On the Permanental Sum of the Tree-Type Polyphenyl System

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.

Keywords:

• Permanental polynomial
• permanental sum
• polyphenyl chain
• polyphenyl spider
• polyphenyl system
• tree-type polyphenyl system

Disclosure statement

The authors declare that there is no conflict of interests regarding the publication of this article.

Additional information

Funding

This work is supported by National Natural Science Foundation of China (NSFC) (No. 11761056), NSF of Qinghai (2020-ZJ-920), Ministry of Education Chunhui Project (No. Z2017047), Education department of Shaanxi province (15JK1135, 2016JK1141), Shaanxi university of technology (SLGQD14-14, SLGKY15-37) and the High-level Personnel of Scientific Research Project of QHMU (No. 2016XJG07).