图的$(H_1,H_2)$ -合并剖分图的谱
Spectra of $(H_1,H_2)$-merged subdivision graph of a graph
摘要：
本文定义了一种三元图运算，它推广了细分图、$R-$图、中心图的构造。此外，它推广了覆盖图的构造（Marius Somodi \emph{et al.}，2017），并因此推广了$Q-$图，全图和拟全图。我们用$[S(G)]^{H_1}_{H_2}$表示这个新的图，其中$G$是一个图，并且$H_1$和$H_2$是对应于$G$的合适的图。此外，我们定义了几个新的一元图操作，成为这种结构的特殊情况。本文给出了几类图$G$、$H_1$和$H_2$的$[S(G)]^{H_1}_{H_2}$的邻接谱和拉普拉斯谱.从这些结果出发，我们导出了由上述一元图运算得到的图的$L$ -谱。作为应用，这些结果使我们能够计算这些图的生成树的数目和Kirchhoff指数。
英文摘要：
In this paper, we define a ternary graph operation which generalizes the construction of subdivision graph, $R-$graph, central graph. Also, it generalizes the construction of overlay graph (Marius Somodi \emph{et al.}, 2017), and consequently, $Q-$graph, total graph, and quasitotal graph. We denote this new graph by $[S(G)]^{H_1}_{H_2}$, where $G$ is a graph and, $H_1$ and $H_2$ are suitable graphs corresponding to $G$. Further, we define several new unary graph operations which becomes particular cases of this construction. We determine the Adjacency and Laplacian spectra of $[S(G)]^{H_1}_{H_2}$ for some classes of graphs $G$, $H_1$ and $H_2$. From these results, we derive the $L$-spectrum of the graphs obtained by the unary graph operations mentioned above. As applications, these results enable us to compute the number of spanning trees and Kirchhoff index of these graphs.
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