论文标题
签名的光谱Turań定理
Signed spectral Turań type theorems
论文作者
论文摘要
签名的图$σ=（g，σ）$是一个图形，其中函数$σ$分配了$ 1 $或$ -1 $的简单图$ g $。 $σ$的邻接矩阵（用$ a（σ）$表示）是在规范上定义的。在最近的一篇论文中，Wang等人。扩展了Hoffman和Cvetković的特征值边界，以供签名图。他们提出了一个与平衡集团数字和签名图最大的特征值有关的开放问题。我们解决了这个开放问题的加强版本。作为副产品，我们为未签名图的最低特征值提供了一些已知经典边界的替代证明。我们扩展了Turán在签名图中的不平等。此外，我们研究了签名图的Bollobás和Nikiforov猜想，并表明猜想对于签名的图不必正确。然而，在某些假设下，猜想仍然存在签名图。最后，我们研究了签名步行数量与签名图最大的特征值之间的一些关系。
A signed graph $Σ= (G, σ)$ is a graph where the function $σ$ assigns either $1$ or $-1$ to each edge of the simple graph $G$. The adjacency matrix of $Σ$, denoted by $A(Σ)$, is defined canonically. In a recent paper, Wang et al. extended the eigenvalue bounds of Hoffman and Cvetković for the signed graphs. They proposed an open problem related to the balanced clique number and the largest eigenvalue of a signed graph. We solve a strengthened version of this open problem. As a byproduct, we give alternate proofs for some of the known classical bounds for the least eigenvalues of the unsigned graphs. We extend the Turán's inequality for the signed graphs. Besides, we study the Bollobás and Nikiforov conjecture for the signed graphs and show that the conjecture need not be true for the signed graphs. Nevertheless, the conjecture holds for signed graphs under some assumptions. Finally, we study some of the relationships between the number of signed walks and the largest eigenvalue of a signed graph.
