Graphs with eigenvalue −1 of multiplicity 2θ(G)+ρ(G)−1

Abstract

Let G be a graph, m(G,λ) be the multiplicity of eigenvalue λ of A(G), θ(G) be the cyclomatic number of G, ρ(G) be the number of pendent vertices of G. Wang et al. proved that m(G,λ)≤2θ(G)+ρ(G)−1 for an arbitrary eigenvalue λ if G is not a cycle, and left an open problem to characterize the extremal graphs attaining the upper bound. We solve the open problem when λ=−1, that is a graph G satisfies m(G,−1)=2θ(G)+ρ(G)−1 if and only if it is obtained from a tree T with m(T,−1)=2θ(T)+ρ(T)−1 by attaching θ(G) cycles of order a multiple of 3 to θ(G) quasi-pendent vertices of T and delete related θ(G) pendent vertices, where ρ(T)≥θ(G).