On the first two eigenvalues of regular graphs

Abstract

Let G be a regular graph with m edges, and let μ1,μ2 denote the two largest eigenvalues of AG, the adjacency matrix of G. We show that, if G is not complete, thenμ12+μ22≤2(ω−1)ωm where ω is the clique number of G. This confirms a conjecture of Bollobás and Nikiforov for regular graphs. We also show that equality holds if and only if G is either a balanced Turán graph or the disjoint union of two balanced Turán graphs of the same size.