Inequalities on Vertex Degrees, Eigenvalues and (Signless) Laplacian Eigenvalues of Graphs

Several inequalities on vertex degrees, eigenvalues, Laplacian eigen-values, and signless Laplacian eigenvalues of graphs are presented in this note. Some of them are generalizations of the inequalities in [2]. We consider only finite undirected graphs without loops or multiple edges. Notation and terminology not defined here follow that in [1]. We use [n] to denote the set of { 1, 2, ..., n}. For each subset I of [n], we use I c to denote [n] − I. Let G be a graph of order n. We assume that the vertices in G are ordered such that d 1 ≥ d 2 ≥ ... ≥ d n , where d i , 1 ≤ i ≤ n, is the degree of vertex v i in G. We define Σ k (G) as Σ n i=1 d k i. The eigenvalues μ 1 (G) ≥ μ 2 (G) ≥ ... ≥ μ n (G) of the adjacency matrix A(G) of G are called the eigenvalues of the graph G. Let D(G) be the diagonal matrix of the degree sequence of G. The eigenvalues λ 1 (G) ≥ λ 2 (G) ≥ ... ≥ λ n−1 ≥ λ n (G) = 0 of L(G) := D(G)−A(G) of G are called the Laplacian eigenvalues of the graph G. The eigenvalues τ 1 (G) ≥ τ 2 (G) ≥ ... ≥ τ n−1 ≥ τ n (G) of Q(G) = D(G) + A(G) of G are called the signless Laplacian eigenvalues of the graph G. In this notes, we will present several inequalities on vertex degrees, eigenvalues, Laplacian eigenvalues, and signless Laplacian eigenvalues of graphs. We need the following Theorem 1 (see Pages 104-108 in [3]) proved by Hoffman and Wielandt in the proofs of our theorems.