Abstract
This study consists of developing some closed and updated formulas derived from multiplicative graph invariants such as general Randic index for , ordinary general geometric-arithmetic (OGA), general version of harmonic index (GHI), sum connectivity index (SI), general sum connectivity index (GSI), 1st and 2nd Gourava and hyper-Gourava indices, (ABC) index, Shegehalli and Kanabur indices, 1st generalised version of Zagreb index (GZI), and forgotten index (FI) for the subdivided Aztec diamond network. Aztec diamond is constructed based on the squares boxes. These square boxes are placed at the centre point and nourish the condition . Furthermore, we put a new vertex of degree-2 at each edge of the small boxes, squares in shapes. A new structure is obtained that has the same properties as its parental graph and is called a subdivided Aztec diamond and symbolised as Saztecn. Subsequently, we compute the multiplicative topological attributes to get some new formulas. For this purpose, a simple, connected, and the finite graph is considered by supposing it Y as the graph of the Saztecn. The order and size have also been discussed in this study and found three different kinds of edges (2, 2), (2, 3), and (2, 4) for computing. The discussion on the networks mentioned above provides us with essential results that can be used in the determination of bio and physio activities and can be interspersed with the molecular compounds and their graphical structures better to understand their physical as well as biological properties.
1. Introduction
The branch of mathematics concerned with graphs, their application, and their correlation with chemical compounds is known as chemical graph theory. In molecular modelling, we use this tool of mathematics. First, the chemical compound’s structure is drawn and compared with its mathematical graphical structure. This theory needs to be used as a mathematical tool to recognise a particular molecular web’s physical and biological features.
Vukicevi’c and Furtula’ developed the 1st GA [1] index in 2009 and formulated as
An OGA invariant [2] was determined in 2011 and symbolised as given in the following real :
In 2017, V.R. Kulli developed 1st and 2nd Gourava and hyper-Gourava indices [3, 4] are computed by
Randic’ index was developed [5] by Milan Randic’ in 1975 and calculated by
Then, Erdos and Bollobas’ invented its general version, familiarised with the general Randic’ index for , where [6], and evaluated by
Zhong [7], in 2012, gave the idea of the harmonic index and familiarised by
Yan determined its generalised form [8] in 2015 and symbolised as
ABC invariant was introduced in 1998 by Estrada et al. [9]; that is,
SK, SK1, and SK2 invariants [10] were introduced by Shegehalli & Kanabur that are
In 2009, Lucic’ described (SI) [11] and computed it as
Then, in 2010, Zhou and Trinajstic generalised it [8, 12] as
Zheng, in 2005, developed the general version of the 1st Zagreb index [13]
Furtula’ and Gutman [14] formulated an invariant known as F-index
2. Material and Methods
Aztec diamond is created based on the square lattices. These square lattices are kept centred at (s, r), satisfying . In addition, we place a new node having 2 as the degree at every edge of the small squares. In this way, we get a new structure known as a subdivided Aztec diamond Saztecn. Next, we evaluate the multiplicative topological attributes in order to obtain new formulas. Let us suppose Y is the graph of the Saztecn. The cardinality of Saztecn with respect to vertices is and with respect to edges, is . There are three different kinds of edges (2, 2), (2, 3), and (2, 4).
3. Results and Discussion
We have implemented various multiplicative graph invariants [15] over the given molecular structures. Figures 1–4 have been depicted for better understanding. We describe two essential components, nodes and edges. Following, some theorems have been constructed with the help of these particular graphical invariants. Let be the cardinality of Y with respect to edge set. The Saztecn is established at the terminal nodes of every edge.
Let’s choose Y as a graph of the subdivided Aztec diamond network Saztecn, defining the terms de and df as the degrees of nodes e and f. We have developed the following theorems.
Theorem 1. For Saztecn, its OGA can be developed as
Proof. With the help of Table 1, we inferBy making some computations, we get
Theorem 2. For Saztecn, 1st and 2nd Gourava descriptors can be developed as
Proof. With the help of Table 1, we inferAfter some calculations, we have
Theorem 3. For Saztecn, 1st and 2nd hyper-Gourava descriptors can be developed as
Proof. With the help of Table 1, we inferBy making some computations, we get
Theorem 4. For Saztecn, GRI can be formulated as
Proof. We know that
Case 1. For , its RI can be formulated asUsing (24) and from Table 1, we getAfter some computations, we get
Case 2. For , its Randic’ index can be computed asUsing (24) and from Table 1, we knowAfter some computations, we get
Case 3. For , its Randic’ index can be computed asUsing (24) and from Table 1, gettingBy doing some calculations, we get
Case 4. For , its Randic’ index can be computed asUsing (24) and from Table 1, we knowBy doing some calculations, we get
Theorem 5. For Saztecn, HI can be developed as
Proof. With the help of Table 1, we inferAfter simplifications, we obtain
Theorem 6. For Saztecn, GHI can be developed as
Proof. With the help of Table 1, we inferBy making some computations, we get
Theorem 7. For Saztecn, the ABC index can be developed as
Proof. With the help of Table 1, we inferBy making some computations, we get
Theorem 8. For Saztecn, SK, SK1, and SK2 descriptors can be developed as
Proof. With the help of Table 1, we inferBy making some calculations, we get
Theorem 9. For Saztecn, SI can be developed as
Proof. With the help of Table 1, we inferBy making some computations, we get
Theorem 10. For Saztecn, GSI can be developed as
Proof. With the help of Table 1, we inferBy making some computations, we get
Theorem 11. For Saztecn, 1st GZI can be developed as
Proof. With the help of Table 1, we infer
Theorem 12. For Saztecn, F-index can be developed as
Proof. With the help of Table 1, we inferBy making some computations, we get
4. Main Findings
(i)Consideration of the molecular structure of the Aztec diamond network(ii)Getting the subdivided version of the Aztec diamond network for n = 1, 2, 3, 4 by inserting a new node at each edge and placing the name of the new derived molecular structure as Saztec4(iii)Association of the mathematical graph with the chemical structure(iv)Vertex labelling for each vertex with their degrees(v)Edge partition of edge set according to their degrees(vi)Computations of the degree of each vertex by constructing the generalised formula(vii)Construction of new closed formulas using many various topological attributes such as general Randic’ index for , GHI, OGAI, SHI, GSHI, 1st and 2nd Gourava and hyper-Gourava descriptors, ABC invariant, SKs’ indices, and F-index5. Conclusions
This work involves inventing many new formulas based on multiplicative graph invariants. We have used many indices such as GRI, OGA, GHI, SI, GSI, 1st and 2nd Gourava and hyper-Gourava indices, ABC index, SKs’ indices, 1st GZI and forgotten index (FI) for Saztecn The above-evaluated formulas can be interspersed with the molecular compounds and their graphical structures to understand their physical and biological properties better. More applications can be investigated for these above-mentioned topological indices.
Data Availability
No data were used in this manuscript.
Additional Points
Future Work. The latest topological indices can be found and applied to more molecular and general mathematical networks.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The researchers would like to thank the Deanship of Scientific Research, Qassim University for funding the publication of this project.