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arXiv · 2506.07264

Refinement of a conjecture on positive square energy of graphs

Saieed Akbari·Hitesh Kumar·Bojan Mohar·Shivaramakrishna Pragada·Shengtong Zhang
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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.

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Saieed Akbari, Hitesh Kumar, Bojan Mohar, Shivaramakrishna Pragada, Shengtong Zhang. 2025-06-08. Refinement of a conjecture on positive square energy of graphs. https://arxiv.org/abs/2506.07264

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