Mn – polynomials of general thorn path graph

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 “ M n – polynomial” of G which denoted by M n (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 M n –polynomial and Hosoya polynomial for general thorn path and obtained some results.