SINGULAR VALUE INEQUALITY AND GRAPH ENERGY CHANGE

By J. Day, Wasin So. 2007

74 citations recorded by the source corpus.

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)

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/5f0e28ab198c6ee901126e2ac160dcd7f0a0ddd6

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In Computational Theory and Mathematics

In Geometry and Topology

Citations in the corpus

Edge addition, singular values, and energy of graphs and matrices (2009, background): “(See [2] for a review of different proofs of (2).”
Beyond graph energy: norms of graphs and matrices (2015, methodology): “s the trace norm of A. This simple observation, made in [32], triggered some sort of a chain reaction. On the one hand, usual tools for norms provided new techniques for graph energy; see, e.g., [2], [9], [10], and [43]. On the other hand, viewed as a trace norm, graph energy was extended to nonsymm”
On incidence energy of a graph (2009, background): “In [28] it is shown that if H is an induced subgraph of the graph G, then E(G \ H) − E(H) ≤ E(G) ≤ E(G \ H) + E(H) .”