Revisiting two classical results on graph spectra

Abstract

Let μ (G) and μmin (G) be the largest and smallest eigenvalues of the adjacency matrix of a graph G. Our main results are: (i) If H is a proper subgraph of a connected graph G of order n and diameter D, then μ (G) - μ (H) > 1/μ (G)2dn. (ii) If G is a connected nonbipartite graph of order n and diameter D, then μ (G) + μmin (G) > 2/μ (G)2dn. For large μ and D these bounds are close to the best possible ones.