Let be a connected graph with vertices set and edges set . The ordinary distance between any two vertices of is a mapping from into a nonnegative integer number such that is the length of a shortest path. The maximum distance between two subsets and of is the maximum distance between any two vertices and such that belong to and belong to . In this paper, we take a special case of maximum distance when consists of one vertex and consists of vertices, . This distance is defined by: where is the order of a graph .
In this paper, we defined – polynomials based on the maximum distance between a vertex in and a subset that has vertices of a vertex set of and – index. Also, we find polynomials for some special graphs, such as: complete, complete bipartite, star, wheel, and fan graphs, in addition to polynomials of path, cycle, and Jahangir graphs. Then we determine the indices of these distances.
