The Sanskruti index of trees and unicyclic graphs

Abstract The Sanskruti index of a graph G is defined as S ( G ) =∑uv∈E(G) s G ( u ) s G ( v ) s G ( u ) + s G ( v ) − 2 3 , $$\begin{align*}S(G)=\sum_{uv\in{}E(G)}{\left(\frac{s_G(u)s_G(v)}{s_G(u)+s_G(v)-2}\right)}^3, \end{align*}$$where sG(u) is the sum of the degrees of the neighbors of a vertex u in G. Let Pn, Cn, Sn and Sn + e be the path, cycle, star and star plus an edge of n vertices, respectively. The Sanskruti index of a molecular graph of a compounds can model the bioactivity of compounds, has a strong correlation with entropy of octane isomers and its prediction power is higher than many existing topological descriptors. In this paper, we investigate the extremal trees and unicyclic graphs with respect to Sanskruti index. More precisely, we show that (1) 512 27 n − 172688 3375 ≤ S ( T ) ≤ ( n − 1 ) 7 8 ( n − 2 ) 3 $\frac{512}{27}n-\frac{172688}{3375}\leq{}S(T)\leq{}\frac{(n-1)^7}{8(n-2)^3}$for an n-vertex tree T with n ≤ 3, with equalities if and only if T ≌ Pn (left) and T ≌ Sn (right); (2) 512 27 n ≤ S ( G ) ≤ ( n − 3 ) ( n + 1 ) 3 8 + 3 ( n + 1 ) 6 8 n 3 $ \frac{512}{27}n\leq{}S(G)\leq{}\frac{(n-3)(n+1)^3}{8}+\frac{3(n+1)^6}{8n^3}$for an n-vertex unicyclic graph with n ≥ 4, with equalities if and only if G ≌ Cn (left) and G ≌ Sn + e (right).


Introduction
In theoretical chemistry, topological indices (or molecular structure descriptors) are utilized as a standard tool to study structure-property relations, especially in quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) applications [9,20]. These topological indices are studied on chemical graphs, whose vertices correspond to the atoms of molecules and edges correspond to chemical bonds [15,[17][18][19]. In past decades, many topological indices have found important relations between the graph structures and physico-chemical properties [2]. Because of their significant applications, they have been widely studied and applied in many contexts, for example with nanostructures [5,10], nanomaterials [14], molecular sciences [13], chemistry networks [11], molecular design [1], drug structure analysis [7], fractal graphs [12], and mathematical chemistry [3]. The literature is exhaustive; for example, one of the indices, the Wiener index, along with its applications, is considered in thousands of papers: as of this writing, the seminal paper of Harold Wiener [20] is cited 3535 times according to Google Scholar.
Application of topological indices in biology and chemistry began in 1947 with the work of Harold Wiener [20], who introduced the Wiener index to show correlations between physico-chemical properties of organic compounds and the index of their molecular graphs. This index reveals the correlations of physico-chemical properties of alkanes, alcohols, amines and their analogous compounds [13]. Estrada et al. [4] proposed what is now a well-known atom-bond connectivity (ABC) index, which provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes. Inspired by applications of the ABC index, Furtula et al. [6] introduced the augmented Zagreb index, whose prediction power was found to be better than that of the ABC index in the study of heat of formation for heptanes and octanes. More recently, Hosamani [10] proposed the Sanskruti index of a molecular graph and showed that it can model the bioactivity of chemical compounds and showed a correlation with entropy of octane isomers that is comparable to or better than some other well-used descrip-tors. More precisely, according to [10], the model entropy = 1.7857S±81.4286 models the data from dataset found at http://www.moleculardescriptors.eu/dataset.htm with correlation coefficient 0.829 and with standard error 17.837. Soon after, the Sanskruti indices of some graph families of interest in chemical graph theory were established [8,16].
Motivated by the new proposed Sanskruti index, we investigate the extremal trees and extremal unicyclic graphs with respect to this topological index. Here, we consider only simple graphs, i.e., undirected graphs without loops and multiple edges. Let G be a graph. We denote by V(G) and E(G) the vertex set and edge set of G, respectively. As usual, Pn, Cn, Sn and Sn + e stand for the path, cycle, star and star plus an edge of n vertices, respectively (see Figure 1). We denote by d G (v) the degree of a vertex v of a graph G and by N G (v) (or simply N(v)) the set of neighbors of v. For two vertices u, v ∈ V(G), the distance between u and v is the length of a shortest path between u and v. We denote by N 2 (v) the set of vertices of distance two from v and by s G (u) the sum of the degrees of the neighbors of u, i.e., s G . Trees are connected graphs without cycles. A vertex in a tree is called a leaf if it has degree one, and a vertex is called a support vertex if it has a leaf neighbor.
In Section 2, we give some definitions and some preliminary observations. The main results are proved in Section 3: first, we give lower and upper bounds for the Sanskruti index on trees and provide the extremal graphs (Theorems 9 and 10), then we give lower and upper bounds for unicyclic graphs and provide the extremal graphs (Theorems 14 and 15).

Preliminaries
The following functions and definitions will be used throughout the paper: For a graph G and an edge uv ∈ E(G), we define and the Sanskruti index of a graph G is defined as Based on the above definitions, the following results are immediate, and the proofs are omitted. Lemma 4 Let t ≠ 2 and Lemma 5 For x ≥ 3 and y ≥ 3, f is an increasing function (as a function of one variable, either x or y). In particular, The following properties that hold on trees will be useful later.

Lemma 6 Let T be an n-vertex tree. Then for any edge uv
Proof. (a) Since T contains no C 3 , we have N(u)∩N(v) = ∅. Suppose to the contrary that N 2 (u)∩N 2 (v) ≠ ∅ and let w ∈ N 2 (u) ∩ N 2 (v). Denote with usw and vtw the shortest u − w path and v − w path. Then s ≠ v. Otherwise w ∈ N(v), a contradiction. Analogously, t ≠ u. Now if s ≠ t, we obtain that uvtws is a cycle of length five in T, a contradiction. If s = t, it follows that uvs is a cycle of length three in T, a contradiction.
It is obvious that the last property also holds for general graphs without triangles and C 4 . We write it as a lemma for later reference.
In a special case used explicitly in a proof later, we have

Lemma 8 Let G be an n-vertex unicyclic graph. If G contains a C 4 , then for any u
, the assertion is obvious.

Extremal trees with respect to Sanskruti index
Theorem 9 Let T be an n-vertex tree with n ≥ 3. Then, we have with equality if and only if T ∼ = Sn.
Proof. By Lemma 6 (c), we have Moreover, Recall that f (x, y) = ( xy x+y−2 ) 3 . From Lemmas 3 and 4 it follows that g(t) is an increasing function on the variable t if t ≥ 4. Note that for any uv ∈ E(T) with at least three vertices, we have s T (u) + s T (v) ≥ 4, then by applying Lemma 3 with t = 2n − 2, we have Conversely, if S(T) = (n−1) 7 8(n−2) 3 , then formula (5) holds for any uv ∈ E(T). It is easy to see that this implies that T must be a star. Proof. First observe that the lower bound holds for trees with 3 ≤ n ≤ 6 vertices (for example, by explicitly comput- Suppose to the contrary that there exists an n such that Fn ≠ ∅, and let n be the minimal number with this property. Let T ′ ∈ Fn and P = x 1 x 2 · · · xp be a longest path in T ′ . Then we claim that Figure 2).

It is obvious that F(4) = 217 27 and that F(x) is an increasing function for
a contradiction. It follows s 3 < 6. Since s 3 ≥ 5, it is obvious that s 3 = 5. But in this case d T ′ (x 4 ) = 1. Thus T is isomorphic to a tree with six vertices, which contradicts with n ≥ 7. So, we have proved that d(x 3 ) = 2 in case x 3 has a pendent P 2 = y 1 z 1 .
Together with Case 1 this means that d(x 3 ) = 2. On the other hand, by Proposition 1, we have Therefore, we have Hence Fn = ∅ for any n ≥ 7, which completes the proof of Theorem 10.

Extremal unicyclic graphs with respect to Sanskruti index
Suppose to the contrary that Gis a graph with maximum Sanskruti index, but GSn + e, We consider the following four cases.
• Case 1: Gcontains a C 3 . Let and for any i ∈ {1, 2, 3} Furthermore, because u 1 and u 2 are not on a cycle, we have for any By Eq. (13), it is impossible that s G (u 1 ) = s G (u 2 ) = n. Therefore, Further, we have and h(v 1 v 2 |G) ≤ f (n + 1, n + 1). It follows ≤ 3 f (n + 1, n + 1) + (n − 3) f (n + 1, n − 1) If the equality S(G) = S(Sn + e) holds, then the equalities in (15) and (16) hold and s G (v 1 ) = s G (v 2 ) = n + 1 and h(u 1 u 2 |G) = f (n − 1, n + 1) for any From these results it follows that G is a graph obtained by adding some pendent vertices to a C 3 = v 1 v 2 v 3 and, in addition, all vertices must be attached to the same vertex in {v 1 , v 2 , v 3 }. Such a graph is isomorphic to Sn + e, which is in contradiction with our assumption. • Case 2: G contains a C 4 .
For any edge uv on the cycle, by Lemma 8, we have and Then we have N 2 (u) ∩ N 2 (v) = ∅, otherwise G contains a C 5 . Together with the assumption that G contains C 4 this is a contradiction with the fact that G is unicyclic. Furthermore, we have N(u) ∩ N(v) = ∅.
Otherwise G contains a C 3 and this is again a contradiction. Since u and v are not in Now we have to consider two separate subcases.
In this case, for any edge uv ∈ E(G) we have s G (u) + s G (v) ≤ 2n − 2 and h(uv|G) ≤ f (n − 1, n − 1). Then < 3 · f (n + 1, n + 1) + (n − 3) · f (n + 1, n − 1) a contradiction. Summing up, we have proved that G does not contain any C k for all k ≥ 3, which contradicts the fact that G is an unicyclic graph. Proof. Suppose G is a graph with minimum Sanskruti index. Let C = v 1 v 2 · · · v k , 3 ≤ k ≤ n, be the unique cycle of G. We consider the following cases. Proof. Otherwise, we assume without loss of generality that Then which means that function q(x, y) is an increasing function on variable y for fixed x ≥ 4. Note that s 2 ≥ 4, so from s 1 ≥ 6 it follows q(s 1 , s 2 ) ≥ q(s 1 , 4) and contradicting with the assumption that G is a graph with minimum Sanskruti index. Thus, s 3 = 5. * Now we have contradicting with the assumption that G is a graph with minimum Sanskruti index.
contradicting with the assumption that G is a graph with minimum Sanskruti index. Thus, we have proved that s 2 ≠ 5. In this case, v i has a leaf neighbor w. (see Figure 3 (right)). Let T = G − {v i v i+1 }, then we have contradicting with the assumption that Gis a graph with minimum Sanskruti index.
This concludes the proof of Theorem 15.

Conclusions and future work
This paper reveals the idea that the structure of a molecular tree or unicyclic graph with minimal Sanskruti index has a path as long as possible. Similarly, a tree or a unicyclic graph with maximal value of Sanskruti index has a path as short as possible, These results may also hold in other families of molecular graphs. Moreover, there are several research avenues that may naturally extend the results of this paper. A natural generalization of trees and unicyclic graphs are cactus graphs, and it may be possible to find the extremal graphs among cacti applying the methods used here. Another idea that may be worth investigation is the following: the exponent 3 in the definition of Sanskruti index seems to be a rather arbitrary and lucky choice. One could replace 3 with arbitrary exponent α > 0 and perhaps obtain similar mathematical properties, and for some other α maybe even better correlation with some chemical properties of the corresponding chemical graphs.

Ethical approval:
The conducted research is not related to either human or animal use.