Degree Distance and Reverse Degree Distance of one Tetragonal Carbon Nanocones

A molecular graph is a simple graph such that its vertices correspond to the atoms and the edges to the bonds. A path of a length n in a graph G is a sequence of n +1 vertices such that from each of these vertices there is an edge to the next vertex in the sequence. Let G be a molecular graph, with the vertex and edge sets of which are represented by V(G) and E(G), respectively. The distance , d(u,v) is defined as the length of the shortest path between u and v in G. D(u) denotes the sum of distances between u and all other vertices of G. For a given vertex u of V(G) its eccentricity, ecc(u) is the largest distance between u and any other vertex v of G. The maximum eccentricity over all vertices of G is called the diameter of G and denoted by Diam(G) and the minimum eccentricity among the vertices of G is called radius of G and denoted by R(G).

,where deg (u) is the degree of u and is the sum of all distances from the vertex u.The reverse degree

INTRODUCTION
A molecular graph is a simple graph such that its vertices correspond to the atoms and the edges to the bonds.A path of a length n in a graph G is a sequence of n +1 vertices such that from each of these vertices there is an edge to the next vertex in the sequence.Let G be a molecular graph, with the vertex and edge sets of which are represented by V(G) and E(G), respectively.The distance , d(u,v) is defined as the length of the shortest path between u and v in G. D(u) denotes the sum of distances between u and all other vertices of G.For a given vertex u of V(G) its eccentricity, ecc(u) is the largest distance between u and any other vertex v of G.The maximum eccentricity over all vertices of G is called the diameter of G and denoted by Diam(G) and the minimum eccentricity among the vertices of G is called radius of G and denoted by R(G).
Research into carbon nanocones (CNC) started almost at the same time as the discovery of carbon nanotubes (CNT) in 1991.Ball studied the closure of (CNT) and mentioned that (CNT) could sealed by a conical cap 1 .The official report of the discovery of isolated CNC was made in 1994, when Ge and Sattler reported their observations of carbon nanocones mixed together with tubules and a flat graphite surface 2 .These are constructed from a graphene sheet by removing a 120° wedge and joining the edges produces a cone with a single tetragonal defect at the apex.
Topological indices are graph invariants and are used for Quantitives Structure-Activity Relationship (QSAR) and Quantitives Structure-Property Relationship (QSPR ) studies 3,4 .The Wiener index of a graph G denoted by W(G) is defined W(G)=½ uv(G) Ð(u).The parameter DD(G) is called the degree distance of G and it was introduced by Dobrynin and Kochetova 5 and Gutman 6 as a graph theoretical descriptor for characterizing alkanes; it can be considered as a weighted version of the Wiener index.It is defined as When G is a tree on n vertices, it has been demonstrated the Wiener index and degree distance are closely related by DD(G)=4W(G)-n(n-1).The reverse degree distance of the graph G is defined as r DD(G)=2q (p-1) Diam (G) -DD (G),where p,q are the number of vertices and the number of edges of G, respectively.Some properties of the reverse degree distance, especially for trees, have been given in 7,8 .There are two reasons for the study of this graph invariant.One is that the reverse degree distance itself is a topological index satisfying the basic requirement to be a branching index and with potential for application in chemistry 8 .The other is the study of the reverse degree distance is actually the study of the degree distance, which is important in both mathematical chemistry and in discrete mathematics.
In this paper, we calculate the degree distance and reverse degree distance of one tetragonal carbon nanocones

RESULTS AND DISCUSSION
The aim of this section is to comput the degree distance and reverse degree distance of one To do this, the following lemmas are necessary.Lemma 1 In the following lemma, the diameter and radius of this nanocone are computed Lemma 2 there are k numbers of vertices and when , see Table 1.For computing the degree distance of one tetragonal carbon nanocones, at first we comput distance sum of all vertices.From Figure 2 (left to right) if u be t-th vertex of G with eccentricity equal to m then we denote its distance Now by using above relations alternatively this proof is completed.¢

Theorem 1
The degree distance of ] , where .Other vertices with the same distance sum are eight numbers.All vertices are of degree 3, except external vertices with eccentricity equal to 4n+2k, where their degrees are equal to 2, when 0 . See Table 2. Thus from Lemma 3 we have , where their degrees are 2, when and for n>6

3
sum by Distance sum of all vertices of C[n] is computed in the following lemma.Lemma Distance sum of all vertices of C[n] is computed by two relations as: u be a vertex of central tetragon with eccentricity equal to 2n+2 then for 1kn, 2k+1 numbers of vertices have distances equal to k and also 3n+2 numbers of vertices have distances equal the distance sum of 4 numbers of vertices is equals to

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These vertices are on the line L in Figure 2. Other vertices with the same distance sum are eight numbers.All vertices are of degree 3, except external vertices with eccentricity equal to