The Sum Degree Distance and the Product Degree Distance of Generalized Transformation Graphs Gab

In this contribution, we consider line splitting graph ) (G Ls of a graph G as transformation graph   G of ab G . We investigate the sum degree distance ) (G DD and product degree distance ) ( * G DD of transformation graph ab G , which are weighted version of Wiener index. The Transformation graphs of ab G are   G ,   G ,   G and   G .


Introduction
Throughout this paper, we consider finite, un-directed, simple, connected, r-regular graphs with vertex set } ,..., , , { = ) ( . For the undefined terminologies we refer [8]. The degree of vertex in a graph G is denoted by deg G (v) or d G (v) and the distance between two vertices i v and j v , denoted by .The degree of edge in a graph G is Topological indices and graph invariants based on distances between vertices of a graph are widely used in mathematical chemistry [2], which are due to their correlations with physical, chemical and thermodynamic parameters of chemical compounds.
One of the oldest and well studied distance based graph invariant associated with a connected graph G is the Wiener number W(G), also termed as Wiener index in chemical or mathematical chemistry literature, which is defined in [13] as the sum of distances over all unordered vertex pairs in G, i.e., Which was first time introduced by Wiener. Initially, the Wiener index W(G) was considered as a molecular structure descriptor used in chemical applications, but soon it attracted the interest of pure mathematicians [1,3,5,14,15].
Eventually a number of modifications of the Wiener index were proposed, which are as follows.
, ( ) (3) * , ( ) The graph invariants defined in (2) and (3) have all been much studied in the past. The invariant  DD was first time introduced by Dobrynin and Kochetova [4] and named as sum-degree distance. Later the same quantity was examined under the name "Schulz index" [7]. For mathematical research on degree distance see [9,12] and the references cited therein. A remarkable property of  DD is that in the case of trees of order n, the identity  DD =4W-n(n-1) holds [10]. Gutman [7] proved that the multiplicative variant of the degree distance, namely (2) from ., . , , obeys an analogous relation: * DD = 4W-(2n-1) (n-1). This latter quantity is sometimes referred to as the "Gutman index" [6], but here we call it product-degree distance . This concept was introduced by Kulli and Biradar in [11].

Generalized Transformation Graphs
respectively. We say that the associativity of  and  is +, if they are adjacent in G otherwise isand the associativity of  and   or  and  is +, if  is the neighborhood point of  or  is neighborhood point of  in G, otherwise is -.
Let ab be a 2-permutation of the set } , {   . We say that  and  corresponds to the first term a of ab, and  ,   E(G). Whereas  and   or  and  corresponds to the both first and second term of ab and ) ( , . The transformation graph ab G of a graph G is the graph with vertex set ) ( v  and  u  to  incident  not  are  which  edges  the  2 Proof. Let G be a (n,m)-graph with regular degree r, then 2. Let G be any (n, m) graph.
3. Let G be any (n, m) graph.

Bulletin of Mathematical Sciences and Applications Vol. 16
Theorem 3.3. For any (n, m) graph G with r ≥ 2, if r = 2 then Proof. Let G be any (n,m)-graph. From Proposition 3.1, G ++ contains 2m vertices and 1)) Applying observation A to the above equation, when r = 2, On simplification, we get (**) (*) and i.e., 2. Let G be any (n,m) graph.
Theorem 3.4. For any (n,m) graph G with r ≥ 2, when r = 2 2. Let G be any (n,m) graph.
2. Let G be any (n,m) graph.
3. Let G be any (n,m) graph.
Theorem 3.6. For any (n,m) graph G,

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On simplification, 2. Let G be any (n,m) graph.
Let G be any (n,m) graph.
Theorem 3.7. For any (n,m) graph G, From (2), we have ) , Applying observation E to the above equation, we get )
3. Let G be any (n,m) graph.
2. Let G be any (n,m) graph.
3. Let G be any (n,m) graph.