Maximizing the spectral radius of graphs with fixed minimum degree and edge connectivity

Abstract

The spectral radius ρ(G) of a graph G is the largest eigenvalue of the adjacency matrix A(G). Suppose a graph G0 maximizes the spectral radius over the class of graphs of order n with fixed minimum degree δ and edge connectivity κ′<δ. In this paper, we mainly show that G0≅Bn,δκ′, where Bn,δκ′ is obtained by adding κ′ edges between Kδ+1 and Kn−δ−1. A property of the adjacency matrix of G0 is also obtained. Moreover, graphs that maximize ρ(G) over the class of graphs with minimum degree δ and edge-connectivity κ′, for κ′=0,1,2,3,δ, are completely determined.