Token图的代数连通度和拉普拉斯谱半径

中文摘要 AI 辅助

对于一个阶为$n$的图$G=(V,E)$以及介于$1$和$\lfloor\frac{n}{2}\rfloor$之间的整数$k$，其token图$F_k(G)$是以$V$的所有$\binom{n}{k}$个$k$-子集为顶点的图，且$F_k(G)$中两个顶点相邻当且仅当它们的对称差是$E$中的一条边。已有研究发现，一个图的代数连通度大于或等于其token图的代数连通度，而一个图的拉普拉斯谱半径小于或等于其token图的拉普拉斯谱半径。此外，利用随机游走的连续马尔可夫链理论，已证明一个图的代数连通度与其token图的代数连通度相等这一猜想。在本文中，我们推导出关于一个图及其添加新边后的图，以及它们的token图的代数连通度的一些结果，从而给出一个组合/代数证明。此外，我们提供了某些条件，在这些条件下，一个图的拉普拉斯谱半径小于其token图的拉普拉斯谱半径。最后，我们刻画了那些与其token图具有相同拉普拉斯谱半径的图，包括树。

英文摘要

For a graph $G=(V,E)$ of order $n$ and an integer $k$ between $1$ and $\lfloor\frac{n}{2}\rfloor$, its token graph $F_k(G)$ is the graph whose vertices consist of the $\binom{n}{k}$ $k$-subsets of $V$, and two vertices of $F_k(G)$ are adjacent whenever their symmetric difference is an edge in $E$. It was found that the algebraic connectivity of a graph is greater than or equal to that of its token graph, while the Laplacian spectral radius of a graph is less than or equal to that of its token graph. Moreover, a conjecture that the algebraic connectivity of a graph coincides with that of its token graph has been proved by using the theory of continuous Markov chains of random walks. In this paper, we derive some results about the algebraic connectivities of a graph and the same graph after adding new edges and their token graphs to obtain a combinatorial/algebraic proof. Besides, we provide some conditions under which the Laplacian spectral radius of a graph is less than that of its token graph. Finally, we characterize the graphs that have the same Laplacian spectral radius as their token graphs, including trees.

发表机构

• Shenzhen University of Information Technology（深圳信息职业技术学院）
• Universitat de Lleida（莱里达大学）
• Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）
• Northwestern Polytechnical University（西北工业大学）

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