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 C5C7 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 ABC4 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. C5C7 Nanosheets and Their Structures
A C5C7 net is a trivalent decoration made by alternating C5 and C7. 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 TUC4C8(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 TUC4C8(S) and TUC4C8(R) nanotubes and V-phenylenic nanotorus in 2011. Al-Fozan et al. [44] computed the Szeged index of C4C8(S) and H-naphtalenic nanosheets (2n, 2m) in 2014. Azari and Iranmanesh [45] computed in Harary index of Pm × Pn (nanosheets), C4 nanotubes, and C4 nanotori in 2014. Ashrafi and Loghman [46] discussed the Padmakar–Ivan (PI) index of TUC4C8(S) nanotubes in 2006.
The topological indices of C5C7 nanostructures have also been studied. Iranmanesh et al. [47] introduced a GAP program to compute the Szeged and PI indices of VC5C7[p,q] and HC5C7[p,q] nanotubes in 2008. Farahani and Zhang [48–53] discussed Randić connectivity index and sum connectivity index of the pent-heptagonal nanotube VAC5C7(S) in 2013. For the nanosheet VC5C7(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 VC5C7(2, 4) has 16mn + 2m + 5n vertices and 24mn + 4n edges. Moreover, it has 6m + 7n vertices with degree 2 and 16mn − 4m − 2n 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 VC5C7(2, 4) by different and , respectively.
(a)
(b)
The nanosheet HC5C7(2, 4) has 16mn + 2m + 4n vertices and 24mn + 3n edges. Moreover, it has 6m + 6n vertices with degree 2 and 16mn − 4m − 2n 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 HC5C7(2, 4) by different and , respectively.
(a)
(b)
3. Main Results
Theorem 1. (i)Let G be a VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(m, n) nanosheet, then
Proof. (i)The proof is calculated based on the edge partition given in Table 2. By the definition of ABC4 index, we have(ii)The proof is calculated based on the edge partition given in Table 4. By the definition of ABC4 index, we have
Theorem 3. (i)Let G be a VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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 VC5C7(m, n) nanosheet, then(ii)Let G be a HC5C7(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