The Graphs Whose Permanental Polynomials Are Symmetric

The permanental polynomial $ \pi (G,x) = \Sigma_{i=0}^n b_i x^{n − i} $ of a graph G is symmetric if $ b_i = b_n−i $ for each $i$. In this paper, we characterize the graphs with symmetric permanental polynomials. Firstly, we introduce the rooted product $ H(K) $ of a graph $ H $ by a graph $ K $, and provide a way to compute the permanental polynomial of the rooted product $ H(K) $. Then we give a sufficient and necessary condition for the symmetric polynomial, and we prove that the permanental polynomial of a graph $ G $ is symmetric if and only if $ G $ is the rooted product of a graph by a path of length one.