## Abstract

The Sanskruti index of a graph *G* is defined as *s _{G}*(

*u*) is the sum of the degrees of the neighbors of a vertex

*u*in

*G*. Let

*P*,

_{n}*C*,

_{n}*S*and

_{n}*S*+

_{n}*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) *n*-vertex tree *T* with *n* ≤ 3, with equalities if and only if *T ≌**P _{n}* (left) and

*T*≌

*S*(right);

_{n}(2) *n*-vertex unicyclic graph with *n* ≥ 4, with equalities if and only if *G ≌**C _{n}* (left) and

*G*≌

*S*+

_{n}*e*(right).

## 1 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 descriptors. 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, *P _{n}*,

*C*,

_{n}*S*and

_{n}*S*+

_{n}*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*.,

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).

## 2 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.

**Proposition 1***Let n* ≥ 3, *and*

**Proposition 2***Let n* ≥ 4, *and*

**Lemma 3***Let t* ≥ 3*, x*, *y* > 0 *and x* + *y* = *t, then f* (*x*, *y*) ≤ *Moreover*, *if and only if*

**Lemma 4***Let t* ≠2 *and**then*

**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 ∈ E*(*T*)*, we have*

*N*(*u*)*∩ N*(*v*) =*∅ and N*_{2}(*u*)*∩ N*_{2}(*v*) =*∅*.Σ

_{v∈N}_{(u)}*|N*(*v*)*∖*{*u*}| =*|N*_{2}(*u*)*|*.*s*(_{T}*u*) =*|N*_{2}(*u*)*|*+*|N*(*u*)*|*=*|N*_{2}(*u*)*|*+*d*(_{T}*u*)*for any u ∈ V*(*T*).

*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.

(b) From the result of part (a), we have (*N*(*v*_{1})*∖* {*u*}) ∩ (*N*(*v*_{2})*∖* {*u*}) = ∅ for any *v*_{1}, *v*_{2} ∈ *N*(*u*). Therefore, Σ _{v}_{∈N(u)}*|N*(*v*)*∖* {*u*}| = |*N*_{2}(*u*)|.

(c) It can be seen that *s _{T}*(

*u*) = Σ

_{v}_{∈N(u)}

*d*(

_{T}*v*) = Σ

_{v}_{∈N(u)}|

*N*(

*v*)| = Σ

_{v}_{∈N(u)}(|

*N*(

*v*)

*∖*{

*u*}| + 1) = Σ

_{v}_{∈N(u)}|

*N*(

*v*)

*∖*{

*u*}| +

*d*(

_{T}*u*). From the result of part (b), we have Σ

_{v}_{∈N(u)}

*|N*(

*v*)

*∖*{

*u*}| =

*|N*

_{2}(

*u*)

*|*, and thus

*s*(

_{T}*u*) = |

*N*

_{2}(

*u*)| +

*d*(

_{T}*u*) for any

*u*∈

*V*(

*T*).

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.

**Lemma 7***Let G be a graph with no triangles and no C*_{4}*. Then s _{G}*(

*u*) =

*|N*

_{2}(

*u*)

*|*+

*|N*(

*u*)

*|*=

*|N*

_{2}(

*u*)

*|*+

*d*(

_{G}*u*)

*for any u ∈ V*(

*G*).

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 ∈ V*(*G*)

*Proof*. If *u ∈ V*(*C*_{4}), we can write *C*_{4} = *uv*_{1}*v*_{2}*v*_{3} and

because (*N*(*v*_{1})*∖ {u}*) *∩* (*N*(*v*_{3})*∖ {u}*) = *{v*_{2}*}*.

For *u ∉**V*(*C*_{4}), the assertion is obvious.

## 3 Main results

### 3.1 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 ≌**S _{n}*.

*Proof*. By Lemma 6 (c), we have *s _{T}*(

*u*) +

*s*(

_{T}*v*) =

*|N*

_{2}(

*u*)

*|*+

*|N*

_{2}(

*v*)

*|*+

*|N*(

*u*)

*|*+

*|N*(

*v*)

*|*for any edge

*uv ∈ E*(

*T*). By Lemma 6 (a),we have

*N*(

*u*)

*∩N*(

*v*) =

*∅*and

*N*

_{2}(

*u*)

*∩N*

_{2}(

*v*) =

*∅*. Therefore,

*s*(

_{T}*u*) +

*s*(

_{T}*v*) =

*|N*

_{2}(

*u*)

*∪ N*

_{2}(

*v*)

*|*+

*|N*(

*u*)

*∪ N*(

*v*)

*|*. Since

*u*,

*v ∉*

*N*

_{2}(

*u*)

*∪ N*

_{2}(

*v*), we have

*|N*

_{2}(

*u*)

*∪ N*

_{2}(

*v*)

*|*≤

*n*− 2. It is clear that

*|N*(

*u*)

*∪ N*(

*v*)

*|*≤

*n*, then we have

Moreover,

Recall 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*= 2

*n*− 2, we have

Conversely, if *uv ∈ E*(*T*). It is easy to see that this implies that *T* must be a star.

**Theorem 10***Let T be an n-vertex tree with n* ≥ 3*, then we have*

with equality if and only if *T ≌**P _{n}*.

*Proof*. First observe that the lower bound holds for trees with 3 ≤ *n* ≤ 6 vertices (for example, by explicitly computing the values for all cases). Therefore, we only need to consider the case *n* ≥ 7. Let *T _{n}* be the family of trees on

*n*vertices and let

*F*= {

_{n}*T|S*(

*T*) ≤

*S*(

*T*)

^{′′}*foranyT*,

^{′′}∈ T_{n}*TP*}. Now we need to prove that

_{n}*F*=

_{n}*∅*for any

*n*≥ 7.

Suppose to the contrary that there exists an *n* such that *F _{n}* =

*∅*, and let

*n*be the minimal number with this property. Let

*T*and

^{′}∈ F_{n}*P*=

*x*

_{1}

*x*

_{2}· · ·

*x*be a longest path in

_{p}*T*. Then we claim that

^{′}**Claim 11**. *d*(*x*_{2}) = 2.

*Proof*. If *d*(*x*_{2}) ≥ 3 write *N*(*x*_{2}) = {*x*_{1}, *x*_{3}, *y*_{1}, *y*_{2}, · · · , *y _{q}*}, where

*q*≥ 1. Now construct a new tree

*T*with

^{′′}*V*(

*T*) =

^{′′}*V*(

*T*) and

^{′}*E*(

*T*) =

^{′′}*E*(

*T*)

^{′}*∪{x*

_{1}

*y*= 1, 2, · · · ,

_{i}|i*q}∖ {x*

_{2}

*y*= 1, 2, · · · ,

_{i}|i*q}*(see Figure 2).

Then we have *s _{T}^{′}* (

*x*

_{3}) >

*s*(

_{T}^{′′}*x*

_{3}) and

*s*(

_{T}^{′}*v*) ≥

*s*(

_{T}^{′′}*v*) for any

*v ∈ V*(

*T*)

^{′}*∖ {x*

_{3}

*}*. By Lemma 5, and by comparing the contributions of pairs of corresponding edges, we have

*S*(

*T*) >

^{′}*S*(

*T*). Tree

^{′′}*T*thus has a smaller value of Sanskruti index than

^{′′}*T*, which contradicts the fact that

^{′}*T*. So,

^{′}∈ F_{n}*d*(

*x*

_{2}) = 2.

**Claim 12**. *d*(*x*_{3}) = 2.

*Proof*. Assume to the contrary that *N*(*x*_{3}) = {*x*_{2}, *x*_{4}, *y*_{1}, *y*_{2}, · · · , *y _{q}*}, where

*q*≥ 1. We should consider two cases:

• Case 1: *y*_{1}is a leaf neighbor of *x*_{3}.

Now let *V*(*T ^{′′}* ) =

*V*(

*T*) and

^{′}*E*(

*T*) =

^{′′}*E*(

*T*)

^{′}*∪ {x*

_{1}

*y*

_{1}

*}∖ {x*

_{3}

*y*

_{1}

*}*. Then we have

and

Note that

From Eqs. (6), (7) and (8) we have that *S*(*T ^{′}*) >

*S*(

*T*), which is a contradiction. Thus, we have already proved that

^{′′}*d*(

*x*

_{3}) = 2 in case

*y*

_{1}is a leaf neighbor of

*x*

_{3}.

• Case 2: *x*_{3} has a pendent *P*_{2} = *y*_{1}*z*_{1}.

Now let *V*(*T ^{′′}* ) =

*V*(

*T*) and

^{′}*E*(

*T*) =

^{′′}*E*(

*T*)

^{′}*∪ {x*

_{1}

*y*

_{1}

*}∖ {x*

_{3}

*y*

_{1}

*}*. Then we have

and *h*(*uv|T ^{′′}* ) ≤

*h*(

*uv|T*) for any

^{′}*uv ∉*{

*x*

_{1}

*x*

_{2},

*x*

_{2}

*x*

_{3},

*x*

_{3}

*y*

_{1},

*y*

_{1}

*z*

_{1},

*x*

_{1}

*y*

_{1}}. Denote

*s*

_{3}≥ 5 and

Define *F*(*x*) is an increasing function for *x* ≥ 4.

If *s*_{3} ≥ 6, since

a contradiction. It follows *s*_{3} < 6. Since *s*_{3} ≥ 5, it is obvious that *s*_{3} = 5. But in this case *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.

**Claim 13**. *d*(*x*_{4}) = 2.

*Proof*. Suppose to the contrary that *d*(*x*_{4}) ≥ 3. Let *T ^{′′}* =

*T*−

^{′}*{x*

_{1}

*}*. By the minimum assumption of

*n*and

*T*,we have

^{′}∈ F_{n}*S*(

*T*)≥

^{′′}*S*(

*P*

_{n}_{−1}). Denote

*s*

_{1}≥ 5 and

*s*

_{2}≥ 4. It can be verified that

On the other hand, by Proposition 1, we have

Therefore, we have

a contradiction. Thus, *d*(*x*_{4}) = 2.

Now we have *i* = 2, 3, 4, and let *T ^{′′′}* =

*T*−

^{′}*{x*

_{1}

*}*. By the minimality assumption on

*n*and

*T*∈

^{′}*F*, we have

_{n}*S*(

*T*) >

^{′′′}*S*(

*P*

_{n}_{−1}). Moreover, it can be seen that

*S*(

*T*) >

^{′}*S*(

*P*), a contradiction with minimality of

_{n}*n*. Hence

*F*=

_{n}*∅*for any

*n*≥ 7, which completes the proof of Theorem 10.

### 3.2 Extremal unicyclic graphs with respect to Sanskruti index

**Theorem 14***Let G be an n-vertex unicyclic graph with n* ≥ 4*, then we have*

with equality if and only if *G ≌**S _{n}* +

*e*.

*Proof*. By Proposition 2, in case *G ≌**S _{n}* +

*e*we have

*S*(

*G*) ≤

*G*≌

*S*+

_{n}*e*.

Suppose to the contrary that *G*is a graph with maximum Sanskruti index, but *GS _{n}* +

*e*,We consider the following four cases.

• Case 1: *G*contains a *C*_{3}.

Let *C*_{3} = *v*_{1}*v*_{2}*v*_{3}. Then for any *u* ∈ *V*(*G*)*∖* {*v*_{1}, *v*_{2}, *v*_{3}} we have

and for any *i* ∈ *{*1, 2, 3*}*

Furthermore, because *u*_{1} and *u*_{2} are not on a cycle, we have for any *u*_{1}*u*_{2} ∉{*v*_{1}*v*_{2}, *v*_{1}*v*_{3}, *v*_{2}*v*_{3}}

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

If the equality *S*(*G*) = *S*(*S _{n}* +

*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

*u*

_{1}

*u*

_{2}∉ {

*v*

_{1}

*v*

_{2},

*v*

_{1}

*v*

_{3},

*v*

_{2}

*v*

_{3}}. 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

*S*+

_{n}*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 *N*_{2}(*u*) *∩ N*_{2}(*v*), we have *|N*_{2}(*u*) *∩ N*_{2}(*v*)*|* ≤ *n* − 2. As *|N*(*u*) *∩ N*(*v*)*|* ≤ *n*, from Eqs. (17-18) it follows

Now we have to consider two separate subcases.

– Case 2.1: There exists no edge *u ^{′}v^{′}* ∈

*E*(

*G*) such that

*s*(

_{G}*u*) =

^{′}*s*(

_{G}*v*) =

^{′}*n*.

In this case, we have *h*(*uv|G*) ≤ *f* (*n* + 1, *n* − 1) for any edge *uv* ∈ *E*(*G*). Then

a contradiction.

– Case 2.2: There exists an edge *uv ∈ E*(*G*) such that *s _{G}*(

*u*) =

*s*(

_{G}*v*) =

*n*.

Since *u ∉**N*_{2}(*u*) *∪ N*(*u*) for any *u ∈ V*(*G*), we have *|N*_{2}(*u*) *∪ N*(*u*)*|* ≤ *n* − 1. Therefore, the equalities hold in Eqs. (17-18) and *uv* must be in *C*_{4}. Then, for any *w ∈ V*(*G*) − *{u*, *v}*, *d*(*w*, *v*) = 1 and *d*(*w*, *u*) = 2 or *d*(*w*, *v*) = 2 and *d*(*w*, *u*) = 1. Let *C*_{4} = *uvst*, *G* is a graph isomorphic to a graph obtained by adding some pendent vertices to *u* or *v* of *C*_{4}. Then there are at most two edges *xy* with *s _{G}*(

*x*) =

*s*(

_{G}*y*) =

*n*, and hence

a contradiction.

From Cases 2.1 and 2.2, it follows that *G* does not contain *C*_{4}.

• Case 3: *G* contains a *C*_{5}.

By Lemma 7, for any *uv ∈ E* (*G*), we have

and

Then *s _{G}*(

*u*) +

*s*(

_{G}*v*) ≤ 2

*n*− 1 and

*h*(

*uv|G*) ≤

*f*(

*n*,

*n*− 1) for any edge

*uv ∈ E*(

*G*). It follows

a contradiction.

• Case 4: *G* contains *C _{k}* for some

*k*≥ 6.

In this case, for any edge *uv ∈ E*(*G*) we have *s _{G}*(

*u*) +

*s*(

_{G}*v*) ≤ 2

*n*− 2 and

*h*(

*uv|G*) ≤

*f*(

*n*− 1,

*n*− 1). Then

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.

**Theorem 15***Let Gbe an n-vertex unicyclic graph with n* ≥ 4*, then we have*

with equality if and only if *G ≌**C _{n}*.

*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.

• Case 1: for any edge *v _{i}v_{i}*

_{+1}in

*C*we have

*s*(

_{G}*v*) = 4 or

_{i}*s*(

_{G}*v*

_{i}_{+1}) = 4.

In this case, we have *d*(*v _{j}*) = 2 for any

*v*in

_{j}*C*, and hence

*G ≌*

*C*.

Observe that if *v _{i} ∈ C* and

*s*(

_{G}*v*) > 4, then there must be a neighbor of

_{i}*v*, say

_{i}*u ∈ C*with

*s*(

_{G}*u*) > 4.

• Case 2: there exists an edge *v _{i}v_{i}*

_{+1}in

*C*such that

*s*(

_{G}*v*)≥ 5 and

_{i}*s*(

_{G}*v*

_{i}_{+1}) ≥ 5.

Now we claim that

**Claim 16**. *s _{G}*(

*v*) = 5 and

_{i}*s*(

_{G}*v*

_{i}_{+1}) = 5.

*Proof*. Otherwise, we assume without loss of generality that *s _{G}*(

*v*

_{i}_{+1}) ≥ 6. Let

*T*=

*G*−

*{v*

_{i}v_{i}_{+1}

*}*, then

*T*is a tree. By Theorem 9 we have

*S*(

*T*) ≥

*S*(

*P*).

_{n}Denote *s*_{3} = *s _{G}*(

*v*),

_{i}*s*

_{1}=

*s*(

_{G}*v*

_{i}_{+1}),

*s*

_{2}=

*s*(

_{G}*v*

_{i}_{+2}).

* First,we have *s*_{3} = 5. Otherwise, *s*_{3} 6. Let

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 *s*_{2} ≠4. Otherwise, we have *d _{G}*(

*v*

_{i}_{+1}) =

*d*(

_{G}*v*

_{i}_{+2}) = 2,

*d*(

_{G}*v*)≥ 4 and thus

_{i}*s*

_{3}≥ 6, contradicting with the above result

*s*

_{3}= 5.

* Also, we have *s*_{2} ≠ 5. Otherwise, *s*_{2} = 5. Then we have *d _{G}*(

*v*

_{i}_{+1}) ≤ 3.

– If *d _{G}*(

*v*

_{i}_{+1}) = 2, then we have

*d*(

_{G}*v*) =

_{i}*d*(

_{G}*v*

_{i}_{+2}) = 3 and both

*v*and

_{i}*v*

_{i}_{+2}have a leaf neighbor. Let the leaf neighbor of

*v*be

_{i}*v*, then we have

^{′}contradicting with the assumption that *G* is a graph with minimum Sanskruti index.

– If *d _{G}*(

*v*

_{i}_{+1}) = 3, then

*d*(

_{G}*v*) =

_{i}*d*(

_{G}*v*

_{i}_{+2}) = 2. Let

*N*(

*v*

_{i}_{+1}) =

*{v*,

_{i}*v*

_{i}_{+2},

*u*and

^{′}}*s*

^{′}_{2}=

*s*(

_{G}*u*). Then

^{′}*s*

^{′}_{2}≥ 4 and

contradicting with the assumption that *G* is a graph with minimum Sanskruti index. Thus, we have proved that *s*_{2} ≠5.

* If *s*_{2} ≥ 6, we exchange the role of *v _{i}* and

*v*

_{i}_{+2}. Since

*s*

_{3}=

*s*(

_{G}*v*) = 5, we also have

_{i}*s*

_{2}=

*s*(

_{G}*v*

_{i}_{+2}) = 5, a contradiction.

Concluding: Claim 16 was proved, we have *s _{G}*(

*v*) = 5 and

_{i}*s*(

_{G}*v*

_{i}_{+1}) = 5.

Continuing with the proof of Theorem 15 in Case 2 it is sufficient to consider the following two cases.

– Subcase 2.1: *d _{G}*(

*v*) = 2 and

_{i}*d*(

_{G}*v*

_{i}_{+1}) = 2.

In this case, obviously *d _{G}*(

*v*

_{i}_{−1}) = 3 and by Claim 16, we have

*s*(

_{G}*v*

_{i}_{−1}) = 5, thus

*v*

_{i}_{−1}has a leaf neighbor

*w*(see Figure 3 (left)). Let

*T*=

*G*−

*{v*

_{i}_{−1}

*v*, then

_{i}}contradicting with the assumption that *G* is a graph with minimum Sanskruti index.

– Subcase 2.2: *d _{G}*(

*v*) = 3 and

_{i}*d*(

_{G}*v*

_{i}_{+1}) = 2.

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 *G*is a graph with minimum Sanskruti index.

This concludes the proof of Theorem 15.

## 4 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.

**Conflict of interest**Authors state no conflict of interest.

## Acknowledgement

This work is supported by the Natural Science Foundation of Guangdong Province under grant 2018A0303130115, and the China Postdoctoral Science Foundation under Grant 2017M621579; the Postdoctoral Science Foundation of Jiangsu Province under Grant 1701081B; Project of Anhui Jianzhu University under Grant no. 2016QD116 and 2017dc03. Research of J. Žerovnik and D. Rupnik Poklukarwas supported in part by Slovenian Research Agency under grants P2-0248, N1-0071 and J1-8155.

## References

[1] Balaban A.T., Can topological indices transmit information on properties but not on structures? J. Comput. Aid. and Mol. Des., 2005, 19, 651–660.10.1007/s10822-005-9010-6Search in Google Scholar

[2] Das K.C., Gutman I., Furtula B., Survey on geometric-arithmetic indices of graphs, MATCH Commun. Math. Comput. Chem., 2011, 65, 595–644.Search in Google Scholar

[3] Dobrynin A.A, Gutman I., Klavžar S., Žigert P., Wiener Index of Hexagonal Systems, Acta Appl. Math., 2001, 72, 247–294.10.1023/A:1016290123303Search in Google Scholar

[4] Estrada E., Torres L., Rodríguez L., Gutman I., An atom bond connectivity index: modelling the enthalpy of formation of alkanes, Indian J. Chem. Sect. A, 1998, 37, 849–855.Search in Google Scholar

[5] Fath-Tabar G.H., Zagreb polynomial and PI indices of some nano structures, Dig. J. Nanomater. Bios, 2009, 4(1), 189–191.Search in Google Scholar

[6] Furtula B., Graovac A., Vukičević D., Augmented Zagreb index, J. Math. Chem., 2010, 48, 370–380.10.1007/s10910-010-9677-3Search in Google Scholar

[7] Gao W., Farahani M.R., Shi L., The forgotten topological index of some drug structures, Acta Medica Mediterr., 2016, 32, 579–585.10.1155/2016/1053183Search in Google Scholar

[8] Gao Y.Y., Farahani M.R., Sardar M.S., Zafar S., On the Sanskruti index of circumcoronene series of benzenoid, Appl. Math., 2017, 8, 520–524.10.4236/am.2017.84041Search in Google Scholar

[9] Gutman I., Trinajstić N., Graph theory and molecular orbitals. Total *π*-electron energy of alternant hydrocarbons, Chem. Phys. Lett., 1972, 17, 535–538.10.1016/0009-2614(72)85099-1Search in Google Scholar

[10] Liu J.B., Pan X.F., Yu L., Li D., Complete characterization of bicyclic graphs with minimal Kirchhoff index, Discrete Appl. Math., 2016, 200, 95–107.10.1016/j.dam.2015.07.001Search in Google Scholar

[11] Hayat S., Wang S., Liu J.-B., Valency-based topological descriptors of chemical networks and their applications, Appl. Math. Model., 2018, 60, 164-178.10.1016/j.apm.2018.03.016Search in Google Scholar

[12] Imran M., Hafi S.E., Gao W., Farahani M.R., On topological properties of Sierpinski networks, Chaos Soliton. Fract., 2017, 98, 199–204.10.1016/j.chaos.2017.03.036Search in Google Scholar

[13] Iranmanesh A., Alizadeh Y., Taherkhani B., Computing the Szeged and PI Indices of *VC*_{5}*C*_{7}*p**q*and *HC*_{5}*C*_{7}*p**q*Nanotubes, Int. J. Mol. Sci., 2008, 9, 131–144.10.3390/ijms9020131Search in Google Scholar
PubMed
PubMed Central

[14] Jagadeesh R., Rajesh Kanna M.R., Indumathi R.S., Some results on topological indices of graphene, Nanomater. Nanotechno., 2016, 6, 1–6.10.1177/1847980416679626Search in Google Scholar

[15] Klein D.J., Lukovits I., Gutman I., On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comp. Sci., 1995, 35, 50–52.10.1021/ci00023a007Search in Google Scholar

[16] Sardar M.S., Zafar S., Farahani M.R., Computing Sanskruti index of the polycyclic aromatic hydrocarbons, Geology, Ecology and Landscapes, 2017, 1, 37–40.10.1080/24749508.2017.1301056Search in Google Scholar

[17] Shao Z.,Wu P., Gao Y., Gutman I., Zhang X., On the maximum ABC index of graphs without pendent vertices, Appl. Math. Comput., 2017, 315, 298-312.10.1016/j.amc.2017.07.075Search in Google Scholar

[18] Shao Z., Wu P., Zhang X., Dimitrov D., Liu J., On the maximum ABC index of graphs with prescribed size and without pendent vertices, IEEE Access. 2018, 6, 27604–27616.10.1109/ACCESS.2018.2831910Search in Google Scholar

[19] Schultz H.P., Topological organic chemistry 1. Graph theory and topological indices of alkanes, J. Chem. Inf. Comp. Sci., 1989, 29, 227–228.10.1021/ci00063a012Search in Google Scholar

[20] Wiener H., Structural determination of paraffin boiling points, J. Am. Chem. Soc., 1947, 1, 17–20.10.1021/ja01193a005Search in Google Scholar PubMed

**Received:**2018-10-13

**Accepted:**2018-11-19

**Published Online:**2019-08-24

© 2019 F. Deng *et al*., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.