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arXiv 2210.05940math.CO
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连通图的距离Seidel矩阵

Distance Seidel matrix of a connected graph

Haritha T, Chithra A.

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AI总结：

本文提出连通图的距离Seidel矩阵，建立其与距离及邻接特征值的关系，刻画最大特征值为3的图，并给出谱半径、能量界及多种图运算下的谱结果与应用。

摘要：

对于连通图$G$，我们提出一种与其距离矩阵和Seidel矩阵相关的新图矩阵概念，称为距离Seidel矩阵$\mathcal{D}^S(G)$。设$\mathcal{D}^S(G)$的特征值为$\partial_{1}^{S}(G) \geq \cdots \geq \partial_{n}^{S}(G)$。本文建立了图的距离Seidel特征值与其距离特征值和邻接特征值之间的关系。我们刻画了所有满足$\partial_{1}^{S}(G)=3$的连通图。同时，我们确定了距离Seidel谱半径和距离Seidel能量的不同界。我们研究了完全二部图因删除一条边而引起的距离Seidel能量变化。此外，我们得到了不同图运算下的距离Seidel谱，例如连接、笛卡尔积、字典序积，以及一元运算如双重图和扩展双重覆盖图。作为应用，我们给出了各类距离Seidel同谱图和距离Seidel整图族。

英文摘要：

For a connected graph $G$, we present the concept of a new graph matrix related to its distance and Seidel matrix, called distance Seidel matrix $\mathcal{D}^S(G)$. Suppose that the eigenvalues of $\mathcal{D}^S(G)$ be $\partial_{1}^{S}(G) \geq \cdots \geq \partial_{n}^{S}(G).$ In this article, we establish a relationship between distance Seidel eigenvalues of a graph with its distance and adjacency eigenvalues. We characterize all the connected graphs with $\partial_{1}^{S}(G)= 3.$ Also, we determine different bounds for the distance Seidel spectral radius and distance Seidel energy. We study the distance Seidel energy change of the complete bipartite graph due to the deletion of an edge. Moreover, we obtain the distance Seidel spectra of different graph operations such as join, cartesian product, lexicographic product, and unary operations like the double graph and extended double cover graph. We give various families of distance Seidel cospectral and distance Seidel integral graphs as an application.

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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