Research Article  Open Access
Topological Indices of the PentHeptagonal Nanosheets VC_{5}C_{7} and HC_{5}C_{7}
Abstract
In this paper, we computed the topological indices of pentheptagonal nanosheet. Formulas for atombond connectivity index, fourth atombond connectivity index, RandiÄ‡ connectivity index, sumconnectivity index, first Zagreb index, second Zagreb index, augmented Zagreb index, modified Zagreb index, hyper Zagreb index, geometricarithmetic index, fifth geometricarithmetic index, Sanskruti index, forgotten index, and harmonic index of pentheptagonal nanosheet have been derived.
1. Introduction
Nanostructures, which have scale less than 100â€‰nm, include nanotubes, nanosheets, and nanoparticles. Nanosheets (twodimensional nanostructures) have large surface and sharp edge, which make them play an important role in many applications such as energy storage [1â€“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 5â€“7 defect in graphene is illustrated. As nanoscalelevel defects introduced into graphene could be extremely useful and exploited to generate novel, innovative, and useful materials and devices [10], the 5â€“8â€“5 [11, 12] and 5â€“7â€“7â€“5 defects are obtained by several approaches [13â€“16]. Naturally, the property of C_{5}C_{7} nanostructures has become an active area of research [17â€“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 atombond connectivity index as
The ABC_{4} index is defined as
The RandiÄ‡ connectivity index (see [24]) is defined on the ground of vertex degrees as
The sumconnectivity 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 geometricarithmetic index (see [29]) is defined as
The is Defined as
For a detailed survey about topological indices and applications of topological indices, we refer [30â€“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 geometricarithmetic index for nanostar dendrimers in 2011. Asadpour et al. [43] computed RandiÄ‡, Zagreb, and ABC indices of TUC_{4}C_{8}(S) and TUC_{4}C_{8}(R) nanotubes and Vphenylenic nanotorus in 2011. AlFozan et al. [44] computed the Szeged index of C_{4}C_{8}(S) and Hnaphtalenic nanosheets (2n, 2m) in 2014. Azari and Iranmanesh [45] computed in Harary index of Pmâ€‰Ã—â€‰Pn (nanosheets), C_{4} nanotubes, and C_{4} nanotori in 2014. Ashrafi and Loghman [46] discussed the Padmakarâ€“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 [48â€“53] discussed RandiÄ‡ connectivity index and sum connectivity index of the pentheptagonal 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)â€‰=â€‰(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)â€‰=â€‰(2, 4)).
(a)
(b)
The nanosheet VC_{5}C_{7}(2, 4) has 16mnâ€‰+â€‰2mâ€‰+â€‰5n vertices and 24mnâ€‰+â€‰4n edges. Moreover, it has 6mâ€‰+â€‰7n vertices with degree 2 and 16mnâ€‰âˆ’â€‰4mâ€‰âˆ’â€‰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 VC_{5}C_{7}(2, 4) by different and , respectively.


(a)
(b)
The nanosheet HC_{5}C_{7}(2, 4) has 16mnâ€‰+â€‰2mâ€‰+â€‰4n vertices and 24mnâ€‰+â€‰3n edges. Moreover, it has 6mâ€‰+â€‰6n vertices with degree 2 and 16mnâ€‰âˆ’â€‰4mâ€‰âˆ’â€‰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 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 RandiÄ‡ connectivity index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the RandiÄ‡ 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 geometricarithmetic index, we have(ii)The proof is calculated based on the edge partition given in Table 3. By the definition of the first geometricarithmetic 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 geometricarithmetic index, we have(ii)The proof is calculated based on the edge partition given in Table 4. By the definition of the fifth geometricarithmetic 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