arXiv subjects

Linfeng Xie

Publications and source records attributed to Linfeng Xie.

4 recordsLinked to original sources

arXiv2026

Index perturbation of signed graphs

Let $\Gamma = (G, \sigma)$ be a signed graph and $v$ a non-isolated vertex of $\Gamma$. Let $\Gamma-v$ denote the graph obtained by deleting the vertex $v$ together with all signed edges incident to it from $\Gamma$, and $d_{\Gamma}(v)$ the degree of $v$ in $\Gamma$. In this paper, we prove that the largest eigenvalue $\lambda_1(\Gamma)$ of $\Gamma$ satisfies \[ \lambda_1(\Gamma) \le \sqrt{\lambda_1^2(\Gamma - v) + 2d_\Gamma(v) - 1}, \] and we also present a refined version of this bound. Moreover, we characterize the extremal signed graphs achieving equality when $\Gamma$ is connected and $d_\Gamma(v)\ge 2$, which are switching equivalent to the balanced complete signed graph.

math.CO↗

arXiv2026

Localization of spectral Tur\'an theorems for signed graphs

In this paper, we extend localized Tur\'an theorems to signed graphs and study the corresponding spectral Tur\'an problems. Firstly, we establish a vertex-localized Tur\'an-type inequality for signed graphs and characterize the extremal graphs reaching this bound. Secondly, we derive a sharp upper bound for the largest eigenvalue of a signed graph via localized Tur\'an parameters, and identify the extremal graphs attaining equality. Finally, we generalize a walk-based local spectral Tur\'an inequality to signed graphs. Our results generalize and improve several existing theorems for unsigned and signed graphs.

math.CO↗

arXiv2026

Spectral Tur\'an problem for $t\mathcal{K}_{4}^{-}$-free unbalanced signed graphs

Let $tK_4$ denote the family of all graphs consisting of $t$ copies of $K_4$ that are allowed to share vertices and $t\mathcal{K}_{4}^{-}$ be the set of all unbalanced signed graphs whose underlying graphs are elements of $tK_4$. In this paper, we characterize the extremal graphs that achieve the maximum index and spectral radius among all $t\mathcal{K}_{4}^{-}$-free unbalanced signed graphs with given order.

math.CO↗

arXiv2025

Bounds for the largest eigenvalue and sum of Laplacian eigenvalues of signed graphs

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 $\Gamma$ of order $n$ and size $m$ contains a balanced triangle if $\lambda_{1}(\Gamma)\ge \sqrt{m-1}$, $\lambda_{1}(\Gamma) \ge |\lambda_{n}(\Gamma)|$ and $\Gamma\not \sim (C_{5}\cup (n-5)K_{1},+)$, where $\lambda_{1}(\Gamma)$ is the largest eigenvalue of the adjacency matrix of $\Gamma$. Thirdly, we confirm a conjecture proposed in [Linear Multilinear Algebra 51 (1) (2003) 21--30] that: if $\Gamma$ is a connected signed graph, then $$ \sum_{i=1}^{k}\mu_{i}(\Gamma) >\sum_{i=1}^{k}d_{i}(\Gamma)~~(1\le k\le n-1), $$ where $\mu_{1}(\Gamma)\ge\mu_{2}(\Gamma)\ge\cdots \ge \mu_{n}(\Gamma)$ are Laplacian eigenvalues of $\Gamma$, and $d_{1}(\Gamma)\ge d_{2}(\Gamma)\ge \dots \ge d_{n}(\Gamma)$ are vertex degrees of $\Gamma$. Finally, we give a lower bound for the sum of the $k$ largest Laplacian eigenvalues of a connected signed graph.

math.CO↗