arXiv 2201.067292022-01-18math.CO

符号图的特征值

Eigenvalues of signed graphs

Dan Li, Huiqiu Lin, Jixiang Meng

备注：: 18 pages, 2 figures

摘要：

符号图的边可以标记为正边或负边。$ρ(M)$表示$Σ$的$M$ -谱半径，其中$M=M(Σ)$是$Σ$的实对称图矩阵。显然，是$ρ(M)=\mbox{max}\{λ_1(M),-λ_n(M)\}$。设$A(Σ)$是$Σ$的邻接矩阵，$(K_n,H^-)$是一个符号完全图，其负边导出一个子图$H$ .本文首先研究谱极图理论中的一个中心问题：在$(K_n,T^-)$中，$T$是生成树的情况下，哪一个有号图的$ρ(A(Σ))$最大？为了回答这个问题，我们分别刻画了在$(K_n,T^-)$中具有最大$λ_1(A(Σ))$和最小$λ_n(A(Σ))$的极值符号图.符号图的另一个有趣的图矩阵是距离矩阵，即由Hameed，Shijin，Soorya，Germina和Zaslavsky定义的$D(Σ)$ [8]。请注意，$A(Σ)=D(Σ)$.本文给出了直径至少为2的符号图$Σ$的最小距离特征值的上界。这一结果意味着Lin [11]证明的一个结果最初是由Aouchiche和Hansen [1]证明的。

英文摘要：

Signed graphs have their edges labeled either as positive or negative. $ρ(M)$ denote the $M$-spectral radius of $Σ$, where $M=M(Σ)$ is a real symmetric graph matrix of $Σ$. Obviously, $ρ(M)=\mbox{max}\{λ_1(M),-λ_n(M)\}$. Let $A(Σ)$ be the adjacency matrix of $Σ$ and $(K_n,H^-)$ be a signed complete graph whose negative edges induce a subgraph $H$. In this paper, we first focus on a central problem in spectral extremal graph theory as follows: Which signed graph with maximum $ρ(A(Σ))$ among $(K_n,T^-)$ where $T$ is a spanning tree? To answer the problem, we characterize the extremal signed graph with maximum $λ_1(A(Σ))$ and minimum $λ_n(A(Σ))$ among $(K_n,T^-)$, respectively. Another interesting graph matrix of a signed graph is distance matrix, i.e. $D(Σ)$ which was defined by Hameed, Shijin, Soorya, Germina and Zaslavsky [8]. Note that $A(Σ)=D(Σ)$ when $Σ\in (K_n,T^-)$. In this paper, we give upper bounds on the least distance eigenvalue of a signed graph $Σ$ with diameter at least 2. This result implies a result proved by Lin [11] was originally conjectured by Aouchiche and Hansen [1].

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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