An interlacing approach for bounding the sum of Laplacian eigenvalues of graphs

Abstract

We apply eigenvalue interlacing techniques for obtaining lower and upper bounds for the sums of Laplacian eigenvalues of graphs, and characterize equality. This leads to generalizations of, and variations on theorems by Grone, and Grone & Merris. As a consequence we obtain inequalities involving bounds for some well-known parameters of a graph, such as edge-connectivity, and the isoperimetric number. Eigenvalue interlacingThroughout this paper, G = (V, E) is a finite simple graph with n = |V | vertices. Theorem 1.1. Let A be a real symmetric n × n matrix with eigenvalues λ 1 ≥ · · · ≥ λ n . For some m < n, let S be a real n × m matrix with orthonormal columns, S ⊤ S = I, and consider the matrix B = S ⊤ AS, with eigenvalues µ 1 ≥ · · · ≥ µ m . Then, (a) the eigenvalues of B interlace those of A, that is,(b) if the interlacing is tight, that is, for some 0 ≤ k ≤ m, λ i = µ i , i = 1, . . . , k, and µ i = λ n−m+i , i = k + 1, . . . , m, then SB = AS.

A bound for the $p$-domination number of a graph in terms of its eigenvalue multiplicities

“…The domination number has also been studied in relation to the Laplacian eigenvalue distribution, see Hedetniem, Jacobs and Trevisan [8]. The Laplacian eigenvalues have also been used to provide bounds for the p-domination number, see Abiad, Fiol, Haemers and Perarnau [1]. While several results are known to connect the domination number with the Laplacian eigenvalues, not much is known about the relation of the domination number with the adjacency spectrum for non-regular graphs.…”