JASC: Journal of Applied Science and Computations ISSN NO: 1076-5131

COMPUTATION OF TOPOLOGICAL INDICES OF

m CENTRAL GRAPH OF DUTCH WINDMILL GRAPH D3 K. Rajam#, U.Mary * # Ph.D Scholar, Department of Mathematics, Nirmala College for Women Coimbatore, Tamilnadu-641049, India. *Associate Professor, Department of Mathematics, Nirmala College for Women Coimbatore, Tamilnadu-641049, India. Abstract: In this paper ,we compute Atom bond connectivity (ABC) index ,Fourth atom bond connectivity (ABC4) index, Sum connectivity index, Randic connectivity index, Geometric arithmetic(GA) index and Fifth geometric arithmetic (GA5) index of central graph of Dutch m windmill graph D3 . AMS Subject Classification: 05C12, 05C90

Keywords: ABC index, ABC4 index, GA index and GA5 index, Line graph, Randic connectivity index, Sum connectivity index. Corresponding author: [email protected] 1. Introduction: All graphs considered here are finite, connected, loopless and without multiple edges. Let G=(V,E) be a graph with n vertices and m edges. Here du represents the degree of a u∈V(G) and uv represents the edge connecting by the vertices u and v. The Dutch wind-mill m graph D3 is also called which is a graph obtained by taking m copies of the cycle C3 with a vertex in common whereas [4] explains more about Dutch wind-mill graph. For a given graph G = (V;E) we do an operation on G, by subdividing each edge exactly once and joining all the non adjacent vertices of G. The graph obtained by this process is called central graph [4] of G denoted by C(G).The work in computation of topological indices of Dutch wind- mill graph and windmill graphs are reported in [2,4]. A topological index also known as a connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph. Topological indices are numerical parameter of a graph which characterize its topology and are usually graph invariant. Further results about some topological indices are explained in [6,7,8].Using these findings we have proved the following results.

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Definition 1.1. Let G=(V,E) be a molecular graph and du is the degree of the vertex u,then ABC

푑푢+푑푣−2 index of G is defined as, ABC(G)=∑푢푣∈퐸(퐺) √ . 푑푢푑푣

Definition 1.2. Let G be a graph then its fourth ABC index is defined as 퐴퐵퐶4(퐺) =

푠푢+푠푣−2 ∑푢푣∈퐸(퐺) √ where su is the sum of the degrees of all neighbors of vertex u in G. 푠푢푠푣

i.e.,su=∑푢푣∈퐸(퐺) 푑푣,similarly for sv.

1 Definition 1.3. For the graph G, Randic index is defined as휒(퐺) = ∑푢푣∈퐸(퐺) . It gives a √푑푢푑푣 quantitative assessment of branching of molecules. Definition 1.4. For a simple connected graph G, its sum connectivity index S(G) is defined as 1 푆(퐺) = ∑푢푣∈퐸(퐺) .This index belongs to a family of Randic like Indices and it was √푑푢+푑푣 introduced by Zhou and Trinajstic. Definition 1.5. Let G be a graph and e= uv be an edge of G then Geometric Arithmetic index is

2√푑푢푑푣 defined as GA(G) = ∑푢푣∈퐸(퐺) and it was introduced by D.Vukicevic. 푑푢+푑푣

Definition 1.6. For a Graph G the Fifth Geometric arithmetic index is defined as 퐺퐴5(퐺) = 2√푠푢푠푣 ∑푢푣∈퐸(퐺) where Su is the sum of the degrees of all neighbors of the vertex u in G. 푠푢+푠푣 풎 2. Structural properties of 푪(푫ퟑ )

푚 푚 The central graph of 퐷3 , 퐶(퐷3 ) has

(i) The number of vertices = 3m (ii) The number of edges = m(2m+1) (iii) m number of vertices of degree 2 (iv) 2m number of vertices of degree 2m.

m 3. Some topological indices of central graph of D3 Theorem:3.1. The atom bond connectivity index of central graph of Dutch windmill graph is 3 m 1 ABC (C (D3 )) = [6푚 + (푚 − 1)2 ]. √2 m Proof: We partition the edges of C(D3 ) into edges of the type E(du,dv).The number of edges of the type E(2,2m) is 6m and E(2m,2m) is 2m(m-1).

푑푢+푑푣−2 Now ABC (G) =∑푢푣∈퐸(퐺) √ . 푑푢푑푣

m 푑푢+푑푣−2 푑푢+푑푣−2 i.e., ABC(C(D3 ))=|퐸2,2푚| ∑푢푣∈퐸(2,2푚) √ + |퐸2푚,2푚| ∑푢푣∈퐸(2푚,2푚) √ 푑푢푑푣 푑푢푑푣

Volume VI, Issue IV, April/2019 Page No:1999 JASC: Journal of Applied Science and Computations ISSN NO: 1076-5131

2+2푚−2 2푚+2푚−2 =6m √ +2m(m-1) √ 2(2푚) 2푚(2푚)

3 m 1 ∴ABC(C(D3 )) = [6푚 + (푚 − 1)2 ]. √2 Theorem 3.2. The Randic index of central graph of Dutch windmill graph is m 휒(C(D3 ))=3√푚+(m-1).

1 Proof: We know that 휒(퐺) = ∑푢푣∈퐸(퐺) √푑푢푑푣

m 1 1 i.e., 휒(C (D3 )) =|퐸2,2푚| ∑푢푣∈퐸(2,2푚) + |퐸2푚,2푚| ∑푢푣∈퐸(2푚,2푚) √푑푢푑푣 √푑푢푑푣

1 1 = 6m √ + 2m(m-1) √ 2(2푚) 2푚(2푚)

m ∴ 휒(C (D3 ) =3√푚+(m-1).

Theorem : 3.3. The Sum connectivity index of central graph of Dutch windmill graph is

m 3√2 푚−1 S(C (D3 ))=푚 [ + ]. √푚+1 √푚 1 Proof: We know that 푆(퐺) = ∑푢푣∈퐸(퐺) √푑푢+푑푣

m 1 1 i.e., S(C (D3 ))=|퐸2,2푚| ∑푢푣∈퐸(2,2푚) + |퐸2푚,2푚| ∑푢푣∈퐸(2푚,2푚) √푑푢+푑푣 √푑푢+푑푣

1 1 =6푚√ + 2푚(푚 − 1)√ 2+2푚 2푚+2푚

m 3√2 푚−1 ∴S(C (D3 )) = 푚 [ + ]. √푚+1 √푚 Theorem: 3.4. The Geometric arithmetic index of central graph of Dutch windmill graph is

m 2푚 2 GA(C (D3 )) = [푚 + 6√푚 − 1]. 푚+1

2√푑푢푑푣 Proof: We know that 퐺퐴(퐺) = ∑푢푣∈퐸(퐺) 푑푢+푑푣

m 2√푑푢푑푣 2√푑푢푑푣 i.e., GA(C(D3 ))=|퐸2,2푚| ∑푢푣∈퐸(2,2푚) + |퐸2푚,2푚| ∑푢푣∈퐸(2푚,2푚) 푑푢+푑푣 푑푢+푑푣

2 2(2푚) 2 2푚(2푚) =6푚 √ + 2m (m-1) √ 2+(2푚) 2푚+(2푚)

푚 =12m √ +2m (m-1) 푚+1

Volume VI, Issue IV, April/2019 Page No:2000 JASC: Journal of Applied Science and Computations ISSN NO: 1076-5131

m 2푚 2 ∴ GA (C (D3 )) = [푚 + 6√푚 − 1]. 푚+1 For finding the fourth atom bond connectivity index and fifth geometric arithmetic index for m central graph of Dutch windmill graph we take edge partition of (C (D3 )) into edges of the ∗ type 퐸푠푢푠푣 where uv is an edge and Su is the sum of the degrees of all neighbors of vertex u in G. i.e., Su = ∑푢푣∈퐸(퐺) 푑푣.similarly for Sv. The following is an edge partition based on degree sum of m neighbors of end vertices of each edge of (C (D3 )).

∗ ∗ 퐸(2(2푚),2(2푚))=2m ; 퐸(2(푚−1)2푚+2(2),2(2푚)) = 4m and ∗ 퐸(2(푚−1)2푚+2(2),2(푚−1)2푚+2(2)) = m(2m-1) . Theorem: 3.5. The fifth Geometric arithmetic index of central graph of Dutch windmill 4 graph is 퐺퐴 (퐶(퐷푚))=2m[푚 + √푚3 − 푚2 + 푚]. 5 3 푚2+1

2√푠푢푠푣 Proof: We know that 퐺퐴5(퐺) = ∑푢푣∈퐸(퐺) 푠푢+푠푣

∗ 2√푠푢푠푣 푚 ∗ 퐺퐴5(퐶(퐷3 ))= |퐸(2(2푚),2(2푚))| ∑푢푣∈퐸 + ((2(2푚),2(2푚))) 푠푢+푠푣

∗ 2√푠푢푠푣 ∗ |퐸(2(푚−1)2푚+2(2),2(2푚))| ∑푢푣∈퐸 + ((2(푚−1)2푚+2(2),2(2푚))) 푠푢+푠푣

∗ 2√푠푢푠푣 |퐸(2(푚−1)2푚+2(2),2(푚−1)2푚+2(2))| ∑푢푣∈퐸∗ . (2(푚−1)2푚+2(2),2(푚−1)2푚+2(2)) 푠푢+푠푣

( ) ( ) 2 2(2푚)2(2푚) 2√(2 푚−1 2푚+2 2 )2(2푚) 퐺퐴 (퐶(퐷푚))= 2푚 √ + 4푚 + 5 3 2(2푚)+2(2푚) 2(푚−1)2푚+2(2)+2(2푚)

2 (2(푚−1)2푚+2(2))(2(푚−1)2푚+2(2)) 2푚(푚 − 1) √ 2(푚−1)2푚+2(2)+2(푚−1)2푚+2(2)

√푚3−푚2+푚 =2m+2m(m-1)+8m 푚2+1 4 ∴ 퐺퐴 (퐶(퐷푚))=2m[푚 + √푚3 − 푚2 + 푚] 5 3 푚2+1 Theorem: 3.6. The fourth atom bond connectivity index of central graph of Dutch windmill m−1 4m3+2m m2−m graph is 퐴퐵퐶 (퐶(퐷푚)) = √ + √ + √8m2 − 8m + 6. 4 3 2 m2−m+1 2m2−2m+2

푠푢+푠푣−2 Proof: We know that 퐴퐵퐶4(퐺) = ∑푢푣∈퐸(퐺) √ 푠푢푠푣

푚 퐴퐵퐶4(퐶(퐷3 )) =

∗ 푠푢+푠푣−2 ∗ 푠푢+푠푣−2 ∗ ∗ |퐸(2(2푚),2(2푚))| ∑푢푣∈퐸 √ | + 퐸(2(2)+2(푚−1)2푚,2(2푚))| ∑푢푣∈퐸 √ ((2(2푚),2(2푚))) 푠푢푠푣 (2(2)+2(푚−1)2푚,2(2푚)) 푠푢푠푣

∗ 푠푢+푠푣−2 + |퐸(2(푚−1)2푚+2(2),2(푚−1)2푚+2(2))| ∑푢푣∈퐸∗ √ (2(푚−1)2푚+2(2),2(푚−1)2푚+2(2)) 푠푢푠푣

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2(2푚)+2(2푚)−2 2(2)+2(푚−1)2푚+2(2푚)−2 =2m√ + 4m √ (2(2푚)2(2푚)) 2(2)+2(푚−1)2푚(2(2푚))

2(푚−1)2푚+2(2)+2(푚−1)2푚+2(2)−2 + 2m(m-1)√ (2(푚−1)2푚+2(2))(2(푚−1)2푚+2(2))

m−1 4m3+2m m2−m ∴ 퐴퐵퐶 (퐶(퐷푚)) = √ + √ + √8m2 − 8m + 6. 4 3 2 m2−m+1 2m2−2m+2

Conclusion:

We found the analytic formula for ABC index,ABC4 index, Sum connectivity index, Randic connectivity index, GA index and GA5 index for central graph of Dutch windmill graph. References [1] Rajesh Kanna M.R., Pradeep Kumar R., Soner Nandappa.Computation of topological indices of windmill graph. International Journal of Pure and Applied Mathematics. Vol 119 No.1 2018, 89—98. [2] Rajesh Kanna M.R., Pradeep Kumar R., Jagadeesh. Computation of topological indices of Dutch windmill graph. Discrete Mathematics. 2016., Vol 6, 74—81. [3] Rajam And Pauline Mary Helen .M, On harmonious coloring of line graph of graph families. International Journal of Statistika and Mathematika Vol 7, 2013, 33—36.

[4] Sameerali pilathottathil And Uma Maheshwari.M, A Study on number of edges in the central 푚 graph of the Dutch–Windmill graph 퐷3 .International Journal of Engineering Technology Management and Applied Sciences.Vol2,2014, 270—275.

[5] Vernold Vivin .J, and K.Thilagavathi,On Harmonious Coloring of Line Graph of Central Graph of paths, Applied Mathematical sciences,Vol.3,2009,No.5,205—214. [6] R.Xing, B.Zhou, F.Dong, On atom bond connectivity index of connected graphs, Discrete Applied Mathematics.,Vol 159,2011,1617—1630. [7]L.Xiao, S.Chen, Z. Gun, Q.Chen, The Geometric –Arithmetic index benzenoidsystems and phenylenes, Int. J. Contemp.Math.Sciences.Vol 5, 2010, 2225—2230. [8] B.Zhou and N. Trinajstic, On general sum -connectivity index, J.Math . Chem., 2010,Vol 47, 210—218.

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