JOURNAL OF COMPUTER AND MATHEMATICAL SCIENCES An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org
ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(4): Pg.352-356
On the Hyper-Wiener Index of Graph Amalgamation Shigehalli V. S1, D. N. Misale2 and Shanmukh Kuchabal3 1
Senior Associate Professor Head, Dept. of Physics Bhaurao Kakatkar College, Belgaum, INDIA. 2 Professor Chairman and Head of the Department 3 Dept. of Mathematics, Rani Channamma University, Vidyasangama, Belagavi, INDIA. (Received on: July 17, 2014) ABSTRACT Let G be the graph. The Wiener Index W(G) is the sum of all distances between vertices of G, where as the Hyper-Wiener ଵ
ଵ
ଶ
ଶ
index WW(G) is defined as WW(G)= W(G)+ ∑d2(u,v). In this paper we prove results on Hyper-Wiener Index of Amalgamation of Compete Graph Kp with common vertex and Amalgamation of cyclic graph C4 with common edge. Keywords: Hyper-Wiener Index, Amalgamation of Compete Graph Kp, Amalgamation of cyclic graph C4.
1. INTRODUCTION
1.1 Cut Method
We have three methods for calculation of the Hyper-Wiener Index of molecular graphs.
The cut method is based on the results from Klavzar, Gutman and Mohar and the calculation of the hyper-Wiener index works for all partial cubes. A graph is a partial cube if it is isomorphic to an isometric subgraph of a hypercube. Let G be a benzenoid graph on vertices. Then an elementary cut C divides G into two components, say G1(C) and G2(C).
(i)
Distance Formula: ଵ WW(G)= ଶ(∑ , + ∑d2(u,v) ) (ii) Cut Method: (iii) The Method of Hosaya Polynomials:
Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)
353
D. N. Misale, et al., J. Comp. & Math. Sci. Vol.5 (4), 352-356 (2014)
Let n1(C) and n2(C) be the number of vertices of G1(C) and G2(C), respectively. Then Wiener index of G can be calculated as: W(G)= ∑ n1(Ci)n2(Ci)
(1)
i
where the summation elementary cuts of G.
goes
over
1.2 Definitions Graph Amalgamation: Let H be a graph then graph Amalgamation G obtained by joining two or more copies of the same graph H either with common vertex or with common edge.
all
The hyper-Wiener index of a benzenoid graph G can be written as:
In this paper we present two main Theorems. The first Theorem proved based on distance formula and second theorem by cut method.
WW(G)=W(G)+WW*(G)
2. MAIN RESULT
(2)
Where WW*(G) consists of a summation over all pairs of elementary cuts. Let C1 and C2 be two elementary cuts of a benzenoid graph G. There are two different cases, as shown in figure (a). With a, b, c and d we denote the number of vertices in the corresponding parts of G. Then the contribution of the pair C1, C2 to WW*(G) is ab+cd in the first case and ab in the second one.
Theorem 2.1 Let Kp be the Compete graph containing p vertices then graph Amalgamation G obtained by joining same graph Kp to each vertex of Kp with common vertex then its Hyper-Wiener index given by WW(G) = {6n2-10n⎪Kp⎪+ ଶ ⎪Kp⎪(n+1)+2n} ଵ
4⎪ ଶ ⎪-
Where n- number of vertices in graph G KpComplete graph containing P vertices ⎪Kp⎪- Cardinality of Compete graph Proof: To find Hyper-Wiener Index of graph we need to find following two parts To find W(G): ଵ W(G)=ଶ∑d(u,v) Figure(a) ଵ
=ଶ {[(1+1+…+1)+(2+2+…+2)+(3+3+…+3)+…. +(1+1+…+1)+(2+2+…+2)+(3+3+…+3)] ⎪Kp⎪-1 times
[n-(2⎪Kp⎪-1)] times
⎪Kp⎪-1 times
[n-(2⎪Kp⎪-1)] times
+[(1+1+….+1)+(2+2+…+2)+…………....+(1+1+.…+1)+(2+2+….+2)]}
Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)
D. N. Misale, et al., J. Comp. & Math. Sci. Vol.5 (4), 352-356 (2014)
2⎪Kp⎪-2 times [n-(2⎪Kp⎪-1)] times
354
2⎪Kp⎪-2 times [n-(2⎪Kp⎪-1)] times
= ଶ {[⎪Kp⎪-1]3+ [n-(2⎪Kp⎪-1)]3+…………………………+ [⎪Kp⎪-1]3+ [n-(2⎪Kp⎪-1)]3} + ଵ
(n-⎪Kp⎪) times ଵ {[2⎪Kp⎪-2] + [n-(2⎪Kp⎪-1)] 2+……………………………...+ [2⎪Kp⎪-2] + [n-(2⎪Kp⎪-1)]2} ଶ
⎪Kp⎪ times = ଶ {(n-⎪Kp⎪) [⎪Kp⎪-1]3+ (n-⎪Kp⎪) [n-(2⎪Kp⎪-1)]3}+ ଶ {[2⎪Kp⎪-2] + [n-(2⎪Kp⎪-1)] 2}⎪Kp⎪ ଵ
ଵ
W(G) = ଶ{n(3n-3⎪Kn⎪) + ⎪Kn⎪(⎪Kn⎪-n)}-----------------------------------------------------(i) ଵ
To find WW*(G): ଵ WW*(G)= ଶ ∑d2(u,v) ଵ
= ଶ{[(1+1+…..+1)+(3+3+..…+3)+…………………+(1+1+…...+1)+(3+3+…..+3)] + ⎪Kp⎪-1 times
[n-(2⎪Kp⎪-1)] times
⎪Kp⎪-1 times [n-(2⎪Kp⎪-1)] times
[(1+1+…+1)+………………………+(1+1+……+1)]} [n-(2⎪Kp⎪-1)] times
[n-(2⎪Kp⎪-1)] times
= ଶ {[⎪Kp⎪-1]+ [n-(2⎪Kp⎪-1)]3+………………………+ {[⎪Kp⎪-1]+ [n-(2⎪Kp⎪-1)]3+ ଵ
n-⎪Kp⎪ times [n-(2⎪Kp⎪-1)]+ [n-(2⎪Kp⎪-1)]+…………………+ [n-(2⎪Kp⎪-1)]} ⎪Kp⎪times WW*(G)= 1)]}..(ii)
ଵ ଶ
{[⎪Kp⎪-1]( n-⎪Kp⎪)+ [n-(2⎪Kp⎪-1)]3 (n-⎪Kp⎪)+ ⎪Kp⎪ [n-(2⎪Kp⎪-
Since WW(G) = W(G)+WW*(G) Combining (i) and (ii) gives Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)
355
D. N. Misale, et al., J. Comp. & Math. Sci. Vol.5 (4), 352-356 (2014)
WW(G) = ଶ{6n2-10n⎪Kp⎪+ 4⎪ ଶ ⎪-⎪Kp⎪(n+1)+2n} ଵ
Illustrations:
n=9, WW(G)=120
n=16, WW(G)=462
Lemma 2.2: In the following theorem there are ଶ number of elementary cuts present in W(G). Lemma 2.3: In the following theorem there are ସ (ଶ − 1) number of elementary cuts present in WW*(G). Theorem 2.4: Let H1,H2,……Hm be the cyclic graph containing four vertices then graph Amalgamation G obtained from joining H1,H2,……Hm with common edge then its Hyper-Wiener Index given by ଵ ሺିଶሻ, ଶ WW(G)=∑ୀଶ, ୀ(ିଶ) + ସn2 +
2∑ మ
ିଵ, ଵ మ
ୀଵ, ୀ ିଵ
ି
+ 2 ∑ିସ భ ୀଶ ଵ + 4 ∑మ ୀଶ ଶ
+….+(n-4) ∑ଶ ୀଶ Where i, j, ଵ , ଶ , … . are 2,4,6,8……… and k, l are 1,2,3,………
Proof: we write the proof using cut method, to find Hyper-Wiener Index of graph we need to find following two parts (i) To find W(G): W(G)=∑ 1(Ci)n2(Ci) =n1(C1)n2(C1)+n1(C2)n2(C2)+……… ….+n1(Ci)n2(Ci) =2(n-2)+4(n-4)+6 (n-6)+…………………+(n-2)2+ଶ ଶ ଵ
ଶ 2 W(G)=∑ିଶ, ୀଶ, ୀିଶ . +ସ n
(i)
(ii) To find WW*(G): Since there are C1C2, C1C3,……………Ci1Ci distinct pair cuts in WW*(G) WW*(G)= C1C2+C1C3+ ……………+Ci-1Ci. = (ଶ − 1)+ (ଶ − 1)+2(ଶ − 2)+2(ଶ − 2)+…..+
Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)
D. N. Misale, et al., J. Comp. & Math. Sci. Vol.5 (4), 352-356 (2014)
( − 1)+ ( − 1) + 2[2+4+6+…+(n-4)] ଶ ଶ +4[2+4+6+…..(n-6)]+………………….+ (n-4)2=2{1( − 1)+2( − 2)+ 3( − 3)…..+
− 1)1}+ 4) ∑ଶ ୀଶ
( ଶ
ଶ
ଶ
2 ∑ିସ భ ୀଶ ଵ
ଶ ∑ି మ ୀଶ ଶ
+ 4
+….+(n-
Since WW(G) = W(G)+WW*(G) Combining (i) and (ii) gives ሺିଶሻ, ଶ ୀ(ିଶ)
WW(G)=∑ୀଶ,
ିଵ, మ
WW*(G)= 2∑
ଵ
+ 2
ୀଵ, ୀ ିଵ మ ି 4 ∑మ ୀଶ ଶ +….+(n-4) ∑ଶ ୀଶ
∑ିସ భ ୀଶ ଵ
2∑ మ +
(ii)
356
ିଵ, ଵ మ
ୀଵ, ୀ ିଵ
ଵ ସ
+ n2 +
ି
+ 2 ∑ିସ భ ୀଶ ଵ + 4 ∑మ ୀଶ ଶ
+….+(n-4) ∑ଶ ୀଶ
Illustrations:
n=8, WW(G)=96 REFERENCES 1. Gordon Cash, Sandi Klavzar, Marko Petkovsek Three methods for calculation of the Hyper-Wiener Index of molecular graphs (2002). 2. H. B. Walikar, D. N. Misale “Resistance Distance Network” Thesis (2002). 3. H. B. Walikar, Smt. V. S. Shigehalli “Radius, Diameter and Other Graph
n=10, WW(G)=205 Theoretic Parameters Concerned with Distances in Graphs” Thesis (2002). 4. Sandi Klavzar, A Bird’s eye view of the cut method and a survey of its applications in chemical graph theory. 5. Shigehalli V. S. and Shanmukh Kuchabal, “Hyper-Wiener Index of Multi-Thorn Even Cyclic Graphs Using Cut-Method” (2014).
Journal of Computer and Mathematical Sciences Vol. 5, Issue 4, 31 August, 2014 Pages (332-411)