Harmonic Status Index of Graphs

The status of a vertex u is defined as the sum of the distances between u and all other vertices of a graph G. In this paper we have defined the harmonic status index of a graph and obtained the bounds for it. Further the harmonic status indices of some graphs are obtained.


Introduction
Several distance based indices of a graph, such as Wiener index [24], distance energy [13,18], hyper Wiener index [21], Harary index [14,17] have been appeared in the literature. In this paper we introduce and study the new index called harmonic status index. Let G be a connected graph of order n and size m. Let V(G) be the vertex set and E(G) be the edge set of G. The edge between the vertices u and v is denoted by uv. The degree of a vertex u is the number of edges incident to it and is denoted by d (u). The distance between two vertices u and v, denoted by d (u, v), is the length of the shortest u-v path in G. The maximum distance between any pair of vertices in G is called the diameter of G and is denoted by diam(G). For graph theoretic terminology, we refer the books [1,2].
The status [11] of a vertex u  V(G) is defined as the sum of its distance from every other vertex in V(G) and is denoted by (u). That is, The Wiener index [24] W(G) of a connected graph G is defined as the sum of the distances between all pairs of vertices of G. That is, For more about the Wiener index one can refer [4,7,10,19,20,23]. The harmonic index of a graph G is defined as [8] Recent results on the harmonic index can be found in [3,5,12,15,16,22,25,26]. Inspired by this definition, we define here harmonic status index of a connected graph G as and obtain the bounds for the harmonic status index. Also we obtain the harmonic status index of some graphs. Further the correlation between the boiling point of paraffins and harmonic status index of the corresponding molecular graph is studied. For a graph given in Fig. 1, (u 1 ) = 5, (u 2 ) = 3, (u 3 ) = 4, (u 4 ) = 4. And

Bounds for the harmonic status index
In this section we obtain the bounds for the harmonic status index of graphs and characterize for the equality of these bonds. (1) Equality on both sides holds if and only if diam(G) ≤ 2.
Proof. Lower bound: For any vertex u of G, there are d(u) vertices which are at distance 1 from u and the remaining (n -1 -d(u)) vertices are at distance at most D. Therefore Upper bound: For any vertex u of G, there are d(u) vertices which are at distance 1 from u and the remaining (n -1 -d(u)) vertices are at distance at least 2. Therefore For equality: If the diameter D is 1 or 2 then the equality holds.
Conversely, let Suppose, D ≥ 3, then there exists at least one pair of vertices u and v such that d(u, v) ≥ 3. Therefore

Bulletin of Mathematical Sciences and Applications Vol. 17
Corollary 1.1. Let G be a connected graph with n vertices, m edges and diam(G) = D. Let  and  be the minimum and maximum degree of the vertices of G respectively. Then

Equality holds if and only if diam(G) ≤ 2.
Proof. For any vertex u of G, d(u) = r. Therefore the result follows from the Theorem 1.

Harmonic status index of some standard graphs
Proposition 2. For a complete graph K n on n vertices, HS(K n ) = n/2.
Proof. For any vertex u of K n , (u) = n -1. Therefore by the definition of harmonic status index, HS(K n ) = n /2.

Proposition 3. For a complete bipartite graph
Proof. The vertex set V(K p,q ) can be partitioned into two independent sets V 1 and V 2 such that for every edge uv of K p,q , the vertex u  V 1 and v  V 2 . Therefore d(u) = q and d(v) = p. The graph K p,q has n = p + q vertices and m = pq edges. Also diam(K p,q ) ≤ 2. Therefore by the equality part of Theorem 1, Proposition 4. For a path P n on n vertices, Proof. If n is even number then for any vertex u of C n , 4 2 2 If n is odd number then for any vertex u of C n , 4 ) 1 ( 2 A wheel W n+1 is a graph obtained from the cycle C n , n ≥ 3 by adding a new vertex and making it adjacent to all the vertices of C n . The degree of a central vertex of W n+1 is n and the degree of all other vertices is 3. Therefore by the equality part of Theorem 1,

Bulletin of Mathematical Sciences and Applications Vol. 17 27
A friendship graph (or Dutch windmill graph) F n , n ≥ 2, is a graph that can be constructed by coalescence n copies of the cycle C 3 of length 3 with a common vertex. It has 2n + 1 vertices and 3n edges. The degree of a coalescence vertex of F n is 2n and the degree of all other vertices is 2. Fig. 3

Harmonic status index of some graphs obtained from the complete graph
In this section we obtain the harmonic status index of the graphs, which were defined in [9].
Therefore, by the equality part of Theorem 1, Proposition 11. Let V k be a k-element subset of the vertex set of the complete graph K n , 2  k  n -1, n  3. The graph Kc n (k) is obtained by deleting from K n all the edges connecting pairs of vertices from V k . Then Proof. The edge set E(Kc n (k)) can be partitioned into two sets E 1 , and E 2 , where E 1 = {uv | d(u) = n k and d(v) = n -1} and E 2 = {uv | d(u) = n -1 and d(v) = n -1}. It is easy to check that |E 1 | = (nk)k and |E 2 | = (n -k)(nk -1) / 2. Also diam(Kc n (k)) = 2.
Therefore, by the equality part of Theorem 1, Proposition 12. Let 3  k  n, n  5. The graph Kd n (k) is obtained by deleting from K n , the edges belonging to a k-membered cycle. Then

Correlation between harmonic status index and boiling point of paraffins
The properties of graphs can be used in the study of quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) of the molecules [6]. In this section we study the correlation between the boiling point (BP) of the paraffin hydrocarbons and the harmonic status index of the corresponding molecular graphs. Using the data of Table 1, the scatter plot between the boiling point (BP) and harmonic status index (HS) of paraffins is depicted in Figure 4.