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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2605.24668 |
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| _version_ | 1866916042327982080 |
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| author | Ning, Bo Zeng, Jing |
| author_facet | Ning, Bo Zeng, Jing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24668 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs Ning, Bo Zeng, Jing Combinatorics 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. |
| title | A Proof of a Conjecture on Positive and Negative Square Energies of Unicyclic Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.24668 |