Bounding the multiplicities of eigenvalues of graph matrices in terms of circuit rank

Let G be a simple undirected graph, θ(G) be the circuit rank of G, ηM(G) be the nullity of a graph matrix $M(G)$, and mM(G,λ) be the multiplicity of eigenvalue λ of $M(G)$. In the case $M(G)$ is the adjacency matrix $A(G)$, (the Laplacian matrix $L(G)$, or the signless Laplacian matrix $Q(G)$) we find bounds to mM(G,λ) in terms of θ(G) when λ is an integer (even integer, respectively). We also demonstrate that when α and λ are rational numbers, similar bounds can be obtained for m_Aα(G,λ), where Aα(G) is the generalized adjacency matrix of G. Distinctively, our bounds involve only θ(G), not a multiple of it. Previous bounds for mA(G,λ) (and later mA_α(G,λ)) in terms of the circuit rank have all included 2θ(G) with the sole exception of the case λ=0. Wong et al. (2022) showed that ηA(Gc)≤ θ(Gc)+1, where Gc 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, as well as to any even eigenvalue of $L(G)$ and $Q(G)$ for any connected graph G. They also showed that ηA(Gc)≤ 1 when every block of the cactus is an odd cycle. This also aligns with a special case of our bound.