Refinement of a conjecture on positive square energy of graphs

TL;DR

This paper investigates a spectral graph theory conjecture related to the sum of squares of positive eigenvalues, verifying it for specific graph classes and proposing a strengthened version with partial proofs.

Contribution

It verifies a conjecture on positive eigenvalue sums for graphs with domination number at most 2 and proposes a stronger conjecture, proving it for claw-free graphs and graphs with diameter 2.

Findings

Verified the conjecture for graphs with domination number ≤ 2.

Strengthened the conjecture to relate to graph size and order.

Proved the strengthened conjecture for claw-free graphs and diameter 2 graphs.

Abstract

Let $G$ be a simple graph of order $n$ with eigenvalues $\lambda_1(G)\geq \cdots \geq \lambda_n(G)$. Define \[s^+(G)=\sum_{\lambda_i >0} \lambda_i^2(G), \quad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2(G).\] It was conjectured by Elphick, Farber, Goldberg and Wocjan that for every connected graph $G$ of order $n$, $s^+(G) \ge n-1.$ We verify this conjecture for graphs with domination number at most 2. We then strengthen the conjecture as follows: if $G$ is a connected graph of order $n$ and size $m \geq n+1$, then $s^+(G) \geq n$. We prove this conjecture for claw-free graphs and graphs with diameter 2.