\lambda$ of a graph matrix $M(G)$, respectively. In the case $M(G)$ is the adjacency matrix $A(G)$, (the Laplacian matrix $L(G)$, the signless Laplacian matrix $Q(G)$) we find bounds to $m_M(G,\lambda)$ in terms of $\theta(G)$ when $\lambda$ is an integer (even integer, respectively). We also show that when $\alpha$ and $\lambda$ are rational numbers similar bounds can be found for $m_{A_{\alpha}}(G,\lambda)$ where $A_{\alpha}(G)$ is the generalized adjaceny matrix of $G$. Our bounds contain only $\theta(G)$, not a multiple of it. Up to now only bounds of $m_A(G,\lambda)$ (and later $m_{A_\alpha}(G,\lambda)$) have been found in terms of the circuit rank and all of them contains $2\theta(G)$. There is only one exception in the case $\lambda=0$. Wong et al. (2022) showed that $\eta_A(G_c)\leq \theta(G_c)+1$, where $G_c$ is a connected cactus whose blocks are even cycles. Our result, in particular, generalizes and extends this result to the multiplicity of any even eigenvalue of A(G) of any even connected graph $G$, and of any even eigenvalue of $L(G)$ and $Q(G)$ of any connected graph $G$. They also showed that $\eta_A(G_c)\leq 1$ when every block of the cactus is an odd cycle. This also corresponds a special case of our bound.