Extremal values for the square energies of graphs

TL;DR

This paper investigates extremal values of the positive and negative square energies of graphs, introducing new tools and proving bounds related to graph eigenvalues, domination number, and edges.

Contribution

It provides new methods for analyzing square energy in graphs and establishes bounds on extremal values related to graph parameters, confirming conjectures up to constants.

Findings

Lower bounds on minimum of positive and negative square energies in terms of domination number.

Lower bounds on positive and negative square energies in terms of number of edges, with optimal exponents.

Verification of a conjecture relating square energies and domination number.

Abstract

Let $G$ be a graph with $n$ non-isolated vertices and $m$ edges. The positive / negative square energies of $G$, denoted $s^+(G)$ / $s^-(G)$, are defined as the sum of squares of the positive / negative eigenvalues of the adjacency matrix $A_G$ of $G$. In this work, we provide several new tools for studying square energy encompassing semi-definite optimization, graph operations, and surplus. Using our tools, we prove the following results on the extremal values of $s^{\pm}(G)$ with a given number of vertices and edges. 1. We have $\min(s^+(G), s^-(G)) \geq n - \gamma \geq \frac{n}{2}$, where $\gamma$ is the domination number of $G$. This verifies a conjecture of Elphick, Farber, Goldberg and Wocjan up to a constant, and proves a weaker version of this conjecture introduced by Elphick and Linz. 2. We have $s^+(G) \geq m^{6/7 - o(1)}$ and $s^-(G) = \Omega(m^{1/2})$, with both exponents being optimal.