Singular value inequality and graph energy change

Abstract

The energy of a graph is the sum of the singular values of its adjacency matrix. A classic inequality for singular values of a matrixsum, including its equality case, is used to study how the energy of a graph changes when edges are removed. One sharp bound and one bound that is never sharp, for the change in graph energy when the edges of a nonsingular induced subgraph are removed, are established. A graph is nonsingular if its adjacency matrixis nonsingular. 1. Singular value inequality for matrix sum. Let X be an n × n complex matrix and denote its singular values by s1(X) ≥ s2(X) ≥ · ·· ≥sn(X) ≥ 0. If X has real eigenvalues only, denote its eigenvalues by λ1(X) ≥ λ2(X) ≥ · ·· ≥λn(X). Define |X| = √ XX ∗ which is positive semi-definite, and note that λi(|X| )= si(X) for all i .W e w riteX ≥ 0t o meanX is positive semi-definite. We are interested in the following singular value inequality for a matrix sum: n � i=1 si(A + B) ≤ n � i=1 si(A )+ n � i=1 si(B)