Bounds for the largest eigenvalue and sum of Laplacian eigenvalues of signed graphs
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866918226050416640 |
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| author | Xie, Linfeng Liu, Xiaogang |
| author_facet | Xie, Linfeng Liu, Xiaogang |
| contents | In this paper, we consider the bounds for the largest eigenvalue and the sum of the $k$ largest Laplacian eigenvalues of signed graphs. Firstly, we give an upper bound on the largest eigenvalue of the adjacency matrix of a signed graph and characterize the extremal graphs that attain this bound. Secondly, we prove that a non-bipartite signed graph $Γ$ of order $n$ and size $m$ contains a balanced triangle if $λ_{1}(Γ)\ge \sqrt{m-1}$, $λ_{1}(Γ) \ge |λ_{n}(Γ)|$ and $Γ\not \sim (C_{5}\cup (n-5)K_{1},+)$, where $λ_{1}(Γ)$ is the largest eigenvalue of the adjacency matrix of $Γ$. Thirdly, we confirm a conjecture proposed in [Linear Multilinear Algebra 51 (1) (2003) 21--30] that: if $Γ$ is a connected signed graph, then
$$
\sum_{i=1}^{k}μ_{i}(Γ) >\sum_{i=1}^{k}d_{i}(Γ)~~(1\le k\le n-1),
$$
where $μ_{1}(Γ)\geμ_{2}(Γ)\ge\cdots \ge μ_{n}(Γ)$ are Laplacian eigenvalues of $Γ$, and $d_{1}(Γ)\ge d_{2}(Γ)\ge \dots \ge d_{n}(Γ)$ are vertex degrees of $Γ$. Finally, we give a lower bound for the sum of the $k$ largest Laplacian eigenvalues of a connected signed graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_01736 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounds for the largest eigenvalue and sum of Laplacian eigenvalues of signed graphs Xie, Linfeng Liu, Xiaogang Combinatorics In this paper, we consider the bounds for the largest eigenvalue and the sum of the $k$ largest Laplacian eigenvalues of signed graphs. Firstly, we give an upper bound on the largest eigenvalue of the adjacency matrix of a signed graph and characterize the extremal graphs that attain this bound. Secondly, we prove that a non-bipartite signed graph $Γ$ of order $n$ and size $m$ contains a balanced triangle if $λ_{1}(Γ)\ge \sqrt{m-1}$, $λ_{1}(Γ) \ge |λ_{n}(Γ)|$ and $Γ\not \sim (C_{5}\cup (n-5)K_{1},+)$, where $λ_{1}(Γ)$ is the largest eigenvalue of the adjacency matrix of $Γ$. Thirdly, we confirm a conjecture proposed in [Linear Multilinear Algebra 51 (1) (2003) 21--30] that: if $Γ$ is a connected signed graph, then $$ \sum_{i=1}^{k}μ_{i}(Γ) >\sum_{i=1}^{k}d_{i}(Γ)~~(1\le k\le n-1), $$ where $μ_{1}(Γ)\geμ_{2}(Γ)\ge\cdots \ge μ_{n}(Γ)$ are Laplacian eigenvalues of $Γ$, and $d_{1}(Γ)\ge d_{2}(Γ)\ge \dots \ge d_{n}(Γ)$ are vertex degrees of $Γ$. Finally, we give a lower bound for the sum of the $k$ largest Laplacian eigenvalues of a connected signed graph. |
| title | Bounds for the largest eigenvalue and sum of Laplacian eigenvalues of signed graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.01736 |