Abstract

Let G be a graph and A(G) the adjacency matrix of G. The polynomial $\pi (G,x)=\mathit{per}(xI-A(G))$ is called the permanental polynomial of G. The permanental sum of G is the sum of the absolute values of the coefficients of $\pi (G,x).$ In this article, we investigate the permanental sum of the tree-type polyphenyl system. We give some inequalities about the permanental sum of tree-type polyphenyl system. And the largest and smallest permanental sums among the tree-type polyphenyl systems and the corresponding extremal graphs are determined. Furthermore, we determine, respectively, the upper and lower bounds of permanental sum of polyphenyl chains and polyphenyl spiders, and the corresponding extremal polyphenyl chains and polyphenyl spiders are determined.

摘要

设G为图，A ( G ) 为G的邻接矩阵。多项式$\pi (G,X)=\mathit{每}(X我-\mathrm{一个}(G))$称为G的永久多项式。G的永久和是系数的绝对值之和$\pi (G,X).$在本文中，我们研究了树型多苯系统的永久总和。我们给出了关于树型多苯系统的永久和的一些不等式。并确定了树型多苯系统中最大和最小的永久和以及对应的极值图。此外，我们分别确定了多苯链和多苯蜘蛛的永久总和的上限和下限，并确定了相应的极值多苯链和多苯蜘蛛。