Some Results on Generalized Degree Distance

In [1], Hamzeh, Iranmanesh Hossein-Zadeh and M. V. Diudea recently introduced the generalized degree distance of graphs. In this paper, we present explicit formulas for this new graph invariant of the Cartesian product, composition, join, disjunction and symmetric difference of graphs and introduce generalized and modified generalized degree distance polynomials of graphs, such that their first derivatives at x = 1 are respectively, equal to the generalized degree distance and the modified generalized degree distance. These polynomials are related to Wiener-type invariant polynomial of graphs.


Introduction
A graph invariant is any function on a graph that does not depend on a labeling of its vertices.Topological indices and graph invariants based on the distances between the vertices of a graph are widely used in theoretical chemistry to establish relations between the structures and the properties of molecules.Topological indices provide correlations with physical, chemical and thermodynamic parameters of chemical compounds [2].In this paper, we only consider simple and connected graphs.Let G be a graph on n vertices and edges.We denote the vertex and the edge set of G by and , respectively.As usual, the distance between the vertices and of G, denoted by for short), is defined as the length of a minimum path connecting them.We let   G d v be the degree of a vertex in G.The eccentricity, denoted by , is defined as the maximum distance from vertex to any other vertex.The diameter of a graph G, denoted by diam G , is the maximum eccentricity over all vertices in a graph G.
The Cartesian product of graphs G and H is a graph such that and b is adjacent to , or and is adjacent to , see [3] for details.Let 1 and 2 be two graphs with disjoint vertex sets and 2 and edge sets and 2 .The join 1 2 is the graph union 1 2 together with all the edges joining and .The composition G G is the graph with vertex set 1 2 V V  and is adjacent to and is adjacent to 2 ), [3, p. 185].The disjunction of graphs G and and  is adjacent to The symmetric difference G of two graphs and The first Zagreb index was originally defined as u [4].The first Zagreb index can be also expressed as a sum over edges of


We refer readers to [5] for the proof of this fact and for more information on Zagreb index.The first Zagreb coindex of a graph G is defined in [6] as: be the number of pairs of vertices of a graph are at distance k ,  be a real number, and , th called the Wiener-at is type invariant of G associated to  , see [7,8 e ] for details.Additively weighted Harary ind x is defined in [9] as Dobrynin and Kochetova in [10] and Gutman in [11] in troduced a new graph invariant with the name degree distance that is defined as follows: In [12], the modified degree distance was defined as follows: is defined as follows in [1].
For every vertex x and real number  ,   , where egree distance (or Schultz i here are many papers for studying this topological index, for example see [13][14][15][16].Also if 1 he d Therefore the study of w topolo portant and we try to obtain some new results related to this topological index.The modified generalized degree distance, denoted by , is defined in [1] as: having the property such that their first derivatives at 1 x  are equal to the generalized degree distance, the m fied generalized degree distance and Wiener-type invariant respectively.These polynomials are defined as follows: The Wiener index of the Cartesian product of graphs w Gutman computed the Szeged index of the Cartesian omplete gra on rtices.Th , .

Main Results
Thes tion is to compute the generalized ve graph operations.We start with a The aim of this sec degree distance for fi lemma which gives some information about the number of vertices and edges of operations on two arbitrary graphs.For a given graph i G , the number of vertices and edges will be denoted by i n and i m , respectively.Lemma 2.1.[3,20] Let G and H be graphs.Then we have: a) 4 .
 tion and symmetric difference of graphs are associative , , ,

2, otherwise
or but not both .

2, otherwise
In Theorem 2.2, we give a formula for the generalized degree distance of the join of two graphs.
Let G be a connected graph with n vertices and m edges.Then The exact formulas

 
H G  for the fan graph K 1 + P n and for the wheel graph are given in the following Corollary.
So the proof of theorem is now completed.□ By composing paths and cycles with various small graphs we can obtain classes of polymer-like graphs.
ow we give the formula of the and Proof.According to definition of , we have the following relations: So we have: This completes the proof.□ Now we prove the theorem that characterizes the generalized degree distance of the symmetric difference of o graphs.
Theorem 2.10.Let G 1 and G 2 be two graphs.Then Proof.We consider four sums as follows: By the definition of , we have: .
In the next theorem we find the generalized degree distance of the Cartesian product of two graphs.Theorem 2.11.Let and be two graphs.Then Proof.Suppose and are two set of vertices of and , respectively.Then by Lemma 2.1 and definition of , , Corollary 2.13.By Theorem 2.9 and Lemma 2.12 we have: . , This completes the proof.□ invariant polynomial and the relatio n the modified generalized degree distance polynom al and Wiener-type invariant polynomial for gr Theorem 2.15.If with vertices an edges, then oof.By definition, we

16 .
If G is a graph with n vertices and m edges, then By definition, we It is obvious from definition that for any 14.In the above theorem, if 1