Mn-polynomials of some special graphs

Abstract

Let be a connected graph with vertices set V=V(G) and edges set E=E(G). The ordinary distance between any two vertices of V(G) is a mapping from into a nonnegative integer number such that d(ν,u) is the length of a shortest (ν-u) - path. The maximum distance between two subsets S and S of is V(G) the maximum distance between any two vertices and such that belong to S and u belong to S. In this paper, we take a special case of maximum distance when S consists of one vertex and S consists of (n-1) vertices, n≥3. This distance is defined by: dmax(ν-S)=max{d(ν,u):u ∊S}, | S| =n-1,3 ≤ n ≤p, ν ∊ V(G), ν ∉ S, where is the order of a graph G. In this paper, we defined Mn- polynomials based on the maximum distance between ν a V(G) vertex in and a subset S that has (n-1)- vertices of a vertex set of G and Mn- index. Also, we find Mn-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.