Some T opological I nvariants of the Möbius Ladder

In this article we compute many topological indices for the family of Möbius ladder. At first we give general closed form of M-polynomial of this family and recover many degree based topological indices out of it. We also compute Zagreb indices and Zagreb polynomials of this family.


Introduction
A graph invariant is a number, a polynomial, or a matrix which uniquely represents the whole graph [2].Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism.Degree based topological indices are of great importance and play a vital role in chemical graph theory see [6,17,18].Wiener [15], working on boiling point of paraffin, introduced the idea of topological index.The Wiener index is originally the first and most studied topological index and is defined as the sum of distances between all pairs of vertices in G, for more details see [5,14,16].Zagreb indices, have been introduced 38 years ago by I. Gutman and N. Trinajstic [20].First Zagreb index M 1 (G) is defined as sum of the squares of degrees of a graph G and second Zagreb M 2 (G) is sum of the product of all degrees corresponding to each edge in G see [20].Second modified Zagreb index is defined by Where d(u) and d(v) are the degrees of vertices u and v respectively see [21].General Randic index of G is defined as sum of (d(u)d(v)) α over all edges uv of G, where d(u) denote the degree of vertex Where α is an arbitrary real number see [22].Symmetric division index is defined by Where d i is the degree of vertex i in Graph G.These indices can help to characterize the chemical and physical properties of molecules see [3,6,15,[17][18][19][21][22][23].Some standard degree based topological indices and the formulas how to compute them from M-polynomial.
The Möbius ladder M n which is a cubic circulant graph with an even number of vertices, formed from an n−cycle by adding edges (called "rungs") connecting opposite pair of vertices in the cycle.It is so-named because (with the exception of M 6 = K 3,3 ) M n has exactly n 2 4-cycles which link together by their shared edges to form a topological Möbius strip.Möbius ladders can also be viewed as a prism with one twisted edge.Two different views of Möbius ladders M n have been shown in Fig. 1.Möbius ladders have many applications in Chemistry, Chemical Stereography, Electronics and Computer Science.For our convenience, we view the Möbius ladder M n which is a cubic circulant graph with an even number of vertices, formed from an n−cycle by adding edges (called "rungs") connecting opposite pair of vertices in the cycle.d be the degrees of vertices in G.We partition the set of vertex V(G) and edge set E(G) of G as follows (∀i, jandk : δ ≤ i, j, k ≤ ): Ghorbani and Azimi defined two new versions of Zagreb indices of a graph G in 2012 [9].The first multiple Zagreb index PM 1 (G), second multiple Zagreb index PM 2 (G) and these indices are defined as: The properties of PM 1 (G), PM 2 (G) indices for some chemical structures have been studied in [4,7,9,11,22,25].The first Zagreb polynomial M 1 (G, x) and second Zagreb polynomial M 2 (G, x) are defined as: The properties of M1(G, x), M2(G, x) polynomials for some chemical structures have been studied in [10].Nowadays there is an extensive research activity on PM 1 (G), PM 2 (G), HM(G), indices, M 1 (G, x), M 2 (G, x) polynomials and their variants, see also [7][8][9]12,13,20,26].

Main Results
Theorem 2. Let M n be Möbius ladder.Then the M-polynomial of M n is M(M n , x, y) = 3nx 3 y 3 Proof.Let M n be Möbius ladder.From the structure of M n Möbius ladder we can see that only one partition By definition of M-polynomial, we can see edge set of M n can be partition as follows: Now we derive topological indices which are directly derivable from M-polynomial.
Theorem 3. Let M n be Möbius ladder.Then PM

16 Definition 1 .
Let G be a graph which is a simple molecular connected graph and where δ and are the minimum and maximum of degree of d v ∀v V(G) and δ = Min{d v |v V(G)} and = Max{d v |v V(G)}, respectively.Now, let G = (V, E) is a graph and let m ij be the number of degrees e = uv of G such that {d v (G), d u (G)} = {i, j}, then the M-polynomial of G define as follows: