Degree-Based Indices of Some Complex Networks

School of Computer Engineering, Anhui Wenda University of Information Engineering, Hefei 231201, China Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan Department of Basic Sciences, Balochistan University of Engineering and Technology Khuzdar, Khuzdar 89100, Pakistan Department of Mathematical Sciences, Fatima Jinnah Women University, Rawalpindi, Pakistan School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, Anhui, China


Introduction and Preliminary Results
A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds between atoms.
In the modern age, network structures have great significance in the field of chemistry, information technology, communication, and physical structures. Each network can be distinguished by a numeric quantity associated with it by defined rules under certain parameters. ese rules are known as topological indices. Any numerical value allocated to a graph, which classifies the structure of a graph, is called a topological index. Popular and well-studied types of topological indices are degree-based topological indices, distance-based topological indices, and counting-related polynomials and indices of graphs. Degree-based topological indices are of great importance among these groups and play a strong role in chemical graph theory and in chemistry. More specifically, to guess the biological activity of various where d i is the degree of each v i and F is a pertinently picked function with the characteristic F(x, y) � F(y, x). A huge amount of topological indices were introduced by various chemists in the advanced studies of indices. For different molecular families, several researchers have shown different computational and theoretical results related to certain topological indices and related them with energies of the graphs. If f 1 , f 2 , . . . , f n are the eigenvalues of the matrix TI, then energy can be defined as e most extensively studied graph energy is the Randic index. J. Rad et al. [3] analyzed the energy (Zagreb energy) and Estrada (Zagreb Estrada) index of a graph, and both are based on the Zagreb matrix's eigenvalues. Furthermore, for these new graph invariants, they define upper and lower limits and relationships between them. e relationship between the Kirchhoff index and Laplacian graph energy is introduced by Das et al. [4]. Milovanovic � et al. [5] gave some lower bound for Kirchhoff index as well some new lower bounds for the Laplacian energy of a graph in the same article. For any connected graph G, Bozkurt et al. [6] acquired an upper bound for distance energy. For the distance energy of connected diameter 2 graphs with given numbers of vertices and edges, they gave an upper bound. In addition, they also provide a lower bound for the distance energy of unicyclic graphs having odd girth. Alikhani et al. [7] compute the ABC index for some families of nanostars and polyphenylene dendrimers. Ba� ca et al. [8] studied the geometric-arithmetic (GA) indices of carbon nanotube networks and fullerene networks. Baig et al. [9,10] computed Omega, Sadhana, and Pl polynomials, for Benzoid nanotubes for the first time. For these interconnection networks, Baig et al. [11] compute the first general indices of Zagreb, ABC, GA, ABC 4 , and GA 5 and give closed formulae of these indices.
In pairs of past decays, the use of graph theoretical methods to explain the chemical structure has gained more and more importance. Since the early periods in which such formalism was used to predict simple properties on simple molecules, such as alkane boiling points, up to the design of novel lead anticancer drugs, for example, considerable progress was made and a long path was covered.
In this paper, by graph G, we always mean the network having vertex set V(G) and edge set E(G). We represent the degree of u ∈ V(G) by d u and We are using notations in the present paper from [12,13]. e atom-bond connectivity (ABC) is the well-known degree-based topological index, which is introduced by Estrada et al. in [14] and defined as Vukičević et al., in [15], introduced a well-known connectivity topological descriptor is geometric-arithmetic (GA) index and defined If we can find the edge partition of these interconnection chemical networks based on the sum of the degrees of the end vertices of each edge in these graphs, only ABC 4 and GA 5 indices can be computed. e fourth version of ABC index is introduced by Ghorbani et al. [16] and is given as follows: e fifth version of GA index is introduced by Graovac et al. [17] and is given as follows: Because of their great importance in chemistry, these topological indices are extensively studied by various mathematicians. Hayat and Imran studied ABC 4 and GA 5 and gave close formulae of these indices for some nanotubes and their corresponding nanotori. ey also give a characterization of k-regular graphs with respect to their GA 5 [18]. In [19], Hayat and Imran compute the (ABC), (GA), and Zagreb indices of VC 5 C 7 [p, q], HC 5 C 7 [p, q] and SC 5 C 7 [p, q] nanotubes. ey also compute ABC 4 and GA 5 for these nanotubes. Similarly, they compute the (ABC 4 ) and (GA 5 ) indices of H-naphtalenic nanotubes and chain silicate, silicate, oxide, hexagonal, and honeycomb networks in [20,21]. is paper deals with a specific organization form of matter. Other forms and description are given and discussed by different authors. For example, for the first time, Ali and Mehdi compute the GA index of TUZC6[p, q], TUAC6[p, q], HC5C7 [p, q], and SC5C7[p, q] nanotubes in [22]. In [23], W. Lin et al. disprove Dimitrov's "mo du lo7conjecture." Shang established new lower bounds for the Gaussian Estrada index in terms of the first Zagrab index and the number of vertices and edges in [24]. Also, Shang obtained the upper and lower bounds for the Laplacian Estrada index of Γ(G) based on the vertex degrees of the graph G in [25]. e rest of the paper breaks as follows. Sections 2-5 contain the degree-based topological indices of enhanced mesh, triangular mesh, star of silicate network, and rhenium trioxide lattice, respectively. In Section 6, we give the conclusion of the paper and pose some open problems.
roughout this paper, (d u , d v ) represents the number of edges of the given graph G with end vertices of each edge having degrees d u and d v , respectively. Similarly, (S u , S v ) denotes the number of edges of the given graph G with end vertices of each edge having degree sum S u and S v , respectively. By degree sum S u , it is meant to be the sum of degrees of all vertices adjacent to the vertex u.

Enhanced Mesh
In this section, we study the degree-based topological descriptors of enhanced mesh network [26].

Construction.
e graph whose vertices correspond to the points in the plane with integer coordinates, x-coordinates in the range 1, 2, . . . , n and y-coordinates in the range 1, 2, . . . , m and two vertices are connected by an edge whenever the corresponding points are at distance 1, is a common form of lattice graph. In other words, for the point set mentioned, it is a unit distance graph. e term n-mesh has also been given to various other types of graphs with a certain structure in the literature, such as the Cartesian product of a number of path graphs. e Cartesian product of paths of order a 1 , a 2 , . . . , a n is an n-mesh M(a 1 , a 2 , . . . , a n ), which is defined as M a 1 , a 2 , . . . , a n � P a 1 × P a 2 × · · · × P a n . (7) For n-mesh, M(a 1 , a 2 , . . . , a n ) has order V M a 1 , a 2 , . . . , a n � a 1 , a 2 , . . . , a n (8) and size E M a 1 , a 2 , . . . , a n � a 1 , a 2 , . . . , a n n − 1 Here, we are going to discuss the enhance 2-mesh network.
and the set is an edge set. An enhanced mesh EM(P a 1 × P a 2 ) is resulted by replacing each 4-cycle of M(P a 1 × P a 2 ) by a wheel W 4 on 4 vertices. us, a wheel W 4 is a graph retrieved by joining the central vertex to each vertex of cycle Figure 1, we see that the number of edges of type (3,4) and (3, 5) are 4 and 8, respectively.
Every vertex that is lying on the boundary of the graph EM(m, n), except the corner vertices, are of degree 5 and the oblique edges which are adjacent to the vertex of degree 4 are edges of type (4,5). ere are total 2(m − 2) + 2(n − 2) � 2m + 2n − 8 � 2(m + n − 4) vertices on the boundary of degree 5, and each vertex induces two edges of type (4,5).
us, the number of edges of type (4, ere are total 2(m + n − 4) vertices on the boundary of the graph G which are of degree 5. Each vertex induces one edge of type (5,8) and two edges of type (4,5). us, the number of edges of type (5, 8) are 4(m + n − 4). Furthermore, there are total 4(m + n − 6) edges of type (5,5). is edge partition of enhanced mesh based on the degrees of end vertices is shown in Table 1. Now, by using this edge partition, we compute the ABC index of enhanced mesh as follows: is implies that

Journal of Mathematics
After simplification, we obtain We will compute the GA index of enhanced mesh, in the following result.
Proof. Using edge partition given in Table 1 and the formula e ABC 4 and GA 5 indices of the enhanced mesh is computed in the next two results. □ Theorem 3. For m, n ≥ 5 and ζ, ς � τ, c 2 ∈ R, the ABC 4 (G) index of enhanced mesh EM(P a 1 × P a 2 ) is ABC 4 (EM(P a 1 × P a 2 )) � ζmn + ςm + τn + c 2 ≈ 1.3054mn − 0.9559(m + n) −   partition of the graph G shown in Figure 1 is computed in Table 2.
Proof. Following the information given in Table 2 and the formula uv∈E(G) 2( ���� S u S v /S u + S v ), we easily get the required proof.

Triangular Mesh
In this section, we are going to study the degree-based topological descriptors for the triangular mesh network [26].
We denote the radix − n triangular mesh network by T n having node set V T n � (x, y) | 0 ≤ x, y ≤ n and 0 ≤ x + y ≤ n , (18) and there exists a mesh arc between nodes (x 1 , y 1 ) and (x 2 , y 2 ) if |x 1 − x 2 | + |y 1 − y 2 | ≤ n − 1 and x 1 + y 1 ≤ x 2 + y 2 . e number of vertices (nodes) in a T n is n(n + 1)/2. e degree of node in the aforementioned network may be 2, 4, or 6. ere exist three vertices of degree 2, which we call as corner vertices. roughout this section, we represent G by the graph of triangular mesh network T n . e graph of triangular mesh T 5 is shown in Figure 2.
}. e edge partition of the graph G based on the degrees of the end vertices lying at distance one from the end vertices of each edge is shown in Table 3.
By using the edge partition above, we calculate the triangular mesh ABC index as follows: is implies that After simplification, we obtain We will compute the GA index of the triangular mesh in the following theorem. □ Theorem 6. For n ≥ 4, Proof. e result is followed by Table 3, and ). e three corner vertices has degree sum 8. e 6 vertices adjacent to the corner vertices has degree sum 16.  e ABC 4 index of the graph G, for n ≥ 8, is computed as Proof. We use the information given in Table 4 to compute the formula for ABC 4 index for G.
After a simple calculation, the above equation can be reduced as  Table 3: e T n edge partition depends on the degree of the end vertices of each edge, where uv ∈ E(G).

Star of Silicate Network
In the present section, we explore the degree-based topological descriptors for the star of silicate network [27].
We define the creation of a new star of a silicate network from the star of the David network in Figure 3. e resulting network is called the n-dimensional star of David network, which is denoted by SD(n).
Step5: replacing each K3 subgraph with a tetrahedron. is network is known as the n dimension star silicate star or star of silicate network star and is denoted by SSL(n). e graph for the SSL(2) silicate network star is shown in Figure 4. roughout this section, we denote the graph of star silicate network SSL(n) by G.
In the graph of star of silicate network, the 6 corner vertices and the central vertex of each tetrahedron is of degree 3. e vertices inserted in step two of the construction have degree 4. All the remaining vertices have degree 6.
us, the set of all distinct degrees d u for u ∈ V(SSL(n)) is 3, 4, 6 { }. e following table give the edge partition of G. e ABC and GA indices of star of silicate network are computed in the next two theorems. Theorem 9. For n ≥ 3, ABC(G) � 12 Proof. We use the information given in Table 5 to compute the ABC index of star of silicate network as follows: By further simplification, we get the following form:

□
Step 1 Step 3 Step 2 Step 4 Proof. e information in Table 5 and the expression uv∈E(G) 2( ) yields the required result. e central vertices of the tetrahedron lying on the corner has degree sum 11. e central vertices of the tetrahedron that are adjacent to the vertices of degree 4 has degree sum 16. e central vertices of the remaining tetrahedron has degree sum 18. e vertices of degree 4 of the tetrahedron lying at corners has degree sum 14. e 4 degree vertices that are adjacent to 6 degree vertices has degree sum 19. e remaining 4 degree vertices has degree sum 17. e vertices of degree 6 have three kinds of degree sum. e vertices adjacent to one vertex of degree 4 have degree sum 26, the vertices adjacent to two vertices of degree 4 have degree sum 28, and the remaining vertices have degree sum 30.
us, the set of all distinct degree sums S u for u ∈ V(SSL(n)) is 11,14,16,17,18,19,26,28,30 { }. From the above information and the construction of the graph, the edge partition is calculated as follows.
e ABC 4 and GA 5 of star of silicate network SSL(n) are given in the following two results.

□
Proof. We use the information given in Table 6 to compute the formula for ABC 4 index for G as follows: is implies that ABC 4 (SSL(n)) � 6