ABSTRACT

The permanental polynomials of graphs received attention in mathematics and were investigated in chemistry at almost the same time. Like the characteristic polynomial of a graph, a Sachs-type theorem was obtained which expresses the coefficients of the permanental polynomial of a graph in terms of the graph structure. This chapter surveys some results on the permanental polynomials of graphs. It presents some methods on computing the permanental polynomial. The chapter provides properties of the roots of the permanental polynomial. It investigates the ability of determining graphs by the permanental polynomial. The relationships between the ordinary and Laplacian permanental polynomials were studied, and various formulas relating the ordinary and Laplacian permanental polynomials were evaluated for their efficiency as algorithm for calculating the Laplacian permanental polynomials of chemical graphs. The chapter concludes with brief discussions of the permanental polynomials of other graph matrices and the immanantal polynomials of graphs.