Multiplicity of Laplacian eigenvalues in terms of cyclomatic number and number of quasi-pendant vertices

Let G=(V(G),E(G)) be a connected graph. The cyclomatic number of G is c(G)=|E(G)|−|V(G)|+1. In this article, we show that for any Laplacian eigenvalue λ≠1, mG(λ)≤2c(G)+q(G), where q(G) denotes the number of quasi-pendant vertices and mG(λ) is the algebraic multiplicity of λ. We further characterize all graphs for which equality holds. A stronger result is also derived, relating the multiplicity of a Laplacian eigenvalue to the cyclomatic number and the number of pendant paths.