Refinement of a conjecture on positive square energy of graphs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908399161049088 |
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| author | Akbari, Saieed Kumar, Hitesh Mohar, Bojan Pragada, Shivaramakrishna Zhang, Shengtong |
| author_facet | Akbari, Saieed Kumar, Hitesh Mohar, Bojan Pragada, Shivaramakrishna Zhang, Shengtong |
| contents | Let $G$ be a simple graph of order $n$ with eigenvalues $λ_1(G)\geq \cdots \geq λ_n(G)$. Define \[s^+(G)=\sum_{λ_i >0} λ_i^2(G), \quad s^-(G)=\sum_{λ_i<0} λ_i^2(G).\] 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_07264 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Refinement of a conjecture on positive square energy of graphs Akbari, Saieed Kumar, Hitesh Mohar, Bojan Pragada, Shivaramakrishna Zhang, Shengtong Combinatorics 05C50, 05C69, 05C76 Let $G$ be a simple graph of order $n$ with eigenvalues $λ_1(G)\geq \cdots \geq λ_n(G)$. Define \[s^+(G)=\sum_{λ_i >0} λ_i^2(G), \quad s^-(G)=\sum_{λ_i<0} λ_i^2(G).\] 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. |
| title | Refinement of a conjecture on positive square energy of graphs |
| topic | Combinatorics 05C50, 05C69, 05C76 |
| url | https://arxiv.org/abs/2506.07264 |