The spectral radius of irregular graphs

The Spectral Radius and the Maximum Degree of Irregular Graphs

Let G be an irregular graph on n vertices with maximum degree ∆ and diameter D. We show that ∆ − λ1 > 1 nD , where λ1 is the largest eigenvalue of the adjacency matrix of G. We also study the effect of adding or removing few edges on the spectral radius of a regular graph. 1 Preliminaries Our graph notation is standard (see West [22]). For a graph G, we denote by λi(G) the i-th largest eigenval...

Spectral radius of bipartite graphs

On the spectral radius of graphs

Let G be a simple undirected graph. For v ∈ V (G), the 2-degree of v is the sum of the degrees of the vertices adjacent to v. Denote by ρ(G) and μ(G) the spectral radius of the adjacency matrix and the Laplacian matrix of G, respectively. In this paper, we present two lower bounds of ρ(G) and μ(G) in terms of the degrees and the 2-degrees of vertices. © 2004 Elsevier Inc. All rights reserved. A...

Walks and the spectral radius of graphs

Given a graph G, write μ (G) for the largest eigenvalue of its adjacency matrix, ω (G) for its clique number, and wk (G) for the number of its k-walks. We prove that the inequalities wq+r (G) wq (G) ≤ μ (G) ≤ ω (G) − 1 ω (G) wr (G) hold for all r > 0 and odd q > 0. We also generalize a number of other bounds on μ (G) and characterize pseudo-regular and pseudo-semiregular graphs in spectral terms.

On the spectral radius of graphs

We characterize the graphs which achieve the maximum value of the spectral radius of the adjacency matrix in the sets of all graphs with a given domination number and graphs with no isolated vertices and a given domination number. AMS Classification: 05C35, 05C50, 05C69