Modified Generalized Degree Distance of Some Graph Operations

The modified generality degree distance, is defined as: ( ) { } ( ) ( ) ( ) ( ) ( ) * , , , G G u v V G H G d u v d u d v λ λ ⊆ = ∑ which is a modification of the generality degree distance. In this paper, we give some computing formulas of the modified generality degree distance of some graph operations, such as, composition, join, etc.


Introduction
Throughout this paper all graphs considered are finite and simple graphs.Let G be such a graph with vertex set to denote the cycle, path and complete graph on n vertices, respectively.Other terminology and notation will be introduced where it is needed or can be found in [1].
A Topological index of a graph is a real number related to the graph; it does not depend on labeling or pictorial representation of a graph.In chemistry, topological index is used for modeling physicochemical, pharmacologic, biological and other properties of chemical compounds [2].One of the oldest and

( )
W G , also termed as Wiener index in chemical or mathematical chemistry literature, which is defined [3] as the sum of distances over all unordered vertex pairs in G, namely, ( ) Dobrynin and Kochetova [4] and Gutman [5] introduced a new graph invariant with the name degree distance or Schultz molecular topological index, which is defined as follows: ( ) In [6], Gutman and Klavzar defined product-degree distance as follow: Note that the degree distance and product-degree distance are degree-weight versions of the Wiener index.We encourage the interested readers to consult [7] [8] [9] for Wiener index.In [4] [10] [11] [12] [13] [14], there are more conclusions for degree distance, which shows that the research of degree distance is a hot topic.In [15], Sagan et al. computed some exact formulae for the Wiener polynomial of various graph operations containing Cartesian product, composition, join, disjunction and symmetric difference of graphs, whose concepts will be presented in later part.In [16], Hamzeh et al. consider the generalized degree distances of some graph operations.The generalized degree distance of a graph is defined as follow [17]: For a real number λ , the modified generalized degree distance, denoted by , is also defined in [17]: , which implies that the generalized degree distance is equal to the degree distance (or Schultz index), and the modified generalized degree distance is equal to the product-degree distance.Therefore the study of this new topological index is important and we try to obtain some new results related to this topological index.
In this paper, we show that the explicit formulas for of some graph operations containing the composition, join, disjunction and symmetric difference of graphs, and we apply the results to compute the modified generality degree distance of some special graphs.
Next, we introduce four types of graph operations: The join V V × and ( ) , .
M G can be also expressed as a sum over edges of G, ( .

Main Results
The purpose of this section is to compute the modified generalized degree distance for four graph operations.We begin with the following crucial lemma related to distance properties of some graph operations.
Lemma 2.1.[18] [21] Let G and H be two graphs, then we have: 2) The join, composition, disjunction and symmetric difference of graphs are associative and all of them are commutative except from composition. 3) Proof.The parts 1) -5) are consequence of definitions and some well-known results of the book of Imrich and Klavzar [18].For the proof of 6) -10) we refer to [21].■ For a given graph i G , we denote i n and i m by the number of vertices and edges, respectively.Then we can obtain the modified generalized degree distance of the join graph G G + , we can partition the set of pairs of vertices of G G + into three subsets 1 2 , A A and 3 A .In 1 A , we collect all pairs of ver- tices u and v such that u is in 1 G and v is in 2 G .Hence, they are adjacent in G G + .The sets 2 A and 3 A are the set of pairs of vertices u and v which are in 1 G and 2 G , respectively.Therefore, we can partition the sum in the formu- la of ( ) By 3) and 7) of Lemma 2.1, we have ( ) ( ) We can observe that ( ) Next, we compute the exact formula for the modified generalized degree distance of the composition of two graphs.Before starting the discussion, we first denote by ( ) , , .

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The complement of G, denoted by G , is the graph with vertex set ( ) V G , in which two distinct vertices are adjacent if and only if they are not adjacent in G, and denoted by m the value of ( ) E G .We use n C , n P and n K

Figure 1 .
Figure 1.The graphs first present the following four sums:

Figure 2
Figure 2. The graphs

n d x d y d u d v n d u d v d x d y d x d y d u d
nd u d x d u n d y nd v d y d v n d x nd u d x d u