标题： 关于令牌图的拉普拉斯谱 显示英文标题

标题： On the Laplacian spectra of token graphs

摘要： We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. 特别是，我们证明对于任何整数$h$和$k$使得$1\le h\le k\le \frac{n}{2}$，图$F_h(G)$的拉普拉斯谱包含在图$F_k(G)$的拉普拉斯谱中。我们还证明双奇图和双约翰逊图可以分别作为完全图$K_n$和星形图$S_{n}=K_{1,n-1}$的标记图得到。 Besides, we obtain a relationship between the spectra of the $k$-token graph of $G$ and the $k$-token graph of its complement $\overline{G}$. This generalizes a well-known property for Laplacian eigenvalues of graphs to token graphs. Finally, the double odd graphs and doubled Johnson graphs provide two infinite families, together with some others, in which the algebraic connectivities of the original graph and its token graph coincide. Moreover, we conjecture that this is the case for any graph $G$ and its token graph. 显示英文摘要

摘要： We study the Laplacian spectrum of token graphs, also called symmetric powers of graphs. The $k$-token graph $F_k(G)$ of a graph $G$ is the graph whose vertices are the $k$-subsets of vertices from $G$, two of which being adjacent whenever their symmetric difference is a pair of adjacent vertices in $G$. In this paper, we give a relationship between the Laplacian spectra of any two token graphs of a given graph. In particular, we show that, for any integers $h$ and $k$ such that $1\le h\le k\le \frac{n}{2}$, the Laplacian spectrum of $F_h(G)$ is contained in the Laplacian spectrum of $F_k(G)$. We also show that the double odd graphs and doubled Johnson graphs can be obtained as token graphs of the complete graph $K_n$ and the star $S_{n}=K_{1,n-1}$, respectively. Besides, we obtain a relationship between the spectra of the $k$-token graph of $G$ and the $k$-token graph of its complement $\overline{G}$. This generalizes a well-known property for Laplacian eigenvalues of graphs to token graphs. Finally, the double odd graphs and doubled Johnson graphs provide two infinite families, together with some others, in which the algebraic connectivities of the original graph and its token graph coincide. Moreover, we conjecture that this is the case for any graph $G$ and its token graph.