# Eccentricity-Based Topological Invariants of Dominating David-Derived Networks

Authors: Muhammad Imran; Mian Muhammad Zobair; Hani Shaker
Journal: Journal of Chemistry
 (2021)

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## 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 graph-invariant. In a QSAR/QSPR study, topological indices are utilized to anticipate the physico-concoction resources and bioactivity of compounds. In this paper, we study some distance-based topological indices such as eccentric connectivity index (ECI), total eccentricity index (TEI), and eccentricity-based Zagreb index for dominating David-derived 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.


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## Body

## 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 physico-concoction 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. Distance-based topological records are of incredible significance and assume an essential part in concoction diagram hypothesis and especially in hypothetical science.LetG be an n-vertex molecular graph with vertex set VG=v1,v2,…,vn and edge set EG. The vertices of G correspond to atoms, and an edge between two vertices corresponds to the chemical bond between these atoms. For a given vertex u∈VG, the eccentricity εu is defined as the largest distance between u and any other vertex v in G.The eccentric-connectivity index of graphG is denoted as ξG. It is a distance-based topological index and is defined as(1)ξG=∑u∈VGdeguεu.When the vertices’ degrees are not taken into account, we obtain the total eccentricity index of graphG defined by(2)ςG=∑u∈VGεu.A new version of Zagreb indices of a molecular graphG defined by Ghorbani and Hosseinzadeh [1] in terms of eccentricity are expressed as follows:(3)M1∗G=∑v∈VGεv2,M2∗G=∑uv∈VGεuεv.This paper first contains a little insight of the graph and algorithm of dominating David-derived networks. Then, the labeling of critical vertices of dominating David-derived network is provided. Then, computation of eccentricities of vertices is given. In the last segment, we process and determine some distance-based topological indices, such as eccentric-connectivity indexECI, total-eccentricity index TEI, and eccentricity-based Zagreb index of dominating David-derived DDDn. For some recent results, see [2, 3].


## 2. Algorithm and Insight into Dominating David-Derived Networks

Simonraj [4] gave a construction algorithm for the dominating David-derived networks, and DDD originated from the graph of star of David. The algorithm for constructing the DDD network (of dimension n) is as follows.Consider a honeycomb networkHCn of dimension n. Subdivide all edges of HCn 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 David-derived network DDDn of dimension n.The graphDDD1 is shown in Figure 1(a) is obtained from the star of David as described in [4]. The graphs DDD3 and DDD5 constructed from the same algorithm are shown in Figures 1(b) and 1(c).Figure 1
(a)DDD1, (b) the graph DDD3 of dominating David-derived networks for n=3, (c) DDD5.
(a)(b)(c)By an easy calculation, one can find the order and size ofDDDn, i.e., VDDDn=15n2−3n+6 and EDDDn=33n2−19n+11.Imran et al. [5, 6] studied the general Randic index, ABC index, and GA index of DDD networks. Liu et al. [7] studied some degree-based indices of David-derived networks DDn, dominating David-derived networks DDDn, and regular triangulene silicate network RTSLn. Farooq et al. [8, 9] studied some degree-based indices of some interconnection networks. Baig et al. [10] studied the Randic index, ABC index, and GA index of DDD networks. Dimitris [11] has also studied star of David which is a family of interwoven molecular inorganic knots, prepared by the employment of naphthalene−2,3−diol in 3d/4f-metal cluster chemistry. David-derived 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 anti-HIV 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 distance-based topological indices such as ECI, TEI, and eccentricity-based Zagreb indices of some families of DD networks discussed in Section4. The ECI gives the best forecast exactness rate with contrast to different indices used in different natural exercises, for example, anti-inflammatory 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 phyisco-chemical properties of these chemical networks, for example, anti-inflammatory activity, anticonvulsant activity, and diuretic activity.


## 3. Main Results

### 3.1. Labeling of Critical Vertices in Dominating David-Derived Networks

Using the symmetry of the graphs of dominating David-derived networksDDDn, it is suffice to calculate the eccentricities of only one quadrant of DDDn. The vertices of one quadrant of DDDn are divided into six groups, namely, a,b,…,f. Each vertex of group x, where x is any of a,b,…,f, is represented by xi,j, where i refers to the column and j refer to the position of vertex x in each column. Here, the index i increases towards the left and j increases downwards. For n=3, one quadrant of DDD2 is labeled in Figure 2.Figure 2
The vertex labeling of the dominating David-derived networkDDD3 (a) and grid representation of DDD5 (b).
(a)(b)There are some vertices inDDDn 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 DDDn. Thus, the vertices in one quadrant of DDDn are further divided into three regions by the vertical axis represented by a dotted line in Figure 2. Regions I and II, respectively, contain the vertices lying above and below the critical vertices and region III contains the rest of the vertices. The critical vertices of the graph DDDn are included in region I. For n=3 and 5, 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 David-Derived NetworkDDDn

The grid vertices ofDDDn are the vertices of the underline grid represented by bold edges in Figure 3. Note that the grid vertices of DDDn only belong to the groups a, c, and e. Let ai,k (resp., ci,k and ei,k) denotes a grid vertex in the ith column of group a (resp., c and e), where k denotes the position of grid vertex in column i. For n=5, the grid vertices lying in region II of DDD5 are also shown in Figure 2. The eccentricities of vertices of DDDn lying in region II are calculated by the help of eccentricities of grid vertices. The eccentricities of vertices of DDDn lying in region II are given by the following cases.Figure 3
The grid vertex labeling of the dominating David-derived networkDDD5 along with critical line in bold.Case 1.
Whenn≡1mod4. The eccentricities of grid vertices ai,k’s are εai,3n/2+9i/2+3k−7=6n+6i+4k−12 for i≡1mod2, εai,3n/2+9i/2+3k−13/2=6n+6i+4k−12 for i≡0mod2, where 1≤i≤n−1/4+1,1≤k≤n/2−2i+5/2. The eccentricities of ai,k’s above and below grid vertices of type a are given by εai,3n/2+9i/2+3k−8=6n+6i+4k−13, where 2≤k≤n/2−2i+5/2 and 1≤i≤n−1/4εai,3n/2+9i/2+3k−6=6n+6i+4k−10, where 1≤k≤n/2−2i+3/2 and 1≤i≤n−1/4
The eccentricities of grid verticesci,k’s are εci,3n/2+9i/2+3k−5=6n+6i+4k−10 for i≡1mod2, εci,3n/2+9i/2+3k−13/2=6n+6i+4k−10 for i≡0mod2, where 1≤i≤n−1/4,1≤k≤n/2−2i+3/2. The eccentricities of ci,k’s above and below grid vertices of type c are given by εci,3n/2+9i/2+3k−4=6n+6i+4k−8, where 1≤k≤n/2−2i+3/2and 1≤i≤n−1/4εci,3n/2+9i/2+3k−6=6n+6i+4k−11, where 2≤k≤n/2−2i+3/2 and 1≤i≤n−1/4.
The eccentricities of grid verticesei,k’s are εei,3n/2+9i/2+3k−4=6n+6i+4k−8 for i≡1mod2, εei,3n/2+9i/2+3k−3=6n+6i+4k−8 for i≡0mod2, where 1≤i≤n−1/4,1≤k≤n/2−2i+1/2. The eccentricities of ei,k’s above and below grid vertices of type e are given by εei,3n/2+9i/2+3k−5=6n+6i+4k−9, where 2≤k≤n/2−2i+1/2 and 1≤i≤n−1/4εei,3n/2+9i/2+3k−3=6n+6i+4k−6, where 1≤k≤n/2−2i+1/2 and 1≤i≤n−1/4.
Similarly, eccentricities ofbi,k, di,k, and fi,k in the ith column are εbi,n+3i+k−3=6n+6i+2k−9, where 1≤i≤n−1/4,1≤k≤n−4i+3.εdi,n+3i+k−2=6n+6i+2k−7, where 1≤i≤n−1/4,1≤k≤n−4i+2.εfi,n+3i+k−1=6n+6i+2k−5, where 1≤i≤n−1/4,1≤k≤n−4i.Case 2.
Whenn≡3mod4, the eccentricities of grid vertices ai,k’s are given by εai,3n/2+9i/2+3k−7=6n+6i+4k−12 for i≡1mod2, εai,3n/2+9i/2+3k−13/2=6n+6i+4k−12 for i≡0mod2, where 1≤i≤n−3/4+1,1≤k≤n/2−2i+5/2. The eccentricities of ai,k’s above and below the grid vertices of type a are εai,3n/2+9i/2+3k−8=6n+6i+4k−13, where 2≤k≤n/2−2i+5/2 and 1≤i≤n−3/4εai,3n/2+9i/2+3k−6=6n+6i+4k−10, where 1≤k≤n/2−2i+3/2 and 1≤i≤n−3/4.
The eccentricities of grid verticesci,k’s are εci,3n/2+9i/2+3k−5=6n+6i+4k−10 for i≡1mod2, εci,3n/2+9i/2+3k−13/2=6n+6i+4k−10 for i≡0mod2, where 1≤i≤n−3/4,1≤k≤n/2−2i+3/2. The eccentricities of ci,k’s above and below the grid vertices of type c are εci,3n/2+9i/2+3k−4=6n+6i+4k−8, where 1≤k≤n/2−2i+3/2 and 1≤i≤n−3/4εci,3n/2+9i/2+3k−6=6n+6i+4k−11, where 2≤k≤n/2−2i+3/2 and 1≤i≤n−3/4.
The eccentricities of grid verticesei,k’s are εei,3n/2+9i/2+3k−4=6n+6i+4k−8 for i≡1mod2, εei,3n/2+9i/2+3k−3=6n+6i+4k−8 for i≡0mod2, where 1≤i≤n−3/4,1≤k≤n/2−2i+1/2. The eccentricities of ei,k’s above and below the grid vertices are εei,3n/2+9i/2+3k−5=6n+6i+4k−9, where 2≤k≤n/2−2i+1/2 and 1≤i≤n−3/4εei,3n/2+9i/2+3k−3=6n+6i+4k−6, where 1≤k≤n/2−2i+1/2 and 1≤i≤n−3/4.
Similarly, eccentricities ofbi,k, di,k, and fi,k in the ith columns are εbi,n+3i+k−3=6n+6i+2k−9, where 1≤i≤n−3/4,1≤k≤n−4i+3;εdi,n+3i+k−2=6n+6i+2k−7, where 1≤i≤n−3/4,1≤k≤n−4i+2; and εfi,n+3i+k−1=6n+6i+2k−5, where 1≤i≤n−3/4,1≤k≤n−4i.


### 3.3. Topological Invariants ofDDDn

Theorem 1.
The eccentric-connectivity index of David-derived networksDDDn for n≡1mod4 and n≥1 is given by(4)−480n62+888n+8418n+12.Proof.
Using symmetry of the networkDDDn, we use only one quadrant of DDDn 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 ai,bi,ci,di,ei,andfi for 1≤i≤n and shown in Table 1 with their eccentricities. Using Table 1, the ECI of DDDn for n≥1 can be written as follows.
After simplification, we obtained the required result. This completes the proof.Table 1
Eccentricities of graphDDDn when n≡1mod4.
εaij = 6n + 6i − 91≤i≤n−1/4+1n−1/4+2≤i≤n1≤j≤3n/2+9i/2−5andiodd,1≤j≤3n/2+9i/2−9/2andieven1≤j≤3n−3i2+12andiodd,1≤j≤3n−3i2+1andievenεaij = 6n + 6i − 81≤i≤n−1/4+1n−1/4+2≤i≤n1≤j≤n+3i−31≤j≤2n−iεaij = 6n + 6i − 71≤i≤n−14+1n−1/4+2≤i≤n1≤j≤3n/2+9i/2−3andiodd,1≤j≤3n/2+9i/2−9/2andieven1≤j≤3n−3i/2+1/2andiodd,1≤j≤3n−3i/2andievenεaij = 6n + 6i − 61≤i≤n−1/4n−1/4+1≤i≤n1≤j≤n+3i−21≤j≤2n−iεaij = 6n + 6i − 51≤i≤n−1/4n−1/4+1≤i≤n1≤j≤3n/2+9i/2−2andiodd,1≤j≤3n/2+9i/2−3/2andieven1≤j≤3n−3i2−32andiodd,1≤j≤3n−3i2−1andievenεaij = 6n + 6i − 41≤i≤n−1/4n−1/4+1≤i≤n−11≤j≤n+3i−11≤j≤2n−i−1Corollary 1.
The total-eccentricity index ofDDDn David-derived networks for n≡1mod4 is given by(5)ξDDDn=755418n+123+50414n+343+112318n+383+368318n+783−33n2−4812n+123+1123456n+43218n2+112864n2+432n−69618n112864n−43212n2+1121728n2−1296n+12012n−25532+112600n2−3168n+546418n+78+1121488n−511218n+782+112−1392n+64818n+382+112−192n2+2160n−17618n+38+112−864n−187218n+122+112−1728n2+3312n+2418n+12+1123024n−2523614n+342+112−1512n2−6228n+3340814n+34+112432n+64812n+122+112864n2−1728n+40812n+12−9612n3+28818n3.Corollary 2.
The first Zagreb-eccentricity index ofDDDn David-derived networks for n≡1mod4 is given by(6)M1∗DDDn=−822405116n+876841924n2−430997524n3+11442985984n3+1244160n2−1057536n−13824018n3+1144373248n4+248832n3−51840n2−152064+26092818n2114411943936+99532818n5+11448957952n2+1990656n−158630418n4+11441492992n4−1492992n3−109008n21389312n−8512012n2+11442985984n4−74649604n3+6220800n2−2419200n+31334412n+1144−995328n−99532812n5+1144−1617408n2+1244160n−19008012n4+11441492992n3−4976640n2+5142528n−102988812n3+1144−401472n4+5172480n3−25766016n2+50038656n−3276729618n+78+1144200736n4−4765248n3+34604160n2−90555840n+7600611218n+782+11441452672n3−20433792n2+80581632n−9217804818n+783+11446033408n−2137190418n+785+1144−4464768n235209728n+6150905618n+784+1144−41472n+13132812n2+−41472n2+179712n−15321612n+−2426112n5+622080n4+41615424n3−107628480n2+220515552n−23789577614n+34+1144−4147212n2+−41472n+4838412n+1446336n4−31181760n3+107639280n2−319616928n+43571980814n+342+11447651584n2−83420928n+22865544014n+344+11447216128n3−47138112n2+231435360n−42260140814n+343+114411943936n−6538060814n+345+1144995328n2−1990656n−107827212n3+−746496n3+1866240n2+20736n−30067212n2+−1492992n4+4478976n3−2032128n2−248832n+62208012n−1605888n4+4773888n3−2711040n2−905472n−102720018n+38+1144497664n+174182412n3+−373248n2−1119744n+61171212n2+−746496n3−1866240n2+2260224n−106790412n+859392n4−3377664n3−354048n2+4248576+333529618n+382+1144−66355212n3+497664n−24883212n2+995328n2−995328n+41472012n+423936n3+2784768n2−5166336n−585254418n+383+1144995328n2−1990656n−19353612n3+−7464096n3+2612736n2−2135808n−14169612n2+−1492992n4+4478976n3−3193344n2+774144n−5068812n−808704n5+590976n4+4677696n3−6591456n2+2104128n−12700818n+18+1144−6144n−292147218n+385+1144−922368n2+1896192n+573318418n+384+1144497664n+124416012n3+−373248n2−373248n+135820812n2+−746496n3−1119744n2+1596672n−49420812n+1088640n4−7091712n3+6394464n2+1924992n−84240018n+122+1144−66355212n3+497664n−74649612n2+995328n2+995328n+30412812n+2032128n3+684288n2−7727616n+81561618n+123+1144−995328n−199065618n+125+1144−1617408n2+5474304n+133056018n+124+37120918n+386+460812n6+460818n+126+188672918n+786+4147218n6+5356814n+346+33378358+884538n6−28948n5+26579324n4.Theorem 2.
The eccentric-connectivity index of David-derived networksDDDn for n≡3mod4 and n≥1 is given by(7)ξDDDn=168014n+143+12420160n−4824014n+142+124−10080n2−2160n+681614n+14+3296318n+183+12424000n+2361618n+182+12411136n2+13152n−2036818n+18+800318n+583+124−6960n−600018n+582+124−10680n2+30576n−1079218n+58−32012n3+1248640n−614412n2+12414400n2−21024n+892812n+1251712n3−74218n2−31993n+65798.Proof.
Using symmetry of the networkDDDn, we use only one quadrant of DDDn 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 ai,bi,ci,di,ei,andfi for 1≤i≤n and shown in Table 2 along with their eccentricities and frequencies. The proof of the theorem is the analogue of Theorem 1.Table 2
Eccentricities of graphDDDn when n≡3mod4.
εaij = 6n + 6i − 9≤i≤n−3/4+1n−3/4+2≤i≤n1≤j≤3n/2+9i/2−5andiodd,1≤j≤3n/2+9i/2−9/2andieven1≤j≤3n−3i/2+1/2andiodd,1≤j≤03n−3i/2+1andievenεaij = 6n + 6i − 81≤i≤n−3/4+1n−3/4+2≤i≤n1≤j≤n+3i−31≤j≤2n−i.εaij = 6n + 6i − 71≤i≤n−3/4+1n−3/4+2≤i≤n1≤j≤3n/2+9i/2−3andiodd,1≤j≤3n/2+9i/2−5/2andieven1≤j≤3n−3i/2+1/2andiodd,1≤j≤3n−3i/2andievenεaij = 6n + 6i − 61≤i≤n−3/4+1n−3/4+2≤i≤n1≤j≤n+3i−21≤j≤2n−i.εaij = 6n + 6i − 51≤i≤n−3/4+1n−3/4+2≤i≤n−11≤j≤3n/2+9i/2−3andiodd,1≤j≤3n/2+9i/2−5/2andieven1≤j≤3n−3i/2+3/2andiodd,1≤j≤3n−3i/2−1andievenεaij = 6n + 6i − 41≤i≤n−3/4n−3/4+1≤i≤n−11≤j≤n+3i−11≤j≤2n−i−1Corollary 3.
The total eccentricity index ofDDDn David-derived networks for n≡3mod4 is given by(8)ξDDDn=50414n+143+1123024n−738014n+142+112−1512n2−468n+1320+14n+141184318n+183+1123792n+288018n+182+1121104n2+1800n−293618n+18+244318n+583+112−1104n−158418n+582+112−1560n2+3816n+20818n+58−9612n3+112864n−14412n2+1121728n2−1008n+7212n+12954n3−4352n2−13854+171.Corollary 4.
The first Zagreb eccentricity index ofDDDn David-derived networks for n≡3mod4 is given by(9)M1∗DDDn=−69342548n+10368n4−12096n3+4464n2−2880n+150812n2+10368n3−22464n2+420160n−316812n3+−194563n−1881618n+585+−339923n2+38016n+249968918n+584+3456n−172812n2+6912n2−6912n+288012n−460812n3+13832n3+304883n2−1849363n−65944318n+583+−2592n2−7776n+396012n2+−10368n4+31104n3−14400n2−672n+335212n+6912n2−13824n−748812n3+6912n2−13824n−748812n3−5616n5+3908n4+312523−1014983n2−8203n−17618318n+58+3731203n−704018n+185+2607203n2−368n−38576918n+184+3456n−288012n2+6912n2−4608n+172812n−460812n3+3408n3+152483n2+109523n−69080318n+183+−2592n−56212n+−5184n3+7776n2−4176n+108012n+3456n−288012n3+9856n4−2280n3+20203n2−24592n+176812918n+182+−5184n3+7776n2−4176n+108012n2−10368n2−23n+14212n+6912n2−4608n+172812n3−784n4−3136n3+40003n2+9408n−18160318n+18+−28812n2+−288n+33612n+10044n4−67932n3+90027n2−48696n+1270614n+142+−288n+33612n2−288n−76212n−16848n5+24408n4+6252n3−21930n2+11276n−179814n+14+82944n−13953614n+145+53136n2−178416n+13420514n+14450112n3−123444n2+136662n−5882414n+143+−6912n+460812n5+−11232n62+8064n−295212n4+89992−1498658n3+62311324n2+334033n4−73339348n5+5356814n+146+460812n6+43520918n+586+596992918n+186884538n6+10368n−32n3−n2+49n−17212n.


## 3.1. Labeling of Critical Vertices in Dominating David-Derived Networks

Using the symmetry of the graphs of dominating David-derived networksDDDn, it is suffice to calculate the eccentricities of only one quadrant of DDDn. The vertices of one quadrant of DDDn are divided into six groups, namely, a,b,…,f. Each vertex of group x, where x is any of a,b,…,f, is represented by xi,j, where i refers to the column and j refer to the position of vertex x in each column. Here, the index i increases towards the left and j increases downwards. For n=3, one quadrant of DDD2 is labeled in Figure 2.Figure 2
The vertex labeling of the dominating David-derived networkDDD3 (a) and grid representation of DDD5 (b).
(a)(b)There are some vertices inDDDn 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 DDDn. Thus, the vertices in one quadrant of DDDn are further divided into three regions by the vertical axis represented by a dotted line in Figure 2. Regions I and II, respectively, contain the vertices lying above and below the critical vertices and region III contains the rest of the vertices. The critical vertices of the graph DDDn are included in region I. For n=3 and 5, 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 David-Derived NetworkDDDn

The grid vertices ofDDDn are the vertices of the underline grid represented by bold edges in Figure 3. Note that the grid vertices of DDDn only belong to the groups a, c, and e. Let ai,k (resp., ci,k and ei,k) denotes a grid vertex in the ith column of group a (resp., c and e), where k denotes the position of grid vertex in column i. For n=5, the grid vertices lying in region II of DDD5 are also shown in Figure 2. The eccentricities of vertices of DDDn lying in region II are calculated by the help of eccentricities of grid vertices. The eccentricities of vertices of DDDn lying in region II are given by the following cases.Figure 3
The grid vertex labeling of the dominating David-derived networkDDD5 along with critical line in bold.Case 1.
Whenn≡1mod4. The eccentricities of grid vertices ai,k’s are εai,3n/2+9i/2+3k−7=6n+6i+4k−12 for i≡1mod2, εai,3n/2+9i/2+3k−13/2=6n+6i+4k−12 for i≡0mod2, where 1≤i≤n−1/4+1,1≤k≤n/2−2i+5/2. The eccentricities of ai,k’s above and below grid vertices of type a are given by εai,3n/2+9i/2+3k−8=6n+6i+4k−13, where 2≤k≤n/2−2i+5/2 and 1≤i≤n−1/4εai,3n/2+9i/2+3k−6=6n+6i+4k−10, where 1≤k≤n/2−2i+3/2 and 1≤i≤n−1/4
The eccentricities of grid verticesci,k’s are εci,3n/2+9i/2+3k−5=6n+6i+4k−10 for i≡1mod2, εci,3n/2+9i/2+3k−13/2=6n+6i+4k−10 for i≡0mod2, where 1≤i≤n−1/4,1≤k≤n/2−2i+3/2. The eccentricities of ci,k’s above and below grid vertices of type c are given by εci,3n/2+9i/2+3k−4=6n+6i+4k−8, where 1≤k≤n/2−2i+3/2and 1≤i≤n−1/4εci,3n/2+9i/2+3k−6=6n+6i+4k−11, where 2≤k≤n/2−2i+3/2 and 1≤i≤n−1/4.
The eccentricities of grid verticesei,k’s are εei,3n/2+9i/2+3k−4=6n+6i+4k−8 for i≡1mod2, εei,3n/2+9i/2+3k−3=6n+6i+4k−8 for i≡0mod2, where 1≤i≤n−1/4,1≤k≤n/2−2i+1/2. The eccentricities of ei,k’s above and below grid vertices of type e are given by εei,3n/2+9i/2+3k−5=6n+6i+4k−9, where 2≤k≤n/2−2i+1/2 and 1≤i≤n−1/4εei,3n/2+9i/2+3k−3=6n+6i+4k−6, where 1≤k≤n/2−2i+1/2 and 1≤i≤n−1/4.
Similarly, eccentricities ofbi,k, di,k, and fi,k in the ith column are εbi,n+3i+k−3=6n+6i+2k−9, where 1≤i≤n−1/4,1≤k≤n−4i+3.εdi,n+3i+k−2=6n+6i+2k−7, where 1≤i≤n−1/4,1≤k≤n−4i+2.εfi,n+3i+k−1=6n+6i+2k−5, where 1≤i≤n−1/4,1≤k≤n−4i.Case 2.
Whenn≡3mod4, the eccentricities of grid vertices ai,k’s are given by εai,3n/2+9i/2+3k−7=6n+6i+4k−12 for i≡1mod2, εai,3n/2+9i/2+3k−13/2=6n+6i+4k−12 for i≡0mod2, where 1≤i≤n−3/4+1,1≤k≤n/2−2i+5/2. The eccentricities of ai,k’s above and below the grid vertices of type a are εai,3n/2+9i/2+3k−8=6n+6i+4k−13, where 2≤k≤n/2−2i+5/2 and 1≤i≤n−3/4εai,3n/2+9i/2+3k−6=6n+6i+4k−10, where 1≤k≤n/2−2i+3/2 and 1≤i≤n−3/4.
The eccentricities of grid verticesci,k’s are εci,3n/2+9i/2+3k−5=6n+6i+4k−10 for i≡1mod2, εci,3n/2+9i/2+3k−13/2=6n+6i+4k−10 for i≡0mod2, where 1≤i≤n−3/4,1≤k≤n/2−2i+3/2. The eccentricities of ci,k’s above and below the grid vertices of type c are εci,3n/2+9i/2+3k−4=6n+6i+4k−8, where 1≤k≤n/2−2i+3/2 and 1≤i≤n−3/4εci,3n/2+9i/2+3k−6=6n+6i+4k−11, where 2≤k≤n/2−2i+3/2 and 1≤i≤n−3/4.
The eccentricities of grid verticesei,k’s are εei,3n/2+9i/2+3k−4=6n+6i+4k−8 for i≡1mod2, εei,3n/2+9i/2+3k−3=6n+6i+4k−8 for i≡0mod2, where 1≤i≤n−3/4,1≤k≤n/2−2i+1/2. The eccentricities of ei,k’s above and below the grid vertices are εei,3n/2+9i/2+3k−5=6n+6i+4k−9, where 2≤k≤n/2−2i+1/2 and 1≤i≤n−3/4εei,3n/2+9i/2+3k−3=6n+6i+4k−6, where 1≤k≤n/2−2i+1/2 and 1≤i≤n−3/4.
Similarly, eccentricities ofbi,k, di,k, and fi,k in the ith columns are εbi,n+3i+k−3=6n+6i+2k−9, where 1≤i≤n−3/4,1≤k≤n−4i+3;εdi,n+3i+k−2=6n+6i+2k−7, where 1≤i≤n−3/4,1≤k≤n−4i+2; and εfi,n+3i+k−1=6n+6i+2k−5, where 1≤i≤n−3/4,1≤k≤n−4i.


## 3.3. Topological Invariants ofDDDn

Theorem 1.
The eccentric-connectivity index of David-derived networksDDDn for n≡1mod4 and n≥1 is given by(4)−480n62+888n+8418n+12.Proof.
Using symmetry of the networkDDDn, we use only one quadrant of DDDn 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 ai,bi,ci,di,ei,andfi for 1≤i≤n and shown in Table 1 with their eccentricities. Using Table 1, the ECI of DDDn for n≥1 can be written as follows.
After simplification, we obtained the required result. This completes the proof.Table 1
Eccentricities of graphDDDn when n≡1mod4.
εaij = 6n + 6i − 91≤i≤n−1/4+1n−1/4+2≤i≤n1≤j≤3n/2+9i/2−5andiodd,1≤j≤3n/2+9i/2−9/2andieven1≤j≤3n−3i2+12andiodd,1≤j≤3n−3i2+1andievenεaij = 6n + 6i − 81≤i≤n−1/4+1n−1/4+2≤i≤n1≤j≤n+3i−31≤j≤2n−iεaij = 6n + 6i − 71≤i≤n−14+1n−1/4+2≤i≤n1≤j≤3n/2+9i/2−3andiodd,1≤j≤3n/2+9i/2−9/2andieven1≤j≤3n−3i/2+1/2andiodd,1≤j≤3n−3i/2andievenεaij = 6n + 6i − 61≤i≤n−1/4n−1/4+1≤i≤n1≤j≤n+3i−21≤j≤2n−iεaij = 6n + 6i − 51≤i≤n−1/4n−1/4+1≤i≤n1≤j≤3n/2+9i/2−2andiodd,1≤j≤3n/2+9i/2−3/2andieven1≤j≤3n−3i2−32andiodd,1≤j≤3n−3i2−1andievenεaij = 6n + 6i − 41≤i≤n−1/4n−1/4+1≤i≤n−11≤j≤n+3i−11≤j≤2n−i−1Corollary 1.
The total-eccentricity index ofDDDn David-derived networks for n≡1mod4 is given by(5)ξDDDn=755418n+123+50414n+343+112318n+383+368318n+783−33n2−4812n+123+1123456n+43218n2+112864n2+432n−69618n112864n−43212n2+1121728n2−1296n+12012n−25532+112600n2−3168n+546418n+78+1121488n−511218n+782+112−1392n+64818n+382+112−192n2+2160n−17618n+38+112−864n−187218n+122+112−1728n2+3312n+2418n+12+1123024n−2523614n+342+112−1512n2−6228n+3340814n+34+112432n+64812n+122+112864n2−1728n+40812n+12−9612n3+28818n3.Corollary 2.
The first Zagreb-eccentricity index ofDDDn David-derived networks for n≡1mod4 is given by(6)M1∗DDDn=−822405116n+876841924n2−430997524n3+11442985984n3+1244160n2−1057536n−13824018n3+1144373248n4+248832n3−51840n2−152064+26092818n2114411943936+99532818n5+11448957952n2+1990656n−158630418n4+11441492992n4−1492992n3−109008n21389312n−8512012n2+11442985984n4−74649604n3+6220800n2−2419200n+31334412n+1144−995328n−99532812n5+1144−1617408n2+1244160n−19008012n4+11441492992n3−4976640n2+5142528n−102988812n3+1144−401472n4+5172480n3−25766016n2+50038656n−3276729618n+78+1144200736n4−4765248n3+34604160n2−90555840n+7600611218n+782+11441452672n3−20433792n2+80581632n−9217804818n+783+11446033408n−2137190418n+785+1144−4464768n235209728n+6150905618n+784+1144−41472n+13132812n2+−41472n2+179712n−15321612n+−2426112n5+622080n4+41615424n3−107628480n2+220515552n−23789577614n+34+1144−4147212n2+−41472n+4838412n+1446336n4−31181760n3+107639280n2−319616928n+43571980814n+342+11447651584n2−83420928n+22865544014n+344+11447216128n3−47138112n2+231435360n−42260140814n+343+114411943936n−6538060814n+345+1144995328n2−1990656n−107827212n3+−746496n3+1866240n2+20736n−30067212n2+−1492992n4+4478976n3−2032128n2−248832n+62208012n−1605888n4+4773888n3−2711040n2−905472n−102720018n+38+1144497664n+174182412n3+−373248n2−1119744n+61171212n2+−746496n3−1866240n2+2260224n−106790412n+859392n4−3377664n3−354048n2+4248576+333529618n+382+1144−66355212n3+497664n−24883212n2+995328n2−995328n+41472012n+423936n3+2784768n2−5166336n−585254418n+383+1144995328n2−1990656n−19353612n3+−7464096n3+2612736n2−2135808n−14169612n2+−1492992n4+4478976n3−3193344n2+774144n−5068812n−808704n5+590976n4+4677696n3−6591456n2+2104128n−12700818n+18+1144−6144n−292147218n+385+1144−922368n2+1896192n+573318418n+384+1144497664n+124416012n3+−373248n2−373248n+135820812n2+−746496n3−1119744n2+1596672n−49420812n+1088640n4−7091712n3+6394464n2+1924992n−84240018n+122+1144−66355212n3+497664n−74649612n2+995328n2+995328n+30412812n+2032128n3+684288n2−7727616n+81561618n+123+1144−995328n−199065618n+125+1144−1617408n2+5474304n+133056018n+124+37120918n+386+460812n6+460818n+126+188672918n+786+4147218n6+5356814n+346+33378358+884538n6−28948n5+26579324n4.Theorem 2.
The eccentric-connectivity index of David-derived networksDDDn for n≡3mod4 and n≥1 is given by(7)ξDDDn=168014n+143+12420160n−4824014n+142+124−10080n2−2160n+681614n+14+3296318n+183+12424000n+2361618n+182+12411136n2+13152n−2036818n+18+800318n+583+124−6960n−600018n+582+124−10680n2+30576n−1079218n+58−32012n3+1248640n−614412n2+12414400n2−21024n+892812n+1251712n3−74218n2−31993n+65798.Proof.
Using symmetry of the networkDDDn, we use only one quadrant of DDDn 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 ai,bi,ci,di,ei,andfi for 1≤i≤n and shown in Table 2 along with their eccentricities and frequencies. The proof of the theorem is the analogue of Theorem 1.Table 2
Eccentricities of graphDDDn when n≡3mod4.
εaij = 6n + 6i − 9≤i≤n−3/4+1n−3/4+2≤i≤n1≤j≤3n/2+9i/2−5andiodd,1≤j≤3n/2+9i/2−9/2andieven1≤j≤3n−3i/2+1/2andiodd,1≤j≤03n−3i/2+1andievenεaij = 6n + 6i − 81≤i≤n−3/4+1n−3/4+2≤i≤n1≤j≤n+3i−31≤j≤2n−i.εaij = 6n + 6i − 71≤i≤n−3/4+1n−3/4+2≤i≤n1≤j≤3n/2+9i/2−3andiodd,1≤j≤3n/2+9i/2−5/2andieven1≤j≤3n−3i/2+1/2andiodd,1≤j≤3n−3i/2andievenεaij = 6n + 6i − 61≤i≤n−3/4+1n−3/4+2≤i≤n1≤j≤n+3i−21≤j≤2n−i.εaij = 6n + 6i − 51≤i≤n−3/4+1n−3/4+2≤i≤n−11≤j≤3n/2+9i/2−3andiodd,1≤j≤3n/2+9i/2−5/2andieven1≤j≤3n−3i/2+3/2andiodd,1≤j≤3n−3i/2−1andievenεaij = 6n + 6i − 41≤i≤n−3/4n−3/4+1≤i≤n−11≤j≤n+3i−11≤j≤2n−i−1Corollary 3.
The total eccentricity index ofDDDn David-derived networks for n≡3mod4 is given by(8)ξDDDn=50414n+143+1123024n−738014n+142+112−1512n2−468n+1320+14n+141184318n+183+1123792n+288018n+182+1121104n2+1800n−293618n+18+244318n+583+112−1104n−158418n+582+112−1560n2+3816n+20818n+58−9612n3+112864n−14412n2+1121728n2−1008n+7212n+12954n3−4352n2−13854+171.Corollary 4.
The first Zagreb eccentricity index ofDDDn David-derived networks for n≡3mod4 is given by(9)M1∗DDDn=−69342548n+10368n4−12096n3+4464n2−2880n+150812n2+10368n3−22464n2+420160n−316812n3+−194563n−1881618n+585+−339923n2+38016n+249968918n+584+3456n−172812n2+6912n2−6912n+288012n−460812n3+13832n3+304883n2−1849363n−65944318n+583+−2592n2−7776n+396012n2+−10368n4+31104n3−14400n2−672n+335212n+6912n2−13824n−748812n3+6912n2−13824n−748812n3−5616n5+3908n4+312523−1014983n2−8203n−17618318n+58+3731203n−704018n+185+2607203n2−368n−38576918n+184+3456n−288012n2+6912n2−4608n+172812n−460812n3+3408n3+152483n2+109523n−69080318n+183+−2592n−56212n+−5184n3+7776n2−4176n+108012n+3456n−288012n3+9856n4−2280n3+20203n2−24592n+176812918n+182+−5184n3+7776n2−4176n+108012n2−10368n2−23n+14212n+6912n2−4608n+172812n3−784n4−3136n3+40003n2+9408n−18160318n+18+−28812n2+−288n+33612n+10044n4−67932n3+90027n2−48696n+1270614n+142+−288n+33612n2−288n−76212n−16848n5+24408n4+6252n3−21930n2+11276n−179814n+14+82944n−13953614n+145+53136n2−178416n+13420514n+14450112n3−123444n2+136662n−5882414n+143+−6912n+460812n5+−11232n62+8064n−295212n4+89992−1498658n3+62311324n2+334033n4−73339348n5+5356814n+146+460812n6+43520918n+586+596992918n+186884538n6+10368n−32n3−n2+49n−17212n.


## 4. Conclusion

Graph theory has been successfully employed through the translation of chemical structures into characteristic numerical descriptors in the chemical graph theory and computational chemistry. In the present study, the relationship of anti-inflammatory activity of David-derived networks with the eccentric connectivity index and the other eccentricity-based topological indices was investigated.In this paper, we give formulae of the eccentric-connectivity index, total-eccentricity index, and an eccentricity-based Zagreb index of dominating David-derived network. Dominating David-derived networks being derived from honeycomb structures has an imperative impact in studying materials with minimal density and high compression properties in chemistry. These structures are also used in studying the tension in different materials which are used in aerospace structures. So, our subject study will be quite helpful in studying the interwoven molecular inorganic knots and metal cluster chemistry. It will also be attributable in slow relaxation of the magnetization and magnetocaloric properties, depending on the metal ion. Investigation of different indices for David-derived networks has the potential to provide the best forecast rate used in different natural exercises, as anti-inflammatory activity, anticonvulsant activity, and diuretic activity.Open Problem 1.
Researchers are invited for open problem to study the same network for the calculation of eccentric-connectivity index, total index, and Zagreb index of David-derived networksDDDn for n≡0mod4 and n≡2mod4 for n≥1 which definitely will change the results and structure of the network.



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