Computation of Topological Indices of Mesh , Grid , Torus and Cylinder

In this paper, we compute ABC index, Randic connectivity index, Sum connectivity index and GA index of grid, extended grid, torus and cylinder. Mathematics Subject Classification: 05C12, 05C90


Introduction
Topological indices are the molecular descriptors that describes the structures of chemical compounds and it help us to predict certain physico-chemical properties like boiling point, enthalpy of vaporization, stability etc.In this paper, we determine the topological indices like atom-bond connectivity index, sum connectivity index, randic connectivity index and geometricarithmetic connectivity index of mesh, grid, torus and cylinder.
All molecular graphs considered in this paper are finite, connected, loop less and without multiple edges.Let G = (V, E) be a graph with n vertices and m edges.The degree of a vertex u ∈ V (G) is denoted by d u and is the number of vertices that are adjacent to u.The edge connecting the vertices u and v is denoted by uv.Using these terminologies, certain topological indices are defined in the following manner.
The Atom-bond connectivity index, ABC index is one of the degree based molecular descriptor, which was introduced by Estrada et al. [5] in late 1990's and it can be used for modeling thermodynamic properties of organic chemical compounds, it is also used as a tool for explaining the stability of branched alkanes [6].Some upper bounds for the atom-bond connectivity index of graphs can be found in [2], The atom-bond connectivity index of chemical bicyclic graphs, connected graphs can be seen in [3,16].For further results on ABC index of trees see the papers [9,11,15,17] and the references cited there in.Definition 1.1.Let G = (V, E) be a molecular graph and d u is the degree of the vertex u, then The first and oldest degree based topological index is Randic index [13] denoted by χ(G) and was introduced by Milan Randic in 1975.It provides a quantitative assessment of branching of molecules.
Sum connectivity index belongs to a family of Randic like indices and it was introduced by Zhou and Trinajstic [19].Further studies on Sum connectivity index can be found in [20,21].Definition 1.3.For a simple connected graph G, its sum connectivity index S(G) is defined as, . The Geometric-arithmetic index, GA(G) index of a graph G was introduced by D. Vukicevic et.al [14].Further studies on GA index can be found in [1,4,18] Definition 1.4.Let G be a graph and e = uv be an edge of G then Geometric-arithmetic index is defined as, 2 Main results if m > 2 and n > 2 Proof.The topological structure of a grid network, denoted by G(m, n), is defined as the cartesian product P m × P n of undirected paths P m and P n .The spectrum of the graph does not depend on the numbering of the vertices.However, here we adopt a particular numbering such that the edges has a pattern which is common for any dimension.We follow the sequential numbering from left to right as shown in the diagram.Row 2 Case 3 : In this case the number of e 2,2 edges is as shown in Figure 3.
Theorem 2.2.Randic index of grid with (m−1) rows and (n−1) columns is given by χ(G(m, n)) Case 2 : Theorem 2.3.Sum Connectivity index of Grid with (m − 1) rows and (n − 1) columns in each row is given by S(G(m, n)) Theorem 2.4.Geometric-arithmetic(GA) index of Grid with (m − 1) rows and (n − 1) columns in each row is given by GA(G(m, n)) Theorem 2.5.Atom bond connectivity index of Extended Grid with (m − 1) rows and (n − 1) columns in each row is given by ABC if m and n = 2 Proof.By making each 4-cycle in a m × n mesh into a complete graph we obtain an architecture called an extended mesh denoted by EX(m, n).The number of vertices in EX(m, n) is mn and the number of edges is 4mn + 3m + 3n + 2. We follow the sequential numbering from left to right.
Consider a extended grid with (m−1) rows and (n−1) columns.Let e i,j denotes the number of edges connecting the vertices of degrees d i and d j .Two-dimensional structure of extended grid as shown in the Figure 4 and it contains e 3,5 , e 3,8 , e 5,5 , e 5,8 and e 8,8 edges.In the above figure e 3,5 , e 3,8 , e 5,5 , e 5,8 and e 8,8 edges are colored in red, purple, green, yellow and black respectively.The number of edges of these types in each row is mentioned in the following table - Consider a two-dimensional structure of Grid with (m − 1) rows and (n − 1) columns as shown in the Figure-1.Let |e i,j | denotes the number of edges connecting the vertices of degrees d i and d j .Case 1 : If m > 2 and n > 2, Grid contains only e 2,3 , e 3,3 , e 3,4 and e 4,4 edges.In the above figure e 2,3 , e 3,3 , e 3,4 and e 4,4 edges are colored in red, blue, green and black respectively.The number of e 2,3 , e 3,3 , e 3,4 and e 4,4 edges in each row is mentioned in the following table.

6 . 2 :
Case If m = 2 and n > 2 In this case Grid contains e 2,2 , e 2,3 and e 3,3 edges.The edges e 2,2 , e 2,3 and e 3,3 are colored in red, blue and black respectively as shown in the Figure 2. The number of e 2,2 , e 2,3 and e 3,3 edges in each row is mentioned in the following table.

Case 2 :
If m = 2 and n > 2 In this case extended grid contains e 3,3 , e 3,5 and e 5,5 edges.The edges e 3,3 , e 3,5 and e 5,5 are colored in red, blue and black respectively as shown in the Figure 5.The number of e 3,3 , e 3,5 and e 5,5 edges in each row is mentioned in the following table.
Case 1 : The atom-bond connectivity index of extended grid for