The multiplicity of an arbitrary eigenvalue of a graph in terms of cyclomatic number and number of pendant vertices

Abstract

Let G be a graph with adjacency matrix A(G). The nullity η(G) of G is the multiplicity of zero as an eigenvalue of A(G), which has received a lot of attention because of its chemical importance. The multiplicity of an arbitrary eigenvalue λ of A(G) is denoted as m(G,λ). In [25], the authors proved that η(G)≤2θ(G)+p(G) for a connected graph G, with equality if and only if G is a cycle of order divisible by 4, where θ(G)=|E(G)|−|V(G)|+1 is the cyclomatic number of G and p(G) is the number of pendant vertices of G. In the present paper, we intend to extend this result from the nullity of G to the multiplicity of an arbitrary eigenvalue of G. Differing from the method in [25], by mainly applying algebraic method we prove the following result: For a connected graph G, m(G,λ)≤2θ(G)+p(G) for an arbitrary eigenvalue λ of G, the equality holds if and only if G is a cycle Cn and λ=2cos2kπn with k=1,2,…,⌈n2⌉−1.