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

Abstract

Let $G$ be a simple undirected graph, $θ(G)$ be the circuit rank of $G$, $η_M(G)$ and $m_M(G,λ)$ be the nullity and the multiplicity of eigenvalue $λ$ 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 $m_M(G,λ)$ in terms of $θ(G)$ when $λ$ is an integer (even integer, respectively). We also show that when $α$ and $λ$ are rational numbers similar bounds can be found for $m_{A_α}(G,λ)$ where $A_α(G)$ is the generalized adjaceny matrix of $G$. Our bounds contain only $θ(G)$, not a multiple of it. Up to now only bounds of $m_A(G,λ)$ (and later $m_{A_α}(G,λ)$) have been found in terms of the circuit rank and all of them contains $2θ(G)$. There is only one exception in the case $λ=0$. Wong et al. (2022) showed that $η_A(G_c)\leq θ(G_c)+1$, 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 $η_A(G_c)\leq 1$ when every block of the cactus is an odd cycle. This also corresponds a special case of our bound.