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.
| Comments: | 8 pages |
| Subjects: | Combinatorics (math.CO) |
| Cite as: | arXiv:2605.24668 [math.CO] |
| (or arXiv:2605.24668v1 [math.CO] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24668 arXiv-issued DOI via DataCite (pending registration) |
From: Bo Ning [view email]
[v1]
Sat, 23 May 2026 17:10:37 UTC (8 KB)
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