Abbr:J.Comp.&Math.Sci.

2014, Vol.5(4): Pg.352-356

An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org

On the Hyper-Wiener Index of Graph Amalgamation

Shigehalli V. S^{1}, D. N. Misale^{2} and Shanmukh Kuchabal^{3}

1Senior Associate Professor Head, Dept. of Physics

Bhaurao Kakatkar College, Belgaum, INDIA.

2Professor

Chairman and Head of the Department

3Dept. 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
index WW(G) is defined as WW(G)= ^{ଵ}

ଶ W(G)+ ^{ଵ}

ଶ ∑d^{2}(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 K_{p}, Amalgamation of cyclic graph C_{4}.

1. INTRODUCTION

We have three methods for calculation of the Hyper-Wiener Index of molecular graphs.

(i) Distance Formula:

WW(G)= ^{ଵ}

ଶ(∑, + ∑d^{2}(u,v) )
(ii) Cut Method:

(iii) The Method of Hosaya Polynomials:

1.1 Cut Method

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 G_{1}(C) and G_{2}(C).

Let n_{1}(C) and n_{2}(C) be the number of
vertices of G_{1}(C) and G_{2}(C), respectively.

Then Wiener index of G can be calculated as:

W(G)= ∑ n_{1}(C_{i})n_{2}(C_{i}) (1)
^{i }

where the summation goes over all elementary cuts of G.

The hyper-Wiener index of a benzenoid graph G can be written as:

WW(G)=W(G)+WW*(G) (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 C_{1}, C_{2} to WW*(G) is
ab+cd in the first case and ab in the second
one.

Figure(a)

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.

In this paper we present two main Theorems. The first Theorem proved based on distance formula and second theorem by cut method.

2. MAIN RESULT

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) = ^{ଵ}

ଶ{6n^{2}-10n⎪Kp⎪+ 4⎪_{}^{ଶ}⎪-

⎪Kp⎪(n+1)+2n}

Where n- number of vertices in graph G K_{p}-
Complete 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):

W(G)=^{ଵ}

ଶ∑d(u,v)

=^{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)]}

2⎪Kp⎪-2 times [n-(2⎪Kp⎪-1)] times 2⎪Kp⎪-2 times [n-(2⎪Kp⎪-1)] times

= ^{ଵ}

ଶ {[⎪Kp⎪-1]3+ [n-(2⎪Kp⎪-1)]3+………+ [⎪Kp⎪-1]3+ [n-(2⎪Kp⎪-1)]3} + (n-⎪Kp⎪) times

ଵ

ଶ {[2⎪Kp⎪-2] + [n-(2⎪Kp⎪-1)] 2+………...+ [2⎪Kp⎪-2] + [n-(2⎪Kp⎪-1)]2}

⎪Kp⎪ times

= ^{ଵ}

ଶ {(n-⎪Kp⎪) [⎪Kp⎪-1]3+ (n-⎪Kp⎪) [n-(2⎪Kp⎪-1)]3}+ ^{ଵ}

ଶ {[2⎪Kp⎪-2] + [n-(2⎪Kp⎪-1)]

2}⎪Kp⎪

W(G) = ^{ଵ}

ଶ{n(3n-3⎪Kn⎪) + ⎪Kn⎪(⎪Kn⎪-n)}---(i) To find WW*(G):

WW*(G)= ^{ଵ}

ଶ ∑d^{2}(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

= ^{ଵ}

ଶ {[⎪Kp⎪-1]+ [n-(2⎪Kp⎪-1)]3+………+ {[⎪Kp⎪-1]+ [n-(2⎪Kp⎪-1)]3+

n-⎪Kp⎪ times

[n-(2⎪Kp⎪-1)]+ [n-(2⎪Kp⎪-1)]+………+ [n-(2⎪Kp⎪-1)]}

⎪Kp⎪times

WW*(G)= ^{ଵ}

ଶ {[⎪Kp⎪-1]( n-⎪Kp⎪)+ [n-(2⎪Kp⎪-1)]3 (n-⎪Kp⎪)+ ⎪Kp⎪ [n-(2⎪Kp⎪- 1)]}..(ii)

Since WW(G) = W(G)+WW*(G) Combining (i) and (ii) gives

WW(G) = ^{ଵ}

ଶ{6n^{2}-10n⎪Kp⎪+ 4⎪_{}^{ଶ}⎪-⎪Kp⎪(n+1)+2n}

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
WW(G)=∑^{ሺିଶሻ, ଶ}ୀଶ, ୀ(ିଶ) + ^{ଵ}

ସn^{2} +

2∑

మିଵ, ଵ

ୀଵ, ୀ^{}_{మ}ି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

W(G)=∑^{ିଶ, ଶ}ୀଶ, ୀିଶ. +^{ଵ}

ସ n^{2} (i)
(ii) To find WW*(G):

Since there are C1C2, C1C3,………Ci- 1Ci distinct pair cuts in WW*(G)

WW*(G)= C1C2+C1C3+ ………+Ci-1Ci.

= (^{}

ଶ− 1)+ (^{}

ଶ− 1)+2(^{}

ଶ− 2)+2(^{}

ଶ− 2)+…..+

(^{}

ଶ− 1)+ (^{}

ଶ− 1) + 2[2+4+6+…+(n-4)]

+4[2+4+6+…..(n-6)]+……….+

(n-4)2=2{1(^{}

ଶ− 1)+2(^{}

ଶ− 2)+ 3(^{}

ଶ− 3)…..+

(^{}

ଶ− 1)1}+ 2 ∑^{ିସ}_{}_{భ}_{ୀଶ}_{ଵ} + 4 ∑^{ି}_{}_{మ}_{ୀଶ}_{ଶ} +….+(n-
4) ∑^{ଶ}_{}_{}_{ୀଶ}_{}

WW*(G)= 2∑

మିଵ, ଵ

ୀଵ, ୀ^{}_{మ}ିଵ + 2 ∑^{ିସ}_{}_{భ}_{ୀଶ}_{ଵ} +
4 ∑^{ି}_{}_{మ}_{ୀଶ}_{ଶ} +….+(n-4) ∑^{ଶ}_{}_{}_{ୀଶ}_{} (ii)

Since WW(G) = W(G)+WW*(G) Combining (i) and (ii) gives

WW(G)=∑^{ሺିଶሻ, ଶ}ୀଶ, ୀ(ିଶ) + ^{ଵ}

ସn^{2} +

2∑

మିଵ, ଵ

ୀଵ, ୀ^{}_{మ}ିଵ + 2 ∑^{ିସ}_{}_{భ}_{ୀଶ}_{ଵ} + 4 ∑^{ି}_{}_{మ}_{ୀଶ}_{ଶ}
+….+(n-4) ∑^{ଶ}_{}_{}_{ୀଶ}_{}

Illustrations:

n=8, WW(G)=96 n=10, WW(G)=205

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

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).