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Computation of Connectivity Indices of Kulli Path Windmill Graph TWMS J. App. Eng. Math. V.8, N.1a, 2018, pp. 178-185 COMPUTATION OF CONNECTIVITY INDICES OF KULLI PATH WINDMILL GRAPH V. R. KULLI1, B. CHALUVARAJU2, H. S. BOREGOWDA3, x (m) Abstract. The Kulli path windmill graph Pn+1 is the graph obtained by taking m ≥ 2 copies of the graph K1 + Pn for n ≥ 4 with a vertex K1 in common. In this paper, we determine Zagreb, hyper-Zagreb, sum connectivity, general sum connectivity, Randic connectivity, General Randic connectivity, atom-bond connectivity, geometric-arithmetic, harmonic and symmetric division deg indices of Kulli path windmill graph. Keywords: Topological indices; Degree based connectivity indices; Windmill graph and Kulli path windmill graph. AMS Subject Classification: 05C05, 05C07, 05C35. 1. Introduction Throughout this paper, we consider simple graphs which are finite, undirected without loops and multiple edges. Let G = (V; E) be a connected graph with vertex set V = V (G) and edge set E = E(G). The degree dG(v) of a vertex v is the number of vertices adjacent to v. The edge connecting the vertices u and v will be denoted by uv. For other undefined notations and terminologies from graph theory, the reader are referred to [7]. A molecular graph is a graph such that its vertices correspond to the atoms and the edges to the bonds. Chemical graph theory is a branch of Mathematical chemistry which has an important effect on the development of the chemical sciences. A single number that can be used to characterize some property of the graph of a molecular is called a topological index for that graph. There are numerous topological descriptors that have found some applications in theoretical chemistry, especially in QSPR/QSAR research. In [6], the first and second Zagreb indices were introduced to take account of the con- tributions of pairs of adjacent vertices. The first and second Zagreb indices of a graph P 2 P G are defined as M1(G) = v2V (G) dG(v) or M1(G) = uv2E(G)[dG(u) + dG(v)] and P M2(G) = uv2E(G)[dG(u)dG(v)]. In [11], Shirdel et al., introduced the first hyper Zagreb index HM1(G) of a graph G. P 2 This index is defined as HM1(G) = uv2E(G)[dG(u) + dG(v)] . In [4], the second hyper P 2 Zagreb index HM2(G) of a graph G is defined as HM2(G) = uv2E(G)[dG(u)dG(v)] . 1 Department of Mathematics, Gulbarga University, Gulbarga - 585 106, India. e-mail: [email protected]; ORCID: https://orcid.org/0000-0002-6881-5201. 2 Department of Mathematics, Bangalore University, J. B. Campus, Bangalore -560 056, India. e-mail: [email protected]; ORCID: https://orcid.org/0000-0002-4697-0059. 3 Department of Studies and Research in Mathematics, Tumkur University, Tumkur-571 103, India. e-mail: [email protected]; ORCID: https://orcid.org/0000-0002-8887-5156. x Manuscript received: November 11, 2016; accepted: December 21, 2016. TWMS Journal of Applied and Engineering Mathematics, Vol.8, No.1a, c I¸sıkUniversity, Department of Mathematics, 2018; all rights reserved. 178 COMPUTATION OF CONNECTIVITY INDICES OF KULLI PATH WINDMILL GRAPH 179 The Randic index or product connectivity index of a graph G is defined as χ(G) = P 1 uv2E(G) p . This topological index was proposed by Randic in [10]. dG(u)dG(v) P 1 The sum connectivity index of a graph G is defined as X(G) = uv2E(G) p . dG(u)+dG(v) This topological index was proposed by Zhou and Trinajstic in [14]. The general Randic connectivity index or second Ka index of a graph G is defined as a P a χ (G) = uv2E(G)[dG(u)dG(v)] . The general sum connectivity index or first Ka index of a P a a graph G is defined as X (G) = uv2E(G)[dG(u) + dG(v)] . The above two topological indices were proposed in [1], [6] and [8]. In [2], Estrada et al. introduced the atom-bond connectivity index, which is defined as q ABC(G) = P dG(u)+dG(v)−2 . uv2E(G) dG(u)dG(v) p 2 d (u)d (v) The geometric-arithmetic index of a graph G is defined as GA(G) = P G G . uv2E(G) dG(u)+dG(v) This index was proposed by Vukicevic and Furtula in [12]. The harmonic index of a graph G is defined on the arithmetic mean as H(G) = P 2 : uv2E(G) dG(u)+dG(v) This index was first appeared in [3]. In [13], Vukicevic and Gasperov posed the symmetric division deg index of a graph G, which is defined as 2 2 X max (dG(u); dG(v)) min (dG(u); dG(v)) X dG(u) + dG(v) SDD(G) = + = : min(dG(u); dG(v)) max(dG(u); dG(v)) dG(u)dG(v) uv2E(G) uv2E(G) (m) The Kulli path windmill graph Pn+1 is the graph obtained by taking m ≥ 2 copies of the graph K1 + Pn for n ≥ 4 with a vertex K1 in common. This graph is shown in Figure-1. (m) (m) The Kulli path windmill graph P2+1 is a friendship graph and it is denoted by F3 . The (m) Kulli path windmill graph P3+1 is the first Kulli path windmill graph. For more details (m) (m) on french windmill graph Fn and Kulli cycle windmill graph Cn+1, refer to [5] and [9], (m) respectively. In this paper, we consider only the Kulli path windmill graphs Pn+1 for m ≥ 2 and n ≥ 4. (m) Figure 1. Kulli path windmill graph Pn+1. 180 TWMS J. APP. ENG. MATH. V.8, N.1A, 2018 2. Results Theorem 2.1. The sum connectivity index of Kulli path windmill graph is 2 3 2 2 X(P (m)) = p − p + p − p m n+1 5 6 mn + 2 mn + 3 1 1 + p + p mn: 6 mn + 3 (m) (m) Proof. Let G = Pn+1, where Pn+1 is a Kulli path windmill graph. By algebraic method, we have jV (G)j = mn + 1 and jE(G)j = 2mn − m. We have three partitions of the vertex set V (G) as follows: V2 = fv 2 V (G): dG(v) = 2g; jV2j = 2m; V3 = fv 2 V (G): dG(v) = 3g; jV3j = mn − 2m; and Vmn = fv 2 V (G): dG(v) = mng; jVmnj = 1: Also we have four partitions of the edge set E(G) as follows: E5 = fuv 2 E(G): dG(u) = 2; dG(v) = 3g; jE5j = 2m; E6 = fuv 2 E(G): dG(u) = 3; dG(v) = 3g; jE6j = mn − 3m; Emn+2 = fuv 2 E(G): dG(u) = mn; dG(v) = 2g; jEmn+2j = 2m; and Emn+3 = fuv 2 E(G): dG(u) = mn; dG(v) = 3g; jEmn+3j = mn − 2m. Now X 1 X(G) = pd (u) + d (v) uv2E(G) G G X 1 X 1 X 1 = p + p + p 2 + 3 3 + 3 mn + 2 uv2E5 uv2E6 uv2Emn+2 X 1 + p mn + 3 uv2Emn+3 1 1 1 = p × 2m + p × (mn − 3m) + p × 2m 5 6 mn + 2 1 + p × (mn − 2m) mn + 3 2 3 2 2 = p − p + p − p m 5 6 mn + 2 mn + 3 1 1 + p + p mn: 6 mn + 3 Theorem 2.2. The general sum connectivity index of Kulli path windmill graph is a (m) a a a a X (Pn+1) = [2(5 ) − 3(6 ) + 2(mn + 2) − 2(mn + 3) ] m + [6a + (mn + 3)a] mn: COMPUTATION OF CONNECTIVITY INDICES OF KULLI PATH WINDMILL GRAPH 181 (m) Proof. Let G = Pn+1 be a Kulli path windmill graph. Now a X a X (G) = [dG(u) + dG(v)] uv2E(G) X X = [2 + 3]a + [3 + 3]a uv2E5 uv2E6 X X + [mn + 2]a + [mn + 3]a uv2Emn+2 uv2Emn+3 = 5a × 2m + 6a × (mn − 3m) + (mn + 2)a × 2m + (mn + 3)a × (mn − 2m) = [2(5a) − 3(6a) + 2(mn + 2)a − 2(mn + 3)a] m + [6a + (mn + 3)a] mn: From the above Theorem, the following results are immediate (m) Corollary 2.1. The first Zagreb index of Pn+1 is m 2 M1(Pn+1) = (mn) + 9mn − 10m: (m) Corollary 2.2. The first hyper Zagreb index of Pn+1 is m 3 2 HM1(Pn+1) = (mn) + 6(mn) + 41mn − 68: Theorem 2.3. The Randic index of Kulli path windmill graph is "p p # 2 2 2 1 1 χ(P (m)) = p − 1 + p − p m + + p mn: n+1 3 mn 3mn 3 3mn (m) (m) Proof. Let G = Pn+1, where Pn+1 is a Kulli path windmill graph. Now X 1 χ(G) = pd (u)d (v) uv2E(G) G G X 1 X 1 X 1 = p + p + p 2 × 3 3 × 3 mn × 2 uv2E5 uv2E6 uv2Emn+2 X 1 + p mn × 3 uv2Emn+3 1 1 1 = p × 2m + p × (mn − 3m) + p × 2m 6 9 2mn 1 + p × (mn − 2m) 3mn "p p # 2 2 2 1 1 = p − 1 + p − p m + + p mn: 3 mn 3mn 3 3mn Theorem 2.4. The general Randic index of Kulli path windmill graph is a (m) a 2a+1 a+1 a a a a χ (Pn+1) = [2 × 6 − 3 + 2 (mn) − 2(3mn) ]m + [9 + (3mn) ]mn: 182 TWMS J. APP. ENG. MATH. V.8, N.1A, 2018 (m) Proof. Let G = Pn+1 be a Kulli path windmill graph. Now a X a χ (G) = [dG(u)dG(v)] uv2E(G) X X X = [2 × 3]a + [3 × 3]a + [mn × 2]a uv2E5 uv2E6 uv2Emn+2 X + [mn × 3]a uv2Emn+3 = 6a × (2m) + 9a × (mn − 3m) + (2mn)a × (2m) + (3mn)a(mn − 2m) = [2 × 6a − 32a+1 + 2a+1(mn)a − 2(3mn)a]m + [9a + (3mn)a]mn: From Theorem 2.4, we have the following results.
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