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

Let G be a simple undirected graph, (G) be the circuit rank of G, M (G) be the nullity of a graph matrix M (G), and mM (G, ) be the multiplicity of eigenvalue 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, ) in terms of (G) when is an integer (even integer, respectively). We also demonstrate that when and are rational numbers, similar bounds can be obtained for m₀_ (G, ), where A_ (G) is the generalized adjacency matrix of G. Distinctively, our bounds involve only (G), not a multiple of it. Previous bounds for mA (G, ) (and later m₀_ (G, ) ) in terms of the circuit rank have all included 2 (G) with the sole exception of the case =0. Wong et al. (2022) showed that A (Gc) (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 A (Gc) 1 when every block of the cactus is an odd cycle. This also aligns with a special case of our bound.