A characterization of the smallest eigenvalue of a graph

Abstract

It is well known that the smallest eigenvalue of the adjacency matrix of a connected d‐regular graph is at least − d and is strictly greater than − d if the graph is not bipartite. More generally, for any connected graph G = (V, E), consider the matrix Q = D + A where D is the diagonal matrix of degrees in the graph G and A is the adjacency matrix of G. Then Q is positive semidefinite, and the smallest eigenvalue of Q is 0 if and only if G is bipartite. We will study the separation of this eigenvalue from 0 in terms of the following measure of nonbipartiteness of G. For any S ⊆ V, we denote by emin(S) the minimum number of edges that need to be removed from the induced subgraph on S to make it bipartite. Also, we denote by cut(S) the set of edges with one end in S and the other in V − S. We define the parameter Ψ as. The parameter Ψ is a measure of the nonbipartiteness of the graph G. We will show that the smallest eigenvalue of Q is bounded above and below by functions of Ψ. For d‐regular graphs, this characterizes the separation of the smallest eigenvalue of the adjacency matrix from −d. These results can be easily extended to weighted graphs.

Eigenvalue bounds for the signless laplacian

“…The matrix D + A is called the signless Laplacian in [16], and it appears very rarely in published papers (see [5]), the paper [13] being almost the only relevant research paper published before 2003. Only recently has the signless Laplacian attracted the attention of researchers [4,8,11,10,16,26].…”

“…The parameter ψ was introduced in [13] as a measure of non-bipartiteness. It is shown that the least eigenvalue q n of the signless Laplacian Q is bounded above and below by functions of ψ.…”

The maximum clique and the signless Laplacian eigenvalues

“…The matrix L = D − A is known as the Laplacian of G and has been studied extensively in literature (see, e.g., [1]). The matrix Q = D + A is called the signless Laplacian in [4] and appears very rarely in published papers (see [1]), the paper [2] being one of the very few research papers concerning this matrix. Since the signless Laplacian is a positive semi-definite matrix, all its eigenvalues are non-negative.…”

On the log-concavity of Laplacian characteristic polynomials of graphs

“…First, we show that the sequences of the moduli of (signless) Laplacian coefficients of graphs are log-concave, and hence unimodal. As a consequence, we obtain some lower and upper bounds on the algebraic connectivity and the least eigenvalue of the signless Laplacian matrix, which is studied in [7] as a measure of non-bipartiteness of a graph. Moreover, we obtain upper bounds for the partial sums of the Laplacian eigenvalues.…”

“…In other words, we have Let η denote the nullity of the matrix Q(G). The following theorem explicitly expresses the connection between η and the structure of G. In [7], the least eigenvalue of Q(G) was studied as a measure of non-bipartiteness of a graph. Here, we obtain a lower bound for this quantity.…”