Energy of graphs with no eigenvalue in the interval (−1,1)

Abstract

The energy of a simple graph G of order n, denoted by E(G), is E(G)=|λ1|+⋯+|λn|, where λ1,…,λn are the eigenvalues of G. Recently, it has been conjectured that if all eigenvalues of G are non-zero, then E(G)≥Δ(G)+δ(G) and the equality holds if and only if G is a complete graph, where δ(G) and Δ(G) are the minimum degree and the maximum degree of vertices of G, respectively. Motivated by this conjecture we find some results for energy of graphs in terms of the number of vertices, the number of edges and the nullity. We prove that if G is a graph with n vertices and m edges such that G has no eigenvalue in the interval (−1,1), then E(G)≥n−1+2m−n+1 and the equality holds if and only if G is a disjoint union of some special complete graphs. In particular, we prove that the above conjecture is true for graphs that have no eigenvalue in the interval (−1,1).