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.
Cvetkovi´ c, Michael Doob, and Horst Sachs,Spectra of graphs: theory and application, Academic press (1979)
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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.