Characterization of Graphs with an Eigenvalue of Large Multiplicity

Abstract EN

Let G be a simple and undirected graph.

The eigenvalues of the adjacency matrix of G are called the eigenvalues of G.

In this paper, we characterize all the n-vertex graphs with some eigenvalue of multiplicity n−2 and n−3, respectively.

Moreover, as an application of the main result, we present a family of nonregular graphs with four distinct eigenvalues.