Some Analytical Properties of the Permanental Polynomial of a Graph

Abstract

Let \(G\) be a graph and \(\pi(G, x)\) its permanental polynomial. A vertex-deleted subgraph of \(G\) is a subgraph \(G – v\) obtained by deleting from \(G\) vertex \(v\) and all edges incident to it. In this paper, we show that the derivative of the permanental polynomial of \(G\) equals the sum of permanental polynomials of all vertex-deleted subgraphs of \(G\). Furthermore, we discuss the permanental polynomial version of Gutman’s problem [Research problem \(134\), Discrete Math. \(88 (1991) 105–106\)], and give a solution.