Further results on the permanental sums of bicyclic graphs

Let $G$ be a graph, and let $A(G)$ be the adjacency matrix of $G$. The permanental polynomial of $G$ is defined as $\pi(G,x)=\mathrm{per}(xI-A(G))$. The permanental sum of $G$ can be defined as the sum of absolute value of coefficients of $\pi(G,x)$. Computing the permanental sum is $\#$P-complete. Any a bicyclic graph can be generated from three types of induced subgraphs. In this paper, we determine the upper bound of permanental sums of bicyclic graphs generated from each a type of induced subgraph. And we also determine the second maximal permanental sum of all bicyclic graphs.