Graphs whose characteristic and permanental polynomials have coefficients of the same magnitude

Let ϕ(G,x)=∑i=0naixn−i and π(G,x)=∑i=0nbixn−i denote the characteristic polynomial and permanental polynomial of a graph GG. In this paper, we consider the family GG of graphs that have corresponding coefficients of the same magnitude, i.e.,  |ai|=|bi||ai|=|bi| for i=0,1,…,ni=0,1,…,n. We prove that the graphs in this family are planar graphs. To characterize the structures of the considered graphs, we introduce the plane ear decomposition. With the help of the plane ear decomposition, we show the characterizations of the bipartite and non-bipartite graphs in this family.