Some results on vertex-edge Wiener polynomials and indices of graphs

The vertex-edge Wiener polynomials of a simple connected graph are defined based on the distances between vertices and edges of that graph. The first derivative of these polynomials at one are called the vertex-edge Wiener indices. In this paper, we express some basic properties of the first and second vertex-edge Wiener polynomials of simple connected graphs and compare the first and second vertex-edge Wiener indices of them with each other. Also, we compute these polynomials and indices for some well-known graphs. Then, we study the relation between the vertex-edge Wiener polynomials of Cartesian product of graphs with the Wiener polynomial and vertex-edge Wiener polynomials of the primary graphs and apply the results to compute the vertex-edge Wiener indices of Cartesian product of graphs. As applications of these results, we present exact formulas for computing the first and second vertex-edge Wiener indices of rectangular grids, C4-nanotubes, C4-nanotori, Hamming graph, and hypercubes.