A Sharp Upper Bound of the Spectral Radius of Graphs

Abstract

Let G be a simple connected graph with n vertices and m edges. Let δ(G) = δ be the minimum degree of vertices of G. The spectral radius p(G) of G is the largest eigenvalue of its adjacency matrix. In this paper, we obtain the following sharp upper bound of p(G): ρ(G)≤ δ - 1 + √(δ + 1)2 + 4(2m - δn)/2. Equality holds if and only if G is either a regular graph or a bidegreed graph in which each vertex is of degree either δ or n - 1. © 2001 Academic Press.