NEW FORMULA OF DEGREE DISTANCE INDEX FOR SOME COMPLEX NETWORKS

Mathematics plays an important role in various fields, one of them is graph theory. Graphs can be used to model many types of relations and processes in many domains such as solving problems related to mathematical chemistry by using topological indices. A topological index of a graph is a number that quantifies the structure of the graph. It is used for modeling the biological and chemical properties of molecules in QSPR (Qualitative Structure-Property Relationships) and QSAR (Qualitative Structure-Activity Relationships) studies. The Degree Distance index DD(G) is one of the important topological indices. In this paper, we are going to determine DD(G) for some complex graphs like: Star vertex’s graph (SV ), Star edge’s graph (SE), and Path’s graph (P).


INTRODUCTION
In mathematics and computer science, graph theory is a branch of discrete combinatorial mathematics that studies graphs. They are mathematical structures used to model the relationships between objects. They can be adopted to model several types of processes in transport are represented by vertices and links between these pages are represented by edges, Essalih [1].
The same approach can be used for social networks, Essalih [1]. In chemistry, molecules are represented by a molecular graph, whose vertices represent atoms and their edges correspond to chemical bonds , Estrada and Bonchev [2]. Some studies have used qualitative structureactivity relationships (QSAR) and qualitative structure-property relationships (QSPR) to define a mathematical relationship between the structure and activity (property) of a molecule, Essalih [1], Zeryouh, El Marraki and Essalih [3], Roy, Kar and Das [4]. Indeed, by using the notion of topological indices, they were able to predict and design chemicals that are more beneficial to health, Roy, Kar and Das [4], Essalih [1]. The concept of topological indices is based on numerical values encoding certain information relating to molecular structure, Essalih [1], Laghridat, Mounir and Essalih [5], Ediz [6], Balaban and Devillers [7]. These values make it possible to establish correlations between the structure of a molecule and its physicochemical properties or its biological activity Essalih [1], Balaban and Devillers [7].
Many types of topological indices have been introduced in theoretical chemistry to measure the topological properties of molecules, such as distance-based topological indices (e.g. Wiener index, Wiener Terminal index, and degree distance index ) or those based on the degree (for example Zagreb index and Randic index), Jamil [8], Wiener [9], Das, Xu, and Nam [10], Zeryouh, EL Marraki and Essalih [11].

PRELIMINARY NOTES
2.1. Graph theory. Let G be a connected graph G = (V, E) where E and V denote respectively the edge set and the vertex set of the graph G. The order of a graph G is its number of vertices |V | denoted by n and the size of graph, denoted by m, is its number of edges |E|.
For vertices u, v ∈ V (G) the distance between two vertices u and v, denoted by d(u, v) is the number of edges of the shortest path connecting these vertices in G.
We denote by deg(u) the degree of the vertex u which is the number of edges incident to u.
The degree distance index of a graph G was considered first in connection with certain chemical applications by Dobrynin and Kochetova [21] and Gutman [20], as a weighted version of the Wiener index, who named it the Schultz index. This name was adopted by most other authors, Dobrynin [22], Schultz and Schultz [23], Zhou [24].
Our objective is to exploit this tool to characterize some specific complex networks such as Star vertex's graph (SV ) , Star edge's graph (SE) and Path's graph (P). Our method relays on the parameter d u G (k) which represents the number of pairs of vertices of G that are at distance k from u (k ≤ D(G)). More precisely, by considering members of a complex network as a set of vertices V(G) and its relations as a set of edges E(G), d u G (1) will represent the degree of u, d u G (2) will represent the number of reachable vertices from the neighbors of u, etc. Laghridat, Mounir and Essalih [5], Essalih, El Marraki and Zeryouh [25]. This parameter has been used in Essalih, El Marraki and Zeryouh [25] to establish a new formula of the degree distance of a graph G. (1)

MAIN RESULTS
In Laghridat, Mounir and Essalih [5], we give a characterisation of some social networks by applying the formula (1). In this work, we improve our results by considering more complex social networks such as : Star vertex's graph (SV ), Star edge's graph (SE) and Path's graph (P).
4.1. The star vertex's graphs. The Star vertex's graph SV is a graph composed of N graphs G i of order n i , connected to each other by a vertex s (see Figure 1 (a)).
Theorem 2. let SV be a star vertex's graph with n vertices and m edges.The Degree Distance index of SV is defined as follows (i = j): With: Proof.  Then: Theorem 3. Let SE be a star edge's graph with n vertices and m edges. The Degree Distance index of SE denoted by DD(SE) is defined as follows (i = j): With: And:    With:

The Path's Graphs. A Path's graph
And: Proof.

PRACTICAL CASES
As complex networks application, we have calculated the degree distance index for some composed graphs, according to the results presented in Laghridat, Mounir and Essalih [5]. In this paragraph, we will calculate the Degree Distance index DD(G) for some specific complex graphs, that are connected to each others in different ways.
5.1. The Star vertex's graph : Star graphs. The Star vertex's graph is a graph composed by N users of the same social network connected to each other by a common friend (vertex).

Corollary 1. Let SS be a Star vertex's graph with n vertices and m edges composed by N same
Star's graph S i such that |V (S i )| = n s and |E(S i )| = m s , i ∈ {1, 2, 3, . . . , N} (see Figure 3). The graphs S i are connected by a common vertex s. We have: • For u ∈ V (S i ) a vertex such that d(u, s) =1 , i ∈ {1, 2, 3, . . . , N} : Moreover, the Degree Distance index of Star vertex's graph SS is :  Figure 4). The Fan graphs F i are connected by a common vertex s and a set of edges {s, u i }. We have: Moreover, the Degree Distance index of Star edge's graph SF is : Proof. By applying Lemma 4.3 and the Theorem 4.4.   Figure   5). The Star graphs S i are connected by a common vertex v i and a set of edges {v i , v i+1 } , i ∈ {1, 2, 3, . . . , N}. We have: • DD(S i ) = (n s − 1)(3n s − 4),

CONCLUSION
Nowadays, social networks have a significant impact on society's behavior and human thinking. To accompany the massive use of such communication models many disciplines are built all around. For example, social network analysis has been introduced to describe and characterize these networks. Thanks to many works, such analysis were formalized into graphs where the vertices represent the accounts and the edges represent the relationship between them. Our contribution provides topological properties to characterize some specific social networks. Indeed, by using topological indices we give the formulas to calculate the Degree Distance index of certain complex graphs connected by a vertex ( Star vertex's graphs) and others connected by an edge (Star edge's graphs) and those connected by a set of edges (Path's graphs).

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.