For 4k-intercyclic bipartite graphs, the permanental polynomial is the modified characteristic polynomial plus 4 times the sum of modified characteristic polynomials of all 4k-cycle deletions.
Enumeration of copermanental graphs
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abstract
Let $G$ be a graph and $A$ the adjacency matrix of $G$. The permanental polynomial of $G$ is defined as $\mathrm{per}(xI-A)$. In this paper some of the results from a numerical study of the permanental polynomials of graphs are presented. We determine the permanental polynomials for all graphs on at most 11 vertices, and count the numbers for which there is at least one other graph with the same permanental polynomial. The data give some indication that the fraction of graphs with a copermanental mate tends to zero as the number of vertices tends to infinity, and show that the permanental polynomial does be better than characteristic polynomial when we use them to characterize graphs.
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Computing the permanental polynomial of $4k$-intercyclic bipartite graphs
For 4k-intercyclic bipartite graphs, the permanental polynomial is the modified characteristic polynomial plus 4 times the sum of modified characteristic polynomials of all 4k-cycle deletions.