Abstract
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.
1 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(α) presents the degree of α ∈ V(B) and
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 ABC4 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.
The first Zagreb index M1 and the second Zagreb index M2 (Reti et al., 2019) for a graph G can be defined as:
The neighborhood first and second Zegreb indices (first defined by Reti et al. (2019)) are as follows:
For more discussion about invariants, see: Ahmad et al. (2017), Alikhani et al. (2014), Akhter and Imran (2016a, 2016b), Akhter et al. (2018, 2019), Bača et al. (2015), Baig et al. (2015a, 2015b), Guirao et al. (2020), Hameed et al. (2019), Hayat and Imran (2014, 2015a, 2015b, 2015c), Imran et al. (2020), Iranmanesh and Zeraatkar (2010), Lin et al. (2014), Manzoor (2015), and Yang et al. (2019).
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, ABC4, GA5, NM1, and NM2 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 W ⊗ F. The vertex set of W ⊗ F is V(W) × V(F) and E(W ⊗ F) = {(w, f)(s,e)|ws ∈ E(W) and fe ∈ E(F)}.
The tensor product of two paths Lp and Lq is described by Lp ⊗ Lq. It is a graph on p × q vertices with vertex set is defined as:
and an edge between the (x1, y1) and (x2, y2) exists if and only if:
The graph Lp ⊗ Lq is known as a generalized Aztec diamond with vertex cardinality pq. L9 ⊗ L10 is depicted in Figure 1. In the next theorem, the ABC index for generalized Aztec diamond graphs has been computed.
The partition of the edges of L9 ⊗ L10 that depend on the degree of every end vertex of each edge.
Theorem 2.1
The ABC index of generalized Aztec diamond graph G = Lp ⊗ Lq 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 Lp ⊗ Lq is derived as:
The cardinality of edges of type (d(m), d(n)) of Lp ⊗ Lq, where m and n are degrees of adjacent vertices of every edge
| (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) |
After simplification of above calculations, we acquire the following result of ABC index:
In Theorem 2.2, we have computed the GA invariant of generalized Aztec diamond.
Theorem 2.2
For
Proof
Again, by using the information given in Table 1, the GA invariant of generalized Aztec diamond is derived follows; since:
From an easy simplification, we acquired:
Similarly, the ABC4 index of the generalized Aztec diamond can be computed by using Table 1.
Theorem 2.3
For p, q ≥ 2, the ABC4 index of Lp ⊗ Lq is calculated as:
Proof
In the ABC4 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 (Sm, Sn) − 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 ABC4 invariant of Lp ⊗ Lq is computed as follows; since:
The cardinality of edges of type (Sm, Sn) of Lp ⊗ Lq, where m and n are sum of degrees of all neighboring vertices of end vertices of each edge
| (Sm, Sn) with mn ∈ E(G) | Number of edges |
|---|---|
| (4, 9) | 4 |
| (6,6) | 4 |
| (6,12) | 8 |
| (8,12) | 4(p + q − 10) |
| (9,8) | 8 |
| (9,16) | 4 |
| (12,12) | 4 |
| (12,16) | 4(p + q − 10) |
| (16,16) | 2 (p − 6)(q − 5) + 8 |
By an easy calculation, the above equation can be written as:
The following theorem gives the GA5 index of generalized Aztec diamond Lp ⊗ Lq.
Theorem 2.4
For p, q ≥ 2, the GA5 index of generalized Aztec diamond G = Lp ⊗ Lq
Proof
By applying the information given in Table 2 on Eq. 2, we get:
After simplification, we get:
Theorem 2.5
For p, q ≥ 2, the NM1 index of Lp ⊗ Lq is calculated as:
Proof
In the NM1 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 2(p + q − 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 − 4) (q − 4) vertices (black color in Figure 1) of degree sum is equal to 16. Now, by applying this information, the NM1 index of Lp ⊗ Lq is computed as follows; since:
By an easy calculation, the above equation can be written as:
The following theorem gives the NM1 index of generalized Aztec diamond Lp ⊗ Lq.
Theorem 2.6
For p, q ≥ 2, the NM2 index of Lp ⊗ Lq is:
Proof
Since:
Therefore, by applying this information given in Table 2, the NM2 index of Lp ⊗ Lq is computed as follows:
3 ABC, GA, ABC4, GA5, NM1, and NM2 indices of tetrahedral diamond lattice
The structure of tetrahedral diamond lattice (Manuel et al., 2015) of t dimension is constructed by t layers, every one interpreted as lt. The starting layer has just a single vertex and the next one have 4 vertices, which is isomorphic to S4. For t ≥ 3, every layer l has
The vertex set of tetrahedral diamond of dimension t has
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 + 1, l ≥ 2, are connected as:
The vertex having label
The vertex with label
Figure 2 depicts a 5-dimension tetrahedral diamond.
The graph of 5-dimension tetrahedral diamond lattice.
In Theorem 3.1, the value of ABC4 invariant of tetrahedral diamond lattice is derived.
Theorem 3.1
For t ≥ 1, the ABC index of tetrahedral diamond lattice G with dimension t is:
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:
The quantity of (m, n)-type edges of tetrahedral diamond G, where m and n are degrees of the adjacent vertices of each edge
| (d(m), d(n)) with mn ∈ E(G) | Number of edges |
|---|---|
| (1,4) | 4 |
| (2,4) | 12(t − 2) |
| (3,4) | 6(t − 2)(t − 3) |
| (4,4) |
|
After computation, we get:
Theorem 3.2
The GA index of G (tetrahedral diamond lattice) of dimension t where t ≥ 1 is given by:
Proof
The GA index formula of tetrahedral diamond lattice G is deduced by using the information from Table 3 as:
After an easy computation, we acquire:
Now we shall calculate the ABC4 and GA5 index for tetrahedral diamond lattice G in upcoming theorems.
Theorem 3.3
The result for ABC4 index of tetrahedral diamond lattice G is:
Proof
Let Sm denote the sum of degrees of all the vertices joined with vertex m. The (Sm, Sn )- type edges are the edges with sum of degrees of neighboring vertices of the end vertices of every edge is Sm and Sn. The formula for ABC4 index of tetrahedral diamond lattice G can be deduced by applying the edge partition of Table 4 and written as:
After simplification, we acquire:
The quantity of (Sm, Sn)-type edges, where m and n are sum of degrees of all neighboring vertices of the end vertices of each edge
| (Sm, Sn) with mn ∈ E(G) | Number of edges |
|---|---|
| (4,7) | 4 |
| (7,8) | 12 |
| (8,10) | 12(t − 3) |
| (12,10) | 4(t − 2)(t − 3) |
| (12,13) | 2(t − 2)(t − 3) |
| (16,13) |
|
In upcoming theorem, we compute the ABC4 index for tetrahedral diamond lattice.
Theorem 3.4
The GA5 index for the tetrahedral diamond lattice G is given by:
Proof
By applying the information from Table 4 in Eq. 5, we derive the GA5 index as:
which can be deduced to:
Theorem 3.5
The result for NM1 index of tetrahedral diamond lattice G is:
Proof
Let Sm denote the sum of degrees of all the vertices joined with vertex m. The degree sum of the vertex in the graph is either 4 or 7 or 8 or 10 or 12 or 16. The computation gives that:
Using this information NM1 index of tetrahedral diamond lattice G can be calculated as follows:
In upcoming theorem, we compute the GA5 index for tetrahedral diamond lattice.
Theorem 3.6
The NM2 index for the tetrahedral diamond lattice G is given by:
4 Conclusion
In this article, some degree depended invariants namely ABC, GA, ABC4, GA5, NM1, and NM2 indices are computed for the networks constructed from generalized Aztec diamonds and tetrahedral diamond lattices. We have found the exact values of these parameters for the said 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.
Acknowledgement
The authors are very grateful to the referees for their constructive suggestions and useful comments, which improved this work very much.
Research funding:
This research is supported by the the UPAR Grant of United Arab Emirates University (UAEU), Al Ain, United Arab Emirates via Grant No. G00002590 and UPAR Grant of UAEU via Grant No. G00003271.
Author contributions:
Syed Ahtsham Ul Haq Bokhary: writing – original draft, writing – review and editing, methodology, formal analysis; Muhammad Imran: writing – original draft, writing – review and editing, visualization, project administration, funding acquisition; Shehnaz Akhter: writing – original draft, investigation, resources; Sadia Manzoor: writing – original draft, data collection.
Conflict of interest:
One of the authors (Muhammad Imran) is a Guest Editor of the Main Group Metal Chemistry's Special Issue “Topological descriptors of chemical networks: Theoretical studies” in which this article is published.
Data availability statement:
The data used to support the findings of this study are included within the article.
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© 2021 Syed Ahtsham Ul Haq Bokhary et al., published by De Gruyter
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