On the Differential Polynomial of a Graph

Abstract

We introduce the differential polynomial of a graph. The differential polynomial of a graph G of order n is the polynomial $$B(G;x):={\sum}_{k=-n}^{\partial(G)}B_k(G)x^{n+k}$$ , where Bk(G) denotes the number of vertex subsets of G with differential equal to k. We state some properties of B(G; x) and its coefficients. In particular, we compute the differential polynomial for complete, empty, path, cycle, wheel and double star graphs. We also establish some relationships between B(G; x) and the differential polynomials of graphs which result by removing, adding, and subdividing an edge from G.