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arXiv · 2605.24668

A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs

Bo Ning·Jing Zeng
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Abstract

Let $G$ be a unicyclic graph of order $n$, and let $k$ be the length of the unique cycle of $G$. For the adjacency eigenvalues of $G$, let $s^{+}(G)$ and $s^{-}(G)$ denote the sums of the squares of the positive and negative eigenvalues, respectively. Akbari, Kumar, Mohar, Pragada, and Zhang conjectured that, when $k$ is odd, the value of $k$ modulo $4$ determines which of $s^+(G)$ and $s^-(G)$ is greater than $n$. More precisely, if $k\equiv 3\pmod 4$, then $s^+(G)>n>s^-(G)$; if $k\equiv 1\pmod 4$, then $s^+(G)<n<s^-(G)$. We confirm this conjecture.

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Bo Ning, Jing Zeng. 2026-05-23. A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs. https://arxiv.org/abs/2605.24668

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