On the Permanental Sum of the Tree-Type Polyphenyl System

Let G be a graph and A(G) the adjacency matrix of G. The polynomial pi(G, x) = 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 p(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.