The Permanental Polynomial

Abstract

This study identifies properties and uses of the permanental polynomial of adjacency matrixes of unweighted chemical graphs. Coefficients and zeroes of the polynomial for several representative structures are provided, and their properties are discussed. A computer program for calculating the permanental polynomial from the adjacency matrix is also described.

Per-spectral characterizations of graphs with extremal per-nullity

“…Cash [5] pointed out that it is interesting to compare the coefficients and roots of the characteristic polynomial and permanental polynomial of a graph. Gutman and Cash [19] and Chen [9] obtained some relations between the coefficients of the permanental and characteristic polynomials of some chemical graphs, such as benzenoid hydrocarbons, fullerenes, and so on.…”

The Permanental Polynomials of Subdivision Graphs

“…Moreover, the computation of the transition amplitude of a quantum circuit can also be encoded as computing the permanent of a matrix [2]. In addition, the constant term of the permanental polynomial of a chemical graph enumerates the close-packed dimers (which is termed as perfect matchings in Mathematics) of a graph, and the coefficients and zeros of permanental polynomials are related to the stability and structure information of chemical graphs [3], [4]. Therefore, it is interesting and exciting to evaluate the permanental polynomials of graphs.…”

On the characterizing properties of the permanental polynomials of graphs

“…(2), one defines the permanental polynomial of G, π(G, x), as the permanent of the characteristic matrix of A(G), i.e., π(G, x) = per(xI − A(G)). (4) In what follows, in parallel to Eq. (3), we write the permanental polynomial in the coefficient form…”

“…Cash developed a computeraided method for the calculation of the permanental polynomials of molecular graphs, and applied it to a variety of benzenoid hydrocarbons [4] and fullerenes [5]. Recently, Belardo et al [1] gave some formulas which express the permanental polynomial of any square matrix in terms of the permanental polynomial of weighted digraphs.…”