International Journal of Molecular Sciences Schultz Index of Armchair Polyhex Nanotubes

The study of topological indices – graph invariants that can be used for describing and predicting physicochemical or pharmacological properties of organic compounds – is currently one of the most active research fields in chemical graph theory. In this paper we study the Schultz index and find a relation with the Wiener index of the armchair polyhex nanotubes T U V C 6 [2p, q]. An exact expression for Schultz index of this molecule is also found.


Introduction
Topological indices are a convenient method of translating chemical constitution into numerical values that can be used for correlations with physical, chemical or biological properties. This method has been introduced by Harold Wiener as a descriptor for explaining the boiling points of paraffins [1][2][3]. If d(u, v) is the distance of the vertices u and v of the undirected connected graph G (i.e., the number of edges in the shortest path that connects u and v) and V (G) is the vertex set of G, then the Wiener index of G is the half sum of distances over all its vertex pairs (u, v): A unified approach to the Wiener topological index and its various recent modifications is presented. Among these modifications particular attention is paid to the Hyper-Wiener, Harary, Szeged, Cluj and Schultz indices as well as their numerous variants and generalizations [4][5][6][7][8][9][10]. The Schultz index of the graph G was introduced by Schultz [14] in 1989 and is defined as follows: where deg(u) is the degree of the vertex u.
The main chemical applications and mathematical properties of this index were established in a series of studies [12][13][14][15]. Also a comparative study of molecular descriptors showed that the Schultz index and Wiener index are mutually related [16][17][18].
Carbon nanotubes, the one-dimensional carbon allotropes, are intensively studied with respect to their promise to exhibit unique physical properties: mechanical, optical electronic etc. [19][20][21]. In [19], Diudea et al. obtained the Wiener index of T U V C 6 [2p, q], the armchair polyhex nanotube (see Figure  1). Here we find a relation between the Schultz index and Wiener index of this molecule. By using this relation we find an exact expression for the Schultz index of the same. The Appendix includes a Maple program [22] to produce the graph of T U V C 6 [2p, q], and to compute the Schultz index of the graph.

Schultz index of armchair polyhex nanotubes
Throughout this paper G := T U V C 6 [2p, q] denotes an arbitrary armchair polyhex nanotube in terms of its circumference 2p and their length q, see Figure 2. At first we consider an armchair lattice and choose a coordinate label for it, as illustrated in Figure 2. The distance of a vertex u of G is defined as the summation of distances between v and all vertices of G. By considering this notation the following lemma gives us a relation between the Schultz and Wiener index of G.  x 11 x 13 x 15 x 10 x 12 x 14 x 17 x 19 x 16 x 18 x 26 x 28 x 27 x 29 x 20 x 22 x 24 x 21 x 23 x 25   Proof: For each k such that 1 ≤ k ≤ q put A k := {u ∈ V (G) | u ∈ level k}( see Figure 2). Then This completes the proof.
To compute the d(u) in the graph G, when u is a vertex in level 1, we first prove the following lemma. . . .
Proof: We calculate the value of w k . We consider that the tube can be built up from two halves collapsing at the polygon line joining x 10 to x q,0 (see Figure 2). The right part is the graph G 1 which consists of vertical polygon lines 0, 1, . . . , p and x 10 is one of the vertices in the first row of the graph G 1 . The left part is the graph G 2 which consists of vertical polygon lines (p + 1), (p + 2), . . . , 2p − 1. We change the indices of the vertices of G 2 in the following way: Figure 3) We must consider two cases: Case 1: If k ≥ p. In the graphs G 1 and for 0 ≤ i < k we have Also in the graphs G 2 and for 1 ≤ i < k we have Case 2: If k < p. First suppose that 1 ≤ i < k. In the graphs G 1 and G 2 we have Now suppose that k ≤ i ≤ p. Then in the graph G 1 we can see that if k is odd, then All of this distances give us For other vertices we can convert those to x 10 by changing transfer vertices and apply a similar argument by choosing suitable G 1 and G 2 and compute w k .
By a straightforward computation (if irem means the positive integer remainder) we can see: So, by Lemma 1, when 1 ≤ k ≤ p, we have Also in the graph G, This leads us to the following corollary: Now suppose that p > q. Then by lemma 2 and equation (1) we have Also if p ≤ q, then by Lemma 1 and equation (1) we have We summarize the above results in the following proposition Proof: See [19].
Since there are 2p vertices on level 1 therefore Finally by replacing d(u) from corollary 1 in the equation (2) the result obtains.   Tables 1 and 2 show the numerical data for the Schultz index in tubes T U V C 6 [2p, q] of various dimensions.