Molecular topological invariants of certain chemical networks


 Topological descriptors are the graph invariants that are used to explore the molecular topology of the molecular/chemical graphs. In QSAR/QSPR research, physico-chemical characteristics and topological invariants including Randić, atom-bond connectivity, and geometric arithmetic invariants are utilized to corelate and estimate the structure relationship and bioactivity of certain chemical compounds. Graph theory and discrete mathematics have discovered an impressive utilization in the area of research. In this article, we investigate the valency-depended invariants for certain chemical networks like generalized Aztec diamonds and tetrahedral diamond lattice. Moreover, the exact values of invariants for these categories of chemical networks are derived.


Introduction
There are many diverse applications of mathematics in electronic and electrical engineering. It relies upon what specific region of electronic and electrical designing you are interested, for instance, there is a masses abstract mathematics invloved in communication, signal processing, network theory, etc. Networks use vertices for communicating with each other. The thoery of networking includes the investigation of the most ideal method of executing a network. Graph theory has given chemists beneficial tools and techniques, such as molecular topological descriptors and molecular topological polynomials. The structure of a molecular graph is constructed with help of molecules and molecular compounds. A molecular graph represents the structural result of a chemical compound in forms of graph theory, in which vertices are denoted by atoms and edges correlate to chemical bonds between their atoms. A newly famous area is cheminformatics which is a common region of mathematics, chemistry, and information science. This new subject describes a relationship between the QSAR and QSPR that are utilized to investigate (in certain degree of accuracy) the theoretical and biological activities of several chemical compounds. In the QSAR/QSPR investigation, the topological invariants like ABC index is known for the prediction of bioactivity of a given chemical compounds.
A topological descriptor is a numeric number achived from different properties of graphs. It also describes the molecular topology of that graph and is not changed under the graph automorphism. The topological indices are generally of two types: one is degree-related and the otheris distance-related. In the degree-related indices, it depends upon the vertex-degree of all vertices and in the distance-related indices, it depends on the distance among the each pair of vertices. It is important to know that a theoretical chemist Wiener (1947) gave the idea of topological indices when he was investigating the characteristics of boiling point of paraffin. Firstly, he termed it as path number but after a bit, it was changed to Wiener index (Wiener, 1947) and the reserch of topological invariants started.
Here, V(B) and E(B) are the vertex and the edge sets of a network B. Some of the terminologies which are used in this paper are given as follows: d(a) presents the degree of a∈V(B) and S a = β ∑ All the terminologies used in this manuscript are acquired from books (Diudea et al. 2001, Gutman andPolansky, 1986). Estrada et al. (1998) gave the concept of renowned degree related topological invariant atom-bond connectivit and described as: The other notable degree dependend invariant is geometric-arithmetic index established by Vukičević et al. (2009) and interpreted as: In both these indices, first we find the possible degree of all the vertices and then partitined the edges of B depending upon the degree of every end vertex adjacent to edge.
Later, Ghorbani and Hosseinzadeh (2010) presented the 4th kind of ABC invariant by generalizing the idea of ABC index. It is interpreted as: Graovac et al. (2011) generalized the geometric index by defining the fifth kind of GA index in a same way as ABC 4 index. The index is written as: In both these indices, first we find the possible degree of all vertices and then partitined the edges of B depending upon the degree of vertices adjacent to every end vertex of each edge.
In this article, we derive the certain degree related molecular topological invariants for chemical networks like generalized Aztec diamonds, tetrahedral diamond lattice, and certain infinite classes of nanostar dendrimers. We compute the analytical formulas for above families of chemically applicable networks.
2 ABC, GA, ABC 4 , GA 5 , NM 1 , and NM 2 indices of generalized Aztec diamonds In this part of the article, we discuss the degree-based topological descriptors for the generalized Aztec diamonds. For any two graphs W and F, the tensor product of the graphs W and F is interpreted by ( )}. The tensor product of two paths L p and L q is described by L p ⊗ L q . It is a graph on p × q vertices with vertex set is defined as: and an edge between the (x 1 , y 1 ) and (x 2 , y 2 ) exists if and only if: The graph L p ⊗ L q is known as a generalized Aztec diamond with vertex cardinality pq. ⊗ L L 9 1 0 is depicted in Figure 1. In the next theorem, the ABC index for generalized Aztec diamond graphs has been computed.

Theorem 2.1
The ABC index of generalized Aztec diamond graph = ⊗ G L L p q for ≥ p q , 2, is: Proof: Firstly, we identify that the graph G has vertices having degree one, two, and four. Thus, the graph G has only edges with end vertices have degree one, two, or four. So possible edges are of type (1,4), (2,2), (2,4), and (4,4), where by edge of type (d(m). d(n)), we mean end vertices of the edge with degrees m and n. In Table 1, all the edges of type (1,4), (2,2), (2,4), and (4,4) are counted. Now, by appying the information shown in Table 1, the ABC index of L p ⊗ L q is derived as: After simplification of above calculations, we acquire the following result of ABC index:

(d(m), d(n)) with mn ∈ E(G) Quantity of edges
(1,4) 4 (2,2) 4 (2,4) 4(p + q -6) (4,4) 2(p -3)(q -3) From an easy simplification, we acquired: Similarly, the ABC 4 index of the generalized Aztec diamond can be computed by using Table 1. Proof: In the ABC 4 index, we first find possible degrees of all the vertices coneected to end vertices of each edge. An edge with sum of the degrees of end vertices is m and n, and is interpreted as (S m , S n ) − type edges. Thus, the possible edges are of type type (4,9), (6,6), (9,8), (8,12), (12,16), (16,16), (6,12), (9,16), and (12,12). In Table 2, all the edges of type (4,9), (6,6), (9,8), (8,12), (12,16), (16,16), (6,12), (9,16), and (12,12) are counted. Now, by applying the information interpreted in Table 2, the ABC 4 invariant of L p ⊗ L q is computed as follows; since: By an easy calculation, the above equation can be written as: Proof: By applying the information given in Table 2 on Eq. 2, we get: Proof: In the NM 1 index, we first find possible degrees sum of all the ertices. The possible degree sum of a vertex is either, 4 or 6 or 8 or 9 or 12. There are 4 vertices (red color in Figure 1) of degree sum is equal to 4. There are 8 vertices (green color in Figure 1) of degree sum is equal to 6. There are + − p q 2( 8) vertices (blue color in Figure 1) of degree sum is equal to 8 and 12. There are 4 vertices (blue color in Figure 1) of degree sum is equal to 9. There are − − p q ( 4)( 4) vertices (black color in Figure 1) of degree sum is equal to 16. Now, by applying this information, the NM 1 index of ⊗ L L p q is computed as follows; since: By an easy calculation, the above equation can be written as: Therefore, by applying this information given in Table 2, the NM 2 index of ⊗ L L p q is computed as follows: The structure of tetrahedral diamond lattice (Manuel et al., 2015) of t dimension is constructed by t layers, every one interpreted as l t . The starting layer has just a single vertex and the next one have 4 vertices, which is isomorphic to There is no any odd cycle in Tetrahedral diamond, so it is a bipartite graph.
The tetrahedral diamond lattice of dimension t is constructed with the help of t-layers in the following ways. The vertex in the first(starting) layer with label 1 is connected with a vertex in layer 2 with label 3. The layers l and + ≥ l l 1, 2, are connected as: in layer l is connected to the vertex which has label . Figure 2 depicts a 5-dimension tetrahedral diamond. In Theorem 3.1, the value of ABC 4 invariant of tetrahedral diamond lattice is derived. Proof: The graph G contains only vertices of degree one, two, three and four. The edges of the graph G of dimension t are of the form (1,4), (2,4), (3,4), and (4,4). The construction of graph indicates that the number of edges of tupe (1,4) are four for every layer. There are 6t-12 vertices of 2 degree, and two edges are induced by an each vertex. So, number of (2,4)-type edges is 12(t-2). Also by induction argument, the number of (3, 4)-type edges is: Note that the quantity of (4,4)-type edges can be obtained by removing all edges of type (1,4), (2,4), (3,4) from G. Thus, the quantity of (4, 4)-type edges is: From Table 3, the formula for ABC index can be deduced to:  After computation, we get: Proof: The GA index formula of tetrahedral diamond lattice G is deduced by using the information from Table 3  Now we shall calculate the ABC 4 and GA 5 index for tetrahedral diamond lattice G in upcoming theorems.

Theorem 3.3
The result for ABC 4 index of tetrahedral diamond lattice G is: In upcoming theorem, we compute the ABC 4 index for tetrahedral diamond lattice.

Theorem 3.4
The GA 5 index for the tetrahedral diamond lattice G is given by: Proof: By applying the information from Table 4 in Eq. 5, we derive the GA 5 index as:  The result for NM 1 index of tetrahedral diamond lattice G is: classes of graphs. This, we believe will work in the favour of researcher working in network science to investigate the underlying topologies of given networks.
In future, it is interesting to design certain new networks and then investigate their topological properties and compute their exact values which can be fruitful in understanding of the underlying topologies.