Applications of a theorem by Ky Fan in the theory of weighted Laplacian graph energy

TL;DR

This paper explores how Ky Fan's theorem can be applied to derive new inequalities and proofs related to the energy of weighted Laplacian graphs, enhancing understanding in spectral graph theory.

Contribution

It introduces novel applications of Ky Fan's theorem to the theory of graph energy, providing new inequalities and alternative proofs for existing results.

Findings

New inequalities for graph energy derived from Ky Fan's theorem

Alternative proofs for known spectral graph inequalities

Enhanced understanding of weighted Laplacian graph properties

Abstract

The energy of a graph $G$ is equal to the sum of the absolute values of the eigenvalues of $G$ , which in turn is equal to the sum of the singular values of the adjacency matrix of $G$. Let $X$, $Y$ and $Z$ be matrices, such that $X+Y= Z$. The Ky Fan theorem establishes an inequality between the sum of the singular values of $Z$ and the sum of the sum of the singular values of $X$ and $Y$. This theorem is applied in the theory of graph energy, resulting in several new inequalities, as well as new proofs of some earlier known inequalities.