NORDHAUS – GADDUM TYPE RESULTS FOR WIENER LIKE INDICES OF GRAPHS

A Nordhaus Gaddum type result is a lower or upper bound on sum or product of a parameter of a graph and its complement. This concept was introduced in 1956 by Nordhaus E. A., Gaddum J. W. Generalized Wiener like indices such as wiener index, Detour index, Reciprocalwiener index, Hararywiener index, Hyperwiener index, ReciprocalDetour index, HararyDetour index and HyperDetour index have been studied in graph theory. In this paper, Nordhaus – Gaddum type results of these indices for k-Sun graph and four regular graph are presented.


Introduction 1:
All graphs considered in this paper are finite, simple and connected. For a graph G = (V, E) with vertices u, v ∈ V, the distance between u and v in G, denoted by dG (u, v), is the length of a shortest (u, v)path in G. The Wiener index [2,3,4] of G is defined as with the summation going over all pairs of distinct vertices of G. The above definition can be further generalized in the following way: (G)=  [1], reciprocal Wiener index and hyperwiener index(WW) [5,6]. The detour index of G is defined as with the summation going over all pairs of distinct vertices of G. The above definition can be further generalized in the following way: (G)= where k≥4 and is any real number.

Proof:
Let G be ksun graph on 2k vertices, where k≥4 and is any real number.
for any real number .
For the ksun graph, the generalized wiener polynomial, the generalized detour polynomial and generalized circular polynomial of ksun graph G are, The 4-sun graph and the wiener detour matrix are shown in The 5-sun graph and the wiener detour matrix are shown in For k =6, the corresponding wiener polynomial, detour polynomial and polynomial of 6sun graph given below, For k =7, the corresponding wiener polynomial, detour polynomial and polynomial of 7sun graph given below, .
x 20 : (   18  17  16  15  14  15  14  13  3 Hence in general, the generalized wiener polynomial, the detour polynomial and the generalized circular polynomial of ksun graph G are given by,

Theorem 2.2.4:
Let ̅ be the complement of ksun graph G. Then the generalized wiener polynomial and detour polynomial for ̅ are respectively given by: Let G be the ksun graph on 2k vertices. Let ̅ be the complement of ksun graph G. Figure 1.7. shows the complement graph ̅ of 5sun graph. The wiener detour matrix of the complement of 5 -Sun graph in Figure 1.8. gives the wiener polynomial and the detour polynomial of complement of 5sun graph.
Hence in general, the generalized wiener and detour polynomial of complement of ksun graph are respectively given by,

Corollary 2.2.5:
Let ̅ be the complement of ksun graph G, then

Corollary 2.2.6:
Let ̅ be the complement of ksun graph G, then

Nordhaus -Gaddum Equation for four regular graph:
In this section the generalized wiener polynomial and generalized detour polynomial of four regular graph and complement of four regular graph are presented and also Nordhaus -Gaddum equation for four regular graph is derived.

Algorithm for Four regular graph:
Input : the number of vertices n of a cyclic graph. Output : the class four regular graph with 2n vertices. Begin Step 1: Take a cycle Cn with vertex set V = {v1, v2,…vn} and e dge set E = {vivi+1 ∪ vnv1: 1≤i≤(n-1)}.
Hence in general, the generalized wiener polynomial of four regular graph G(Cn) is, Hence in general, the generalized wiener polynomial of four regular graph G(Cn) is  Let ̅ be the complement of four regular graph G. Then the generalized wiener polynomial and detour polynomial for ̅ are respectively given by:

Proof:
Let G be the four regular graph and ̅ be the complement of G.  : In similar mannar, when k =5, 6,7,8., the corresponding wiener polynomials of complement graph ̅ of four regular graph given below, : (  ;  45  ) : Hence in general, the generalized detour polynomial of complement of four regular graph is.

Conclusions:
In 1956, Nordhaus E. A., Gaddum J. W. [7] introduced the bounds involving the chromatic number (G) of a graph G and its complement. Many authors studied [8,9] Nordhaus-Gaddum bounds for domination number, connected domination number, total domination number and also there has been many publications on Nordhaus-Gaddum type results for indices like Gutman wiener index, Steiner index, Krichhoff index. This paper deals with Nordhaus -Gaddum equations for wiener like indices to ksun graph four regular graph.