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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.

Let G = (V(G), E(G)) be any connected simple graph and S be any subset of V(G) such that the cardinality of S is n – 1, n ≥ 3. The maximum distance between a vertex ν, ν ∈ V(G) and S(S ⊆ V (G) is the maximum distance between ν and u for all u ∈ S such that the vertex ν is not belong to S, that is: d max ( v , S ) = max { d ( v , u ) : u ∈ S } , | S | = n − 1 , 3 ≤ n ≤ p , v ∉ S . . The maximum polynomial “Mn – polynomial” of G which denoted by Mn (G;x) and defined by: M n ( G ; x ) = ∑ k = m δ max ⁡ ( G , n ) C n ( G , k ) x K , , where m = min{d max (ν,S),ν ∈ V – S, S ⊆ V} and C n(G,K) be the number of pairs (ν,S), S ⊆ V(G), |S| = n – 1,3 ≤ n ≤ p, p = |V(G)| such that d max (ν,S) = k, for each m ≤ k ≤ δ max (G,n) and δ max (G,n) = max u∈V {d max (u,S)}. In this paper, we find the Mn –polynomial and Hosoya polynomial for general thorn path and obtained some results.

The basis number, b(G) ,of a graph G is defined to be the smallest positive integer k such that G has a k-fold basis for its cycle space. We investigate an upper bound for) (

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