#### Abstract

In this paper, we computed the topological indices of pent-heptagonal nanosheet. Formulas for atom-bond connectivity index, fourth atom-bond connectivity index, Randić connectivity index, sum-connectivity index, first Zagreb index, second Zagreb index, augmented Zagreb index, modified Zagreb index, hyper Zagreb index, geometric-arithmetic index, fifth geometric-arithmetic index, Sanskruti index, forgotten index, and harmonic index of pent-heptagonal nanosheet have been derived.

#### 1. Introduction

Nanostructures, which have scale less than 100 nm, include nanotubes, nanosheets, and nanoparticles. Nanosheets (two-dimensional nanostructures) have large surface and sharp edge, which make them play an important role in many applications such as energy storage [1–3], optoelectronics [4], catalysis [5], and bioelectronics [6]. Graphene [7], silicone [8], and borophene [9] are typical nanosheets. Graphene is the basic nanosheet that exhibits the honeycomb lattice structure. Because of the unique structural, electrical, optical, and mechanical properties, graphene nanosheets drew dramatic attention of academic and industrial research [10].

People found some defects in graphene nanosheets made through experiments. In Figure 1, the 5–7 defect in graphene is illustrated. As nanoscale-level defects introduced into graphene could be extremely useful and exploited to generate novel, innovative, and useful materials and devices [10], the 5–8–5 [11, 12] and 5–7–7–5 defects are obtained by several approaches [13–16]. Naturally, the property of C_{5}C_{7} nanostructures has become an active area of research [17–22].

In chemical graph theory, topological indices are very useful to describe molecules or nanostructures. The vertices of the graph correspond to the atoms of molecules, and the edges correspond to chemical bonds.

In this paper, we only consider undirected graphs without loops and multiple edges. Let *G* be a graph. We denote *V*(*G*) and *E*(*G*) as the vertex set and edge set of *G*, respectively. We denote by the degree of the vertex of graph *G* and the set of neighbors of . We denote by the set of edges with degrees of end vertices *a* and *b*, i.e.,

Let the sum of degrees of neighboring vertices of *u* be denoted by , which is defined as

We define

In 1998, Estrada et al. [23] defined the atom-bond connectivity index as

The ABC_{4} index is defined as

The Randić connectivity index (see [24]) is defined on the ground of vertex degrees as

The sum-connectivity index is defined (see [25]) as

The first Zagreb index is defined as the sum of squares of the vertex degrees and of vertices *u* and in graph *G* (see [26]). The mathematical formula of first Zagreb index is

The second Zagreb index is defined as the sum of squares of the vertex degrees and of vertices *u* and in graph *G* (see [26]). The mathematical formula of second Zagreb index is

Augmented Zagreb index AZI(*G*) (see [27]) is defined as

Modified Zagreb index (see [28]) is defined as

In 2013, Shirdel et al. [28] introduced hyper Zagreb index as

The geometric-arithmetic index (see [29]) is defined as

The is Defined as

For a detailed survey about topological indices and applications of topological indices, we refer [30–38] and references therein. Recently, motivated by the previous research on topological descriptors and their applications, Hosamani [39] proposed the Sanskruti index which can be utilized to guess the bioactivity of chemical compounds and shows good correlation with entropy of octane isomers.

The Sanskruti index of graph *G* is defined as

The forgotten index is defined as the sum of cubic of the vertex degrees in *G*. Mathematical formula of forgotten index is [40]

The harmonic index (see [41]) is defined as

#### 2. C_{5}C_{7} Nanosheets and Their Structures

A C_{5}C_{7} net is a trivalent decoration made by alternating C_{5} and C_{7}. It can cover either a cylinder or a torus and certainly able to cover a sheet.

The topological indices of nanostructures have been studied extensively. Graovac et al. [42] discussed the GA index of TUC_{4}C_{8}(S) nanotube in 2009. Graovac et al. [42] computed fifth geometric-arithmetic index for nanostar dendrimers in 2011. Asadpour et al. [43] computed Randić, Zagreb, and ABC indices of TUC_{4}C_{8}(S) and TUC_{4}C_{8}(R) nanotubes and V-phenylenic nanotorus in 2011. Al-Fozan et al. [44] computed the Szeged index of C_{4}C_{8}(S) and H-naphtalenic nanosheets (2*n*, 2*m*) in 2014. Azari and Iranmanesh [45] computed in Harary index of P*m* × P*n* (nanosheets), C_{4} nanotubes, and C_{4} nanotori in 2014. Ashrafi and Loghman [46] discussed the Padmakar–Ivan (PI) index of TUC_{4}C_{8}(S) nanotubes in 2006.

The topological indices of C_{5}C_{7} nanostructures have also been studied. Iranmanesh et al. [47] introduced a GAP program to compute the Szeged and PI indices of VC_{5}C_{7}[p,q] and HC_{5}C_{7}[p,q] nanotubes in 2008. Farahani and Zhang [48–53] discussed Randić connectivity index and sum connectivity index of the pent-heptagonal nanotube VAC_{5}C_{7}(S) in 2013. For the nanosheet VC_{5}C_{7}(*m*, *n*), we denote the number of pentagons in the first row by *n*; in this nanotube, the first four rows of vertices and edges are repeated alternatively and we denote the number of this repetition by *m* (see, e.g.,Figure 2(a) with (*m*, *n*) = (2, 4)).

For the nanosheet HC5C7(*m*, *n*), we denote the number of heptagons in the first row by *n*. In this nanotube, the first four rows of vertices and edges are repeated alternatively and we denote the number of this repetition by *m* (see, e.g., Figure 2(b) with (*m*, *n*) = (2, 4)).

**(a)**

**(b)**

The nanosheet VC_{5}C_{7}(2, 4) has 16*mn* + 2*m* + 5*n* vertices and 24*mn* + 4*n* edges. Moreover, it has 6*m* + 7*n* vertices with degree 2 and 16*mn* − 4*m* − 2*n* vertices with degree 3. The edge partitions according to different and are presented in Tables 1 and 2, respectively. We computed some topological indices according to different types of and . As an example, Figures 3(a) and 3(b) show the edge partition of the nanosheet VC_{5}C_{7}(2, 4) by different and , respectively.

**(a)**

**(b)**

The nanosheet HC_{5}C_{7}(2, 4) has 16*mn* + 2*m* + 4*n* vertices and 24*mn* + 3*n* edges. Moreover, it has 6*m* + 6*n* vertices with degree 2 and 16*mn* − 4*m* − 2*n* vertices with degree 3. We computed some topological indices according to different types of and . The edge partitions according to different and are presented in Tables 3 and 4, respectively. As an example, Figures 4(a) and 4(b) show the edge partition of the nanosheet HC_{5}C_{7}(2, 4) by different and , respectively.

**(a)**

**(b)**

#### 3. Main Results

Theorem 1. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of ABC index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of ABC index, we have

Theorem 2. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 2. By the definition of ABC_{4} index, we have(ii)The proof is calculated based on the edge partition given in Table 4. By the definition of ABC_{4} index, we have

Theorem 3. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of the Randić connectivity index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the Randić connectivity index, we have

Theorem 4. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of the sum connectivity index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the sum connectivity index, we have

Theorem 5. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of the first Zagreb index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the first Zagreb index, we have

Theorem 6. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of the second Zagreb index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the second Zagreb index, we have

Theorem 7. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let G be a*HC_{5}C_{7}(*m*,*n*)*nanosheet, then**Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of the augmented Zagreb index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the augmented Zagreb index, we have

Theorem 8. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of the modified Zagreb index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the modified Zagreb index, we have

Theorem 9. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of the hyper Zagreb index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the hyper Zagreb index, we have

Theorem 10. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of the first geometric-arithmetic index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the first geometric-arithmetic index, we have

Theorem 11. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 2. By the definition of the fifth geometric-arithmetic index, we have(ii)The proof is calculated based on the edge partition given in Table 4. By the definition of the fifth geometric-arithmetic index, we have

Theorem 12. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 2. By the definition of the Sanskruti index, we have(ii)The proof is calculated based on the edge partition given in Table 4. By the definition of the Sanskruti index, we have

Theorem 13. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of the forgotten topological index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the forgotten topological index, we have

Theorem 14. (i)*Let G be a VC_{5}C_{7}(m, n) nanosheet, then*(ii)

*Let*

*G*be a HC_{5}C_{7}(*m*,*n*) nanosheet, then*Proof. *(i)The proof is calculated based on the edge partition given in Table 1. By the definition of the harmonic index, we have