Research Article  Open Access
Muhammad Imran, Mian Muhammad Zobair, Hani Shaker, "EccentricityBased Topological Invariants of Dominating DavidDerived Networks", Journal of Chemistry, vol. 2021, Article ID 8944080, 10 pages, 2021. https://doi.org/10.1155/2021/8944080
EccentricityBased Topological Invariants of Dominating DavidDerived Networks
Abstract
A topological index is a numerical descriptor of the molecular structure based on certain topological features of the corresponding molecular graph. Topological indices are scientific contemplations of a graph that outline its subatomic topology and are graphinvariant. In a QSAR/QSPR study, topological indices are utilized to anticipate the physicoconcoction resources and bioactivity of compounds. In this paper, we study some distancebased topological indices such as eccentric connectivity index (ECI), total eccentricity index (TEI), and eccentricitybased Zagreb index for dominating Davidderived networks (DD network) and provide exact formulae of the said indices. These outcomes are valuable to organize the science of hidden topologies of this network.
1. Introduction and Preliminary Results
Graph theory has given chemists a decent variety of helpful apparatuses in terms of topological indices. Atoms and molecular compounds are regularly displayed by a molecular graph. An atomic graph is a delineation of the basic equation of a synthetic compound as far as graph hypothesis whose vertices deliver a connection between the molecules of compound and edges relate to synthetic bonds. In the QSAR/QSPR contemplate, topological files are utilized to anticipate physicoconcoction properties and bioactivity of the substance mixes. A topological index is a number related with a graph that portrays the topology of diagram; furthermore, it is invariant under graph automorphism. Distancebased topological records are of incredible significance and assume an essential part in concoction diagram hypothesis and especially in hypothetical science.
Let be an vertex molecular graph with vertex set and edge set . The vertices of correspond to atoms, and an edge between two vertices corresponds to the chemical bond between these atoms. For a given vertex , the eccentricity is defined as the largest distance between and any other vertex in .
The eccentricconnectivity index of graph is denoted as . It is a distancebased topological index and is defined as
When the vertices’ degrees are not taken into account, we obtain the total eccentricity index of graph defined by
A new version of Zagreb indices of a molecular graph defined by Ghorbani and Hosseinzadeh [1] in terms of eccentricity are expressed as follows:
This paper first contains a little insight of the graph and algorithm of dominating Davidderived networks. Then, the labeling of critical vertices of dominating Davidderived network is provided. Then, computation of eccentricities of vertices is given. In the last segment, we process and determine some distancebased topological indices, such as eccentricconnectivity index , totaleccentricity index , and eccentricitybased Zagreb index of dominating Davidderived . For some recent results, see [2, 3].
2. Algorithm and Insight into Dominating DavidDerived Networks
Simonraj [4] gave a construction algorithm for the dominating Davidderived networks, and originated from the graph of star of David. The algorithm for constructing the network (of dimension ) is as follows.
Consider a honeycomb network of dimension . Subdivide all edges of by inserting a new vertex in each edge. Then, connect every pair of vertices lying at distance four in each hexagon. After removing the original hexagonal network from the resulting graph, subdivide each horizontal edge into two edges to obtain the dominating Davidderived network of dimension .
The graph is shown in Figure 1(a) is obtained from the star of David as described in [4]. The graphs and constructed from the same algorithm are shown in Figures 1(b) and 1(c).
(a)
(b)
(c)
By an easy calculation, one can find the order and size of , i.e., and .
Imran et al. [5, 6] studied the general Randic index, index, and index of networks. Liu et al. [7] studied some degreebased indices of Davidderived networks , dominating Davidderived networks , and regular triangulene silicate network . Farooq et al. [8, 9] studied some degreebased indices of some interconnection networks. Baig et al. [10] studied the Randic index, index, and index of networks. Dimitris [11] has also studied star of David which is a family of interwoven molecular inorganic knots, prepared by the employment of naphthalenediol in metal cluster chemistry. Davidderived networks are being investigated for possible uses in information sciences and other fields. They have huge range of applications in nanoscience, biology, and chemistry. Bajaj et al. [12, 13] studied topological models for the prediction of antiHIV activity of acylthiocarbamates. For more detailed study on the indices, see [14–18].
In this paper, we compute and derive closed analytical formulae for the distancebased topological indices such as ECI, TEI, and eccentricitybased Zagreb indices of some families of DD networks discussed in Section 4. The ECI gives the best forecast exactness rate with contrast to different indices used in different natural exercises, for example, antiinflammatory activity, anticonvulsant activity, and diuretic activity. Thus, these indices have the potential to be used in QSAR/QSPR studies. The obtained results are useful in to explore the certain phyiscochemical properties of these chemical networks, for example, antiinflammatory activity, anticonvulsant activity, and diuretic activity.
3. Main Results
3.1. Labeling of Critical Vertices in Dominating DavidDerived Networks
Using the symmetry of the graphs of dominating Davidderived networks , it is suffice to calculate the eccentricities of only one quadrant of . The vertices of one quadrant of are divided into six groups, namely, . Each vertex of group , where is any of , is represented by , where refers to the column and refer to the position of vertex in each column. Here, the index increases towards the left and increases downwards. For , one quadrant of is labeled in Figure 2.
(a)
(b)
There are some vertices in which exhibit different patterns of eccentricities as compared to the eccentricities of other vertices. These vertices are referred to as critical vertices of the graph . Thus, the vertices in one quadrant of are further divided into three regions by the vertical axis represented by a dotted line in Figure 2. Regions and , respectively, contain the vertices lying above and below the critical vertices and region contains the rest of the vertices. The critical vertices of the graph are included in region . For and , the edges incidented with the critical vertices are represented by bold lines in Figure 2. The three regions are also labeled in the same figure.
3.2. Eccentricities of Dominating DavidDerived Network
The grid vertices of are the vertices of the underline grid represented by bold edges in Figure 3. Note that the grid vertices of only belong to the groups , , and . Let (resp., and ) denotes a grid vertex in the column of group (resp., and ), where denotes the position of grid vertex in column . For , the grid vertices lying in region of are also shown in Figure 2. The eccentricities of vertices of lying in region are calculated by the help of eccentricities of grid vertices. The eccentricities of vertices of lying in region are given by the following cases.
Case 1. When . The eccentricities of grid vertices ’s are for , for , where . The eccentricities of ’s above and below grid vertices of type are given by where and , where and
The eccentricities of grid vertices ’s are for , for , where . The eccentricities of ’s above and below grid vertices of type are given by where and , where and .
The eccentricities of grid vertices ’s are for , for , where . The eccentricities of ’s above and below grid vertices of type are given by , where and where and .
Similarly, eccentricities of , , and in the column are where where where .
Case 2. When , the eccentricities of grid vertices ’s are given by for , for , where . The eccentricities of ’s above and below the grid vertices of type are where and where and .
The eccentricities of grid vertices ’s are for , for , where . The eccentricities of ’s above and below the grid vertices of type are where and , where and .
The eccentricities of grid vertices ’s are for , for , where . The eccentricities of ’s above and below the grid vertices are , where and , where and .
Similarly, eccentricities of , , and in the columns are , where where and , where
3.3. Topological Invariants of
Theorem 1. The eccentricconnectivity index of Davidderived networks for and is given by
Proof. Using symmetry of the network , we use only one quadrant of as labeled in Figure 2. We take one representative from a set of vertices which has same degree and eccentricity. These representatives are labeled by for and shown in Table 1 with their eccentricities. Using Table 1, the of for can be written as follows.
After simplification, we obtained the required result. This completes the proof.

Corollary 1. The totaleccentricity index of Davidderived networks for is given by
Corollary 2. The first Zagrebeccentricity index of Davidderived networks for is given by
Theorem 2. The eccentricconnectivity index of Davidderived networks for and is given by
Proof. Using symmetry of the network , we use only one quadrant of as labeled in Figure 2. We take one representative from a set of vertices which has the same degree and eccentricity. These representatives are labeled by for and shown in Table 2 along with their eccentricities and frequencies. The proof of the theorem is the analogue of Theorem 1.
