Computing Some Degree-Based Topological Indices of Honeycomb Networks

Guangzhou Kangda Vocational Technical College, Guangzhou, Guangdong 510000, China Department of Sciences and Humanities, National University of Computer and Emerging Sciences, Lahore Campus, B-Block, Faisal Town, Lahore, Pakistan Department of Mathematics, University of Gujrat, Hafiz Hayat Campus, Gujrat, Pakistan Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore (RCET), Pakistan


Introduction
Chemical graph theory/mathematical chemistry has made easier for the scientists to predict physicochemical properties of different chemical structures using topological indices. A numeric value assigned to a molecular graph (a graph portraying chemical compounds) is recognized as the topological index, and it remains unchanged under the isomorphism of the graph. For a given period, there is an increasing interest in topological indices that reflect some structural features of molecular compounds. Topological indices hold attention because they correlate well with certain chemical properties of molecular graphs. Also, they play an important role in the quantitative structure-activity relationship (QSAR) and quantitative structure property relationship (QSPR) studies [1]. To overview the structureactivity relationship, topological indices are required to effectively characterize structural features and bioactivity of chemical compounds [2][3][4]. In the current study, under discussion, molecular structures are nanostructures due to their design and applications.
All the graphs in present study are simple, undirected, finite, and connected. For a vertex w ∈ V(G), we use the notation N(w) for the set containing the vertices adjacent to w. e degree of a vertex w is the cardinality of the set N(w) and is denoted by d w . Let S(w) denote the sum of degrees of the vertices adjacent to w. In other words, S(w) � uw∈E(G) d u and N(t) � v ∈ V(G) | tv ∈ E(G) { }. For undefined terminologies related to graph theory, the author can read [5][6][7]. Consider the following general graph invariant: (G) f(S(v), S(w)). (1) Some special cases of the above invariants I have already been appeared in mathematical chemistry. For example, if we take f(S(v), S(w)) � S(v)S(w) or 1/ then I gives neighborhood second Zagreb index NM 2 [8] and the first extended first-order connectivity index 1 χ 1 [9,10], respectively. ese indices are defined as 1 Toropov et al. [11] proved that extended molecular connectivity indices have good correlation with the boiling point of chemical structures. Detail about theoretical and computation aspects of some families of chemical structures can be viewed in [12][13][14][15][16].
In this report, we are interested in five well-known chemical structures, name as oxide network (OX n ), honeycomb network (HC n ), silicate network (SL n ), chain silicate network (CS n ), and hexagonal network (HX n ). As far as authors are aware, neighborhood second Zagreb index and the first extended first-order connectivity index of oxide network (OX n ), silicate network (SL n ), chain silicate network (CS n ), hexagonal network (HX n ), and honeycomb network (HC n ) have not been investigated until now, and therefore, the study under consideration is an addition in the aforesaid direction.

Motivation
Since topological indices are useful to predict the physicochemical properties of chemical compounds, therefore, it is important to analyze the correlation of the topological indices investigated. Yousaf et al. [17] proved that the correlation of entropy and eccentric factor of octane isomers with neighborhood second Zagreb index and the first extended first-order connectivity index has high accuracy. In addition, topological indices are moreover used for discrimination against octane isomers. e discrimination ability of an index has remarkable application within the coding and the computer processing of molecular structures (isomers).
Denote by x q,r (H) (or x q,r , when terms are not confusing) the number of edges in H with end-vertex degrees q and r. Two graphs H 1 and H 2 are called edge-equivalent graphs if they are satisfying x q,r (H 1 ) � x q,r (H 2 ) for all q and r with 1 ≤ q ≤ r ≤ Δ. A topological index TI is called edgeequivalent topological index if it satisfies TI(H 1 ) � TI(H 2 ) for every pair of edge-equivalent graphs H 1 and H 2 . We notice that the bond incident degree indices [18,19] (BID indices for short [20]) are edge-equivalent topological indices. General form of the BID indices is where g(q, r) is a bivariate symmetric function. It is worth noting that most of the degree-based topological indices used in mathematical chemistry are BID indices. ese indices may be produced from equation (4) based on the selection of g(q, r).
To study the discriminatory performance of the topological index NM 2 and 1 χ 1 , the class of octane isomers are considered which are representing the 8-vertex trees. ese 18 molecular graphs of octane isomers are shown in Figure 1. us, the topological index NM 2 and 1 χ 1 does not belong to the class of edge-equivalent topological indices. Subsequently, we can presume that the topological indices NM 2 and 1 χ 1 are identified by a better discriminatory power than the conventional BID indices. Particularly, Wang et al. [21] proved that all the molecular graphs of 18 octane isomers have different NM 2 and 1 χ 1 values. It is worth mentioning that, in certain specific cases, the discriminatory capacity of NM 2 and 1 χ 1 is limited, but still better than that of the BID indices. is can be explained by examining the corresponding edgeequivalent graphs illustrated in Figure 2: we have NM 2 (D 1 ) � 451 and 1 χ 1 (D 1 ) � 1.09, but NM 2 (D 2 ) � NM 2 (D 3 ) � 448 and 1 χ 1 (D 2 )� 1 χ 1 (D 3 ) � 1.08 (of course, these three graphs are identical for any arbitrary BID index; in other words, with the help of an arbitrary BID index, it is difficult to distinguish between the graphs D 1 , D 2 , and D 3 ).

Main Results and Discussion
Multiprocessor interconnection networks are frequently needed to join many identical processor-memory pairs that are repeated, each one is referred to as a processing node. Message passing rather than shared memory is frequently used for synchronization of a communication between the processing nodes for the execution of the program. Multiprocessor interconnection networks are most attracted networks due to the accessibility of powerful and inexpensive microprocessors and memory chips. By repeating regular polygons, periodic plane tessellations can be easily built. For direct interconnection networks, this design is very important as it offers high global performance. Some parallel networks that originate from popular meshes are silicate, chain silicate, hexagonal, honeycomb, and oxide networks. Such networks have very attractive topological properties that have been investigated in various ways in [22][23][24][25][26][27][28][29][30][31][32][33].

Oxide Networks.
Oxide networks are important in investigating silicate networks. Oxide networks are obtained by the deletion of silicon nodes from a silicate network (see Figure 3). An oxide network of dimension n is denoted as OX n . e order and size of OX n is 9n 2 + 3n and 18n 2 , respectively. In oxide networks OX n , the edge set can be partitioned into six sets depending on degree summation of neighbors of end vertices.
is partition is presented in Table 1.

Complexity
In the following theorems, we investigate 1 χ 1 and NM 2 of OX n .

Theorem 1.
e 1 χ 1 index of oxide networks OX n is given by Proof. We use equation (2) and Table 1 to calculate the expression for 1 χ 1 index of the oxide networks OX n : □ Theorem 2. e NM 2 index of oxide networks OX n is given by Proof. To calculate the expression for neighborhood second Zagreb index of oxide networks OX n , we use equation (3) and Table 1 to get the required result: □ 3.2. Honeycomb Networks. In material sciences, honeycomb networks have vital role in different fields especially in computer graphics, cellular phone base stations, and image processing and in representing benzenoid hydrocarbons. Honeycomb networks are shaped by the iterative use of hexagonal tiles in a specific design. We denote the n-dimensional honeycomb network with n hexagons between central and boundary hexagon by HC n . HC n is constructed by adding layer of hexagon around HC n−1 (see Figure 4). e order and size of HC n is 6n 2 and 9n 2 − 3n, respectively. Table 2 shows partition of edges. By using partition of edges of HC n , we investigate 1 χ 1 and NM 2 of HC n in the following theorems.
Theorem 3. e 1 χ 1 index of Honeycomb network HC n is given by Proof. We use equation (2) and Table 2 to calculate the expression for 1 χ 1 index of the Honeycomb network HC n :  e NM 2 index of Honeycomb network HC n is given by Proof. To calculate the expression for NM 2 index of Honeycomb network HC n , we use equation (3) and Table 2, to get the required result: + 6(n − 1)(7 × 9) + 9n 2 − 21n + 12 (9 × 9) � 729n 2 + 903n + 324.  Figure 7. From a chemical perspective, the oxygen atoms are in fact the corner vertices of tetrahedron SiO 4 and silicon atom is its central vertex. Graphically, we describe the central atom as silicon vertex, corner atoms as oxygen vertices, and edges are bonds between them. Figure 8 illustrates a tetrahedron of SiO 4 . We denote a silicate array with n hexagons between center and border of silicate sheet by SL n . Figure 9 illustrates a three-dimensional silicate network. e order and size of SL n is 15n 2 + 3n and 36n 2 , respectively. In the following theorems, the silicate networks are analyzed through certain graph invariants.
In the next theorem, we computed 1 χ 1 and NM 2 of the SL n in the following way.

Complexity
Proof. We use equation (2) and Table 3 to calculate the expression for 1 χ 1 index of the silicate networks: □ Theorem 6. e NM 2 index of silicate networks is given by Proof. To calculate the expression for NM 2 index of silicate networks, we use equation (3) and Table 3 to get the required result: □ 3.4. Chain Silicate Networks. Chain silicate is acquired by linear arrangement of tetrahedron. n-dimensional chain silicate networks are characterized as follows. A network of n-dimensional chain silicate stands for CS n is acquired by linear arrangement of n tetrahedron. e order and size of CS n with n > 1 is 3n + 1 and and 6n, respectively. A network of n-dimensional chain silicate is illustrated in Figure 10.
Tables 4 and 5 suggest such a partition that we use to compute certain topological indices. Now, within the following theorems, we analyzed the chain silicate networks CS n via topological indices.
e following results provide the investigation of certain topological indices of CS n .
ere are three cases that need to be examined in order to prove these outcomes. First, we demonstrate this result for n � 2. In the graph G � CS 2 , the edge partition is of the type (S(12), S(12)) and (S(12), S(18)) and each of this partition has six edges. en, For the case n � 3, the edge partition CS n is presented in Table 5. Now, 1 χ 1 (CS 3 ) can be computed as follows: Let us now turn to the third case of n > 3. To calculate the expression for 1 χ 1 index of the chain silicate networks CS n , we use equation (2) and Table 4 to obtain the required result: e NM 2 index of the chain silicate networks CS n is given by Proof. ere are three cases that need to be examined in order to prove these outcomes. First, we demonstrate this result for n � 2. In the graph G � CS 2 , the edge partition is of the type (S(12), S(12)) and (S(12), S(18)) and each of this partition has six edges. en, For the case n � 3, the edge partition CS n is presented in Table 5. Now, NM 2 (CS 3 ) can be computed as Let us now turn to the third case of n > 3. To calculate the expression for the NM 2 index of the chain silicate networks CS n , we use equation (3) and Table 4 to obtain the required result:   (S(v)S(w)), where vw ∈ E(CS n ) Cardinality (12, 12) 6 (12, 21) 6 (15,15) n − 2 (15,21) 4 (15,24) 4n − 12 (21, 24) 2 (24,24) n − 4

Hexagonal Networks.
It is a known fact that tilling a plane by regular polygon can be done only with regular triangles, hexagons, and squares. A hexagonal network can be obtained by using triangles (see Figure 11). A hexagonal network with n vertices in each side of the hexagon is denoted by HX n . Let n > 1; then, order and size of HX n is 3n 2 − 3n + 1 and 9n 2 − 15n + 6, respectively. Figure 11 depicts the graph of HX 6 . Tables 6-8 depict a partition that we use to compute the topological indices of hexagonal networks HX n for n ≥ 3. Now, we investigate aforementioned indices of hexagonal networks HX n . In the following theorems, we investigate 1 χ 1 and NM 2 of HX n .

Theorem 9.
e 1 χ 1 index of hexagonal networks HX n is given by Proof.
ere are four cases that need to be examined in order to prove these outcomes. First, we demonstrate this result for n � 2. In the graph G � HX 2 , the edge partition is of the type (S(10), S(10)) and (S(10), S(18)) and each of this partition has six edges. en, For the case n � 3, the edge partition HX 3 is presented in Table 6. Now, 1 χ 1 (HX 3 ) can be computed as For the case n � 4, the edge partition HX 4 is presented in Table 7. Now, 1 χ 1 (HX 4 ) can be computed as Let us now turn to the case of n > 4. To calculate the expression for 1 χ 1 index of hexagonal networks HX n , we use equation (2) and Table 8 to obtain the required result: Proof. ere are four cases that need to be examined in order to prove these outcomes. First, we demonstrate this result for n � 2. In the graph G � HX 2 , the edge partition is of the type (S(10), S(10)) and (S(10), S(18)) and each of this partition has six edges.
en, For the case n � 3, the edge partition HX n is presented in Table 6. Now, NM 2 (HX 3 ) can be computed as For the case n � 4, the edge partition HX n is presented in Let us now turn to the third case of n > 4. To calculate the expression for NM 2 index of hexagonal networks HX n , we use equation (3) and Table 8 to obtain the required result:

Conclusion
e aim of this paper is to discuss the degree-based topological indices of several nanotube networks. In this paper, we determined the neighborhood second Zagreb index and the first extended first-order connectivity index for oxide network (OX n ), silicate network (SL n ), chain silicate network (CS n ), and hexagonal network (HX n ). Also, we determined the neighborhood second Zagreb index and the first extended first-order connectivity index for honeycomb network (HC n ). In this paper, results achieved demonstrate the promising opportunities to apply for chemical sciences. It is very helpful to know more about nanotube networks that have become the central interest of fundamental sciences and study their topological indices that will be completely useful to understand their underlying topologies.

Data Availability
No data were used to support the findings of the study.

Conflicts of Interest
On behalf of all authors, the corresponding author states that there are no conflicts of interest.