# Computation of Irregularity Indices of Certain Computer Networks

Authors: Jiangnan Liu; Lulu Cai; Abaid ur Rehman Virk; Waheed Akhtar; Shahzad Ahmed Maitla; Yang Wei
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

A graph is said to be a regular graph if all its vertices have the same degree; otherwise, it is irregular. In general, irregularity indices are used for computational analysis of nonregular graph topological composition. The creation of irregular indices is based on the conversion of a structural graph into a total count describing the irregularity of the molecular design on the map. It is important to be notified how unusual a molecular structure is in various situations and problems in structural science and chemistry. In this paper, we will compute irregularity indices of certain networks.


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## Body

## 1. Introduction

In mathematics, graph theory can be used to describe different types of graphs that are computational structures. It is also used to model sensible item connections [1, 2]. An irregularity index is a statistical value connected with a graph that defines a graph’s irregularity. The theory of networks is a part of computer science and network engineering graph theory.A topological invariantTOPG is referred to as a graph G irregularity index if the topological invariant TOPG≥0 and TOPG=0 if G is a regular graph. However, if all its vertices have the same degree, it is said that the graph is regular. Topological invariant TOPG is a numeric value of a molecular structure of a chemical compound. Nonetheless, the creation of irregular indices is based on the conversion of a structural graph into a total count describing the irregularity of the molecular design on the map [3, 4]. Many networks including silicate, chain silicate, oxide, hexagonal, and honeycomb networks are identical to networks of atomic or chemical structure. There are very important unusual characteristics in such networks.In this paper, we are concerned with simple connected graphs symbolized byGV,E, where VG and EG represent the set of vertices and edges of G, respectively. The degree of a vertex u of a graph G is the count of first neighbors of u. And uv represents an edge for G, connecting vertices u and v [5, 6].Graph theory was established in 1736 when Leonhard Euler presented “Solutio problematic as situspertinntis geometries” (the solution of a problem related to place theory). Wiener is the pioneer of topological indices; he discovered the first topological index and found out the boiling point of a compound (paraffin, a member of the alkane family) in 1947. It was named as path number, but latterly, it was renamed as the Wiener index [7].(1)WG=12∑u,v∈VGdu,dv.Wiener also invented the Wiener polarity index. Milan Randic invented the first and oldest degree-based index named as the Randic index in 1975 [8]:(2)R−1/2G=∑uv∈EG1dudv.The graph invariant denoted byM1G is called the first Zagreb index, which is equal to the sum of square of the degrees of the vertices of a graph; it was introduced by Trinjastic and Gutman in 1972 [9]. M1G is linked with sum of quantities in the field of chemical graph theory. M1G is known as the Gutman index; M1G is bounded and attains lower and upper bound [10]:(3)M1G=∑uv∈EGdu+dv.The second Zagreb index is a graph invariant denoted byM2G which is defined as the aggregate of the product of degrees of connected pairs of vertices of the molecular compound, and it was introduced by Trinjastic and Gutman in 1972.(4)M2G=∑uv∈EGdu⋅dv.Bo-Zhou and Ivan Gutman presented the upper bound for these Zagreb indices w.r.t the min-max degree [11, 12].Nevertheless, the emergence of new topological descriptors with extremely detecting capacity retains a concern without deformation for the scientific world [13, 14]. Therefore, there is a great willingness to change novel graph invariants with enormous detecting power combined with insignificant degeneration. In this paper, we compute irregularity indices for certain networks.


## 2. Irregularity Indices

All these selected irregularity indices belong to the family of degree-based topological indices. Tamas Reti et al. selected these irregularity indices as a molecular descriptor in the QSPR study to predict physicochemical properties of octane isomers [15]. The selected irregularity indices for certain networks are represented by(5)VARG=M1Gn−2mn2,ALG=∑uv∈Edu−dv,IR1G=FG−2mnM1G,IR2G=M2Gm−2mn,IRFG=FG−2M2G,IRFWG=IRFGM2G,IRAG=∑uv∈Edu−1/2−dv−1/22,IRBG=∑uv∈Edu1/2−dv1/22,IRCG=∑uv∈Edudvm−2mnIRDIFG=∑uv∈Edudv−dvdu,IRLG=∑uv∈Elndu−lndv,IRLUG=∑uv∈Edu−dvmindu,dv,IRLFG=∑uv∈Edu−dvdudv,IRLAG=2∑uv∈Edu−dvdu+dv,IRD1G=∑uv∈Eln1+du−dv,IRGAG=∑uv∈Elndu+dv2dudv.


## 3. Main Results and Discussion

In this section of the paper, we will discuss about the silicate network, chain silicate network, oxide network, hexagonal network, and honeycomb network briefly and will compute the irregularity indices for these networks.


### 3.1. Silicate Network

Silicates are the most popular, significant, and most complex mineral class. TetrahedronSiO4 is the basic chemical unit of silicates. By fusing metal oxides or metal carbonates with sand, silicates are produced. The fact is that all silicates contain tetrahedron of SiO4. For chemistry, the corner vertices of tetrahedron SiO4 reflect oxygen ions, and the silicon ion is represented by the middle vertex. We name the corner vertices as oxygen nodes in graph theory and the middle vertex as the silicon node. Although the tetrahedron is arranged linearly, chain silicates are produced (refer Figure 1).Figure 1
Structure of the silicate network.The total number of vertices and edges inSLn is 15n2+3n and 36n2, respectively.Theorem 1.
The irregularity indices for the silicate networkSLn for n>1 are(6)VARSLn=54n2+16n−225n+12,ALSLn=3n36n+6,IR1SLn=9072n3−3690n2+486n5n+1,IR2SLn=324n−9012n−24n5n+1,IRFSLn=3n54n+18,IRFWSLn=54n+18324n−90,IRASLn=0.1716n3n+1,IRBSLn=3.0883n3n+1,IRCSLn=30.7279n−3.72796n−24n5n+1,IRDIFSLn=27n2+9n,IRLSLn=0.693118n2+6n,IRLUSLn=6n3n+1,IRLFSLn=4.2426n3n+1,IRLASLn=4n3n+1,IRD1SLn=8.3178n3n+1,IRGASLn=0.3533n3n+1.Proof.
By using the edge partition based on degrees of end vertices of each edge of the silicate networkSLn given in Table 1, we compute the irregularity indices of the silicate network SLn, and the computations are given as follows:(7)VARSLn=M1SLnn−2mn2=3n126n−223n5n+1−236n23n5n+12=54n2+16n−225n+12,ALSLn=∑uv∈Edu−dv=3−36n+3−6182+6n+6−618n2−12n=3n36n+6,IR1SLn=FSLn−2mnM1SLn=2106n2−486n−236n2378n2−66n3n5n+1=9072n3−3690n2+486n5n+1,IR2SLn=M2SLnm−2mn=3n324n−9036n2−236n23n5n+1=324n−9012n−24n5n+1,IRFSLn=FSLn−2M2SLn=2106n2−486n−2972n2−270n=3n54n+18,IRFWSLn=IRFSLnM2SLn=3n54n+183n324n−90=54n+18324n−90,IRASLn=∑uv∈Edu−1/2−dv−1/22=13−1326n+13−16218n2+6n+16−162182−12n=0.1716n3n+1,IRBSLn=∑uv∈Edu1/2−dv1/22=3−326n+3−6218n2+6n+6−6218n2−6n=3.0883n3n+1,IRCSLn=∑uv∈Edudvm−2mn=96n+18n2+6n18+18n2−12n3636n2−72n23n5n+1=30.7279n−3.72796n−24n5n+1,IRDIFSLn=∑uv∈Edudv−dvdu=6n33−33+18n2+6n36−63+18n2−12n66−66=27n2+9n,IRLSLn=∑uv∈Elndu−lndv=6nln3−ln3+18n2+6nln3−ln6+18n2−12nln6−ln6=0.693118n2+6n,IRLUSLn=∑uv∈Edu−dvmindu,dv=6n3−33+6n3n+13−63+18n2−12n6−66=6n3n+1.
Specific values of irregularity indices ofSLn for different values of parameters are given in Table 2.Table 1
Partition of the edge set ofSLn which is based on degrees of end vertices.
du,dv3,33,66,6Number of edges6n18n2+6n18n2−12nTable 2
Values of irregularity indices forSLn.
Irregularity indicesn = 1n = 2n = 3n = 4n = 5VARSLn1.33331.86782.00002.05442.0828ALSLn126468102618002790IR1SLn9785344.363613324.500024929.142940160.7692IR2SLn0.41590.45820.44970.44110.4344IRFSLn216756162028084320IRFWSLn0.30770.22580.20410.19400.1882IRASLn0.68642.40245.14808.923213.7280IRBSLn12.353243.236292.6490160.5916247.6640IRCSLn0.50000.44700.41420.39450.3816IRDIFSLn36126270468720IRLSLn16.634458.2204124.7580216.2472332.6880IRLUSLn2484180312480IRLFSLn16.970459.3964127.2780220.6152339.4080IRLASLn1656120208320IRD1SLn33.2712116.4492249.5340432.5256665.4240IRGASLn1.41324.946210.599018.371628.2640(8)IRLFSLn=∑uv∈Edu−dvdudv=6n3−39+18n2+6n6−318+18n2−12n6−636=4.2426n3n+1,IRLASLn=2∑uv∈Edu−dvdu+dv=26n3−36+218n2+6n6−39+218n2−12n6−612=4n3n+1,IRD1SLn=∑uv∈Eln1+du−dv=ln1+3−36n+ln1+3−618n2+6n+ln1+6−618n2−12n=8.3178n3n+1,IRGASLn=∑uv∈Elndu+dv2dudv=6nln3+329+18n2+6nln3+6218+18n2−12nln6+6236=0.3533n3n+1.


### 3.2. Chain Silicate Network

Chain silicate network is obtained when tetrahedra are organized in a sequence. Ann-dimensional chain silicate network is represented by CSn, and it is generated by sequentially organizing n tetrahedra. An n-dimensional chain silicate network is shown in Figure 2.Figure 2
Structure of the chain silicate network.The edge partition of the chain silicate network is given in Table3.Table 3
Edge partition ofCSn.
du,dv, where uvϵG3,33,66,6Number of edgesn+422n−1n−2The total number of vertices and edges inCSn is 3n+1 and 6n, respectively.Theorem 2.
The irregularity indices for the chain silicate networkCSn for n>1 are(9)VARCSn=18n2−183n+12,ALCSn=12n−6,IR1CSn=162n2−13n+1,IR2CSn=913n−86n−12n3n+1,IRFCSn=36n−18,IRFWCSn=4n−213n−8,IRACSn=0.028574n−2,IRBCSn=0.51474n−2,IRCCSn=3n+4+324n−2+6n−26n−12n3n+1,IRDIFCSn=6n−3,IRLCSn=0.69314n−2,IRLUCSn=4n−2,IRLFCSn=22n−1,IRLACSn=432n−1,IRD1CSn=4n−2ln4,IRGACSn=2n−1ln32.Proof.
By using the edge partition based on degrees of end vertices of each edge of the chain silicate networkCSn given in Table 3, the computations for irregularities indices are given as follows:(10)VARCSn=M1CSnn−2mn2=183n−13n+1−26n3n+12=18n2−183n+12,ALCSn=∑uv∈Edu−dv=3−3n+4+3−64n−2+6−6n−2=12n−6,IR1CSn=FCSn−2mnM1CSn=270n−162−6n254n−183n+1=162n2−1623n+1,IR2CSn=M2CSnm−2mn=913n−86n−26n3n+1=913n−86n−12n3n+1,IRFCSn=FCSn−2M2CSn=270n−162−1813n−8=36n−18,IRFWCSn=IRFCSnM2CSn=182n−1913n−8=4n−213n−8,IRACSn=∑uv∈Edu−1/2−dv−1/22=13−132n+4+13−1624n−1+16−162n−2=0.028574n−2,IRBCSn=∑uv∈Edu1/2−dv1/22=3−32n+4+3−624n−26−62n−2=0.51474n−2,IRCCSn=∑uv∈Edudvm−2mn=9n+4+4n−218+n−2366n−12n3n+1=3n+4+324n−2+6n−26n−12n3n+1,IRDIFCSn=∑uv∈Edudv−dvdu=n+433−33+4n−236−63+n−266−66=6n−3,IRLCSn=∑uv∈Elndu−lndv=n+4ln3−ln3+4n−2ln3−ln6+n−2ln6−ln6=0.69314n−2,IRLUCSn=∑uv∈Edu−dvmindu,dv=n+43−33+4n−23−63+n−26−66=4n−2,IRLFCSn=∑uv∈Edu−dvdudv=n+43−39+4n−26−318+n−26−636=22n−1,IRLACSn=2∑uv∈Edu−dvdu+dv=2n+43−36+24n−26−39+2n−26−612=2n−143,IRD1CSn=∑uv∈Eln1+du−dv=ln1+3−3n+4+ln1+3−64n−2+ln1+6−6n−2=4n−2ln4,IRGACSn=∑uv∈Elndu+dv2dudv=n+4ln3+329+4n−2ln3+6218+n−2ln6+6236=2n−1ln32.
Specific values of irregularity indices for different values of involved parameters for the chain silicon network are given in Table4.Table 4
Values of irregularity indices forCSn.
Irregularity indicesn = 1n = 2n = 3n = 4n = 5VARCSn0.00001.10201.44001.59761.6875ALCSn618304254IR1CSn0.000069.4286129.6000186.9231243.0000IR2CSn−0.26140.24570.33370.36690.3852IRFCSn185490126162IRFWCSn0.40000.33330.32260.31820.3158IRACSn0.05720.17160.28600.40030.5147IRBCSn1.02943.08825.14707.20589.2646IRCCSn−0.08580.19270.25700.28260.2956IRDIFCSn39152127IRLCSn1.38624.15866.93109.703412.4758IRLUCSn26101418IRLFCSn1.41424.24267.07119.899512.7279IRLACSn1.33334.00006.66679.333312.0000IRD1CSn2.77268.317813.862919.408124.9533IRGACSn0.75202.25613.76025.26436.7683


### 3.3. Oxide Network

Oxide networks play a crucial role in the analysis of silicate networks. If we detach silicone vertices out of a silicate network, we get an oxide networkOXn that is referred to as an n-dimensional oxide network as shown in Figure 3.Figure 3
Structure of the oxide network.There are two types of edge partition in the oxide networkOXn centered on degrees of end vertices. Table 5 shows the edge partition for OXn.Table 5
Edge partition for the oxide network.
du,dv, where uvϵG2,44,4Number of edges12n18n2−12nThe total number of vertices and edges inOXn is 9n2+3n and 18n2, respectively.Theorem 3.
The irregularity indices of oxide networksOXn for n>1 are(11)VAROXn=72n23n−19n2+3n2,ALOXn=72n,IR1OXn=144n3n−13n+1,IR2OXn=163n−13n−12n3n+1,IRFOXn=48n,IRFWOXn=7243n−1,IRAOXn=0.5136n,IRBOXn=4.1177n,IRCOXn=222+6n−43n−12n3n+1,IRDIFOXn=18n,IRLOXn=8.3177n,IRLUOXn=12n,IRLFOXn=12n2,IRLAOXn=8n,IRD1OXn=13.1833n,IRGAOXn=0.7068n.Proof.
By using the edge partition based on degrees of end vertices of each edge of the oxide networkOXn given in Table 5, we have the following computations for irregularities of the oxide network OXn:(12)VAROXn=M1OXnn−2mn2=24n6n−13n3n+1−2.18n23n3n+12=72n23n−19n2+3n2,ALOXn=∑uv∈Edu−dv=2−412n+4−418n2−12n=72n,IR1OXn=FOXn−2mnM1OXn=576n2−144n−18n2224n6n−19n2+3n=144n3n−13n+1,IR2OXn=M2OXnm−2mn=96n3n−118n2−218n23n3n+1=163n−13n−12n3n+1,IRFOXn=FOXn−2M2OXn=576n2−144n−2288n2−96n=48n,IRFWOXn=IRFOXnM2OXn=28n96n3n−1=7243n−1,IRAOXn=∑uv∈Edu−1/2−dv−1/22=12−14212n+14−14218n2−12n=0.5136n,IRBOXn=∑uv∈Edu1/2−dv1/22=2−4212n+4−4218n2−12n=4.1177n,IRCOXn=∑uv∈Edudvm−2mn=222+6n−43n−12n3n+1,IRDIFOXn=∑uv∈Edudv−dvdu=12n24−42+18n244−44=18n.
Specific values of irregularity indices for different values of involved parameters for the oxide network are given in Table6.Table 6
Values of irregularity indices forOXn.
Irregularity indicesn = 1n = 2n = 3n = 4n = 5VAROXn1.00000.81630.64000.52070.4375ALOXn72144216288360IR1OXn72205.7143345.6000487.3846630.0000IR2OXn0.26603.87447.713711.626515.5718IRFOXn4896144192240IRFWOXn0.14580.05830.03650.02650.0208IRAOXn0.51361.02721.54082.05442.5680IRBOXn4.11778.235412.353116.470820.5885IRCOXn0.21890.18090.13960.11240.0938IRDIFOXn1836547290IRLOXn8.317716.635424.953133.270841.5885IRLUOXn1224364860IRLFOXn8.485316.970625.455833.941142.4264IRLAOXn816243240IRD1OXn13.183326.366639.549952.733265.9165IRGAOXn0.70681.41362.12042.82723.5340(13)IRLOXn=∑uv∈Elndu−lndv=12nln2−ln4+18n2ln4−ln4=8.3177n,IRLUOXn=∑uv∈Edu−dvmindu,dv=12n2−42+18n24−44=12n,IRLFOXn=∑uv∈Edu−dvdudv=12n4−28+18n24−416=12n2,IRLAOXn=2∑uv∈Edu−dvdu+dv=212n4−26+218n24−48=8n,IRD1OXn=∑uv∈Eln1+du−dv=ln1+4−212n+ln1+4−418n2=13.1833n,IRGAOXn=∑uv∈Elndu+dv2dudv=12nln4+228+18n2ln4+4216=0.7068n.


### 3.4. Hexagonal Network

It is well known that there are three normal plane tilings with the same regular polygon type as triangle, hexagon-shaped, and square structures. Triangular tiling is used in the design of hexagonal networks. Ann-dimensional hexagonal network is commonly referred to as HXn, where n is each side’s number of hexagonal vertices. A hexagonal network HXn is shown in Figure 4.Figure 4
Structure of the hexagonal network.Table7 shows the edge partition of the hexagonal network HXn. The total number of vertices and edges in hexagonal networks HXn is 3n2−3n+1 and 9n2−15n+6, respectively.Table 7
Edge partition for the hexagonal network.
du,dv, where uvϵG3,43,64,44,66,6Number of edges1266n−312n−292−33n+30Theorem 4.
The irregularity indices of the hexagonal networkHXn for n>1 are(14)VARHXn=72n3+243n2+15n−303n2−3n+12,ALHXn=64n−3,IR1HXn=720n3−2142n2+1878n−4623n2−3n+1,IR2HXn=108n2−268n+1563n2−5n+2−18n2−30n+123n2−3n+1,IRFHXn=68n−5,IRFWHXn=28n−5108n2−268n+156,IRAHXn=0.1008n−0.0418,IRBHXn=2.4240n−0.8981,IRCHXn=412+182+8n−24+4n−824+2n2−33n+303n2−5n+2−18n2−30n+123n2−3n+1,IRDIFHXn=10n−4,IRLHXn=4.8656n+2.1201,IRLUHXn=6n−2,IRLFHXn=4.8990n−20.0913,IRLAHXn=168n−7635,IRDIHXn=13.1833n−9.7311,IRGAHXn=0.2449n−0.0126.Proof.
By using the edge partition based on degrees of end vertices of each edge of the hexagonal networkHXn given in Table 7, we have the following computations for irregularities of the hexagonal network HXn:(15)VARHXn=M1HXnn−2mn2=108n2−228n+1143n2−3n+1−29n2−15n+63n2−3n+12=72n3+243n2+15n−303n2−3n+12,ALHXn=∑uv∈Edu−dv=3−412+3−66+4−46n−18+4−612n−24+6−69n2−33n+30=64n−3,IR1HXn=FHXn−2mnM1HXn=648n2−1560n+906−29n2−15n+6108n2−228n+1143n2−3n+1=720n3−2142n2+1878n−4623n2−3n+1,IR2HXn=M2HXnm−2mn=324n2−804n+4689n2−15n+6−29n2−15n+63n2−3n+1=108n2−268n+1563n2−5n+2−18n2−30n+123n2−3n+1,IRFHXn=FHXn−2M2HXn=648n2−1560n+906−2324n2−804n+468=68n−5,IRFWHXn=IRFHXnM2HXn=68n−5324n2−804n+468=28n−5108n2−268n+156,IRAHXn=∑uv∈Edu−1/2−dv−1/22=13−14212+13−1626+14−1426n−18+14−16212n−24+16−1629n2−33n+30=0.1008n−0.0418,IRBHXn=∑uv∈Edu1/2−dv1/22=3−4212+3−626+4−6212n−24=2.4240n−0.8981,IRCHXn=∑uv∈Edudvm−2mn=1212+618+6n−1816+12n−2424+9n2−33n+3069n2−15n+6−29n2−15n+63n2−3n+1=412+182+8n−24+4n−824+29n2−33n+303n2−5n+2−18n2−30n+123n2−3n+1,IRDIFHXn=∑uv∈Edudv−dvdu=1234−43+636−63+12n−2446−64=10n−4,IRLHXn=∑uv∈Elndu−lndv=12ln3−ln4+6ln3−ln6+12n−24ln4−ln6=4.8656n+2.1201,IRLUHXn=∑uv∈Edu−dvmindu,dv=123−43+63−63+12n−244−64=6n−2,IRLFHXn=∑uv∈Edu−dvdudv=123−412+66−318+12n−246−424=4.8990n−20.0913,IRLAHXn=2∑uv∈Edu−dvdu+dv=2124−37+266−39+212n−246−410=168n−7635.
Specific values of irregularities of the hexagonal network for different values of involved parameters are given in Table8.Table 8
Values of irregularity indices forHXn.
Irregularity indicesn = 1n = 2n = 3n = 4n = 5VARHXn30031.591811.48486.22794.0633ALHXn6305478102IR1HXn−6.000069.4286280.7368509.6757743.9016IR2HXn∞0.17700.38960.33770.2846IRFHXn1866114162210IRFWHXn−1.50000.42310.11730.06650.0462IRAHXn0.05900.15980.26060.36140.4622IRBHXn1.52593.94996.37398.797911.2219IRCHXn∞0.15680.28870.25290.2146IRDIFHXn616263646IRLHXn2.74557.611112.476717.342322.2079IRLUHXn410162228IRLFHXn2.80777.706712.605717.504722.4037IRLAHXn2.62867.428612.228617.028621.8286IRD1HXn3.452216.635529.818843.002156.1854IRGAHXn0.23230.47720.72210.96701.2119(16)IRD1HXn=∑uv∈Eln1+du−dv=ln1+3−412+ln1+3−66+ln1+4−612n−24=13.1833n−9.7311,IRGAHXn=∑uv∈Elndu+dv2dudv=12ln3+4212+6ln3+6218+12n−24ln4+6224=0.2449n−0.0126.


### 3.5. Honeycomb Network

Honeycomb networks are commonly used as an illustration of benzoid hydrocarbons in chemistry in digital effects, cell phone transmitters, and image analysis. If we use dynamically hexagonal tiling in a specific pattern, honeycomb networks will be formed. Ann-dimensional honeycomb network is referred to as HCn, where n is the number of hexagons between the hexagon core and the boundary. Honeycomb network HCn is built from HCn−1 by adding a hexagon sheet across the boundary of HCn−1 (refer Figure 5).Figure 5
Structure of the honeycomb network.Table9 shows the edge partition of the honeycomb network HCn. The total number of vertices and edges in the honeycomb network HCn is 6n2 and 9n2−3n.Table 9
Edge partition of the honeycomb network.
du,dv, where uvϵG2,22,33,3Number of edges612n−19n2−15n+6Theorem 5.
The irregularity indices of the honeycomb networkHCn for n>1 are(17)VARHCn=n−1n2,ALHCn=12n−1,IR1HCn=30n−1,IR2HCn=27n2−21n+2n3n−1−3n−1n,IRFHCn=12n−1,IRFWHCn=14n−127n2−21n+2,IRAHCn=0.2016n−1,IRBHCn=1.2120n−1,IRCHCn=4+46n−1+9n2−15n+6n3n−1−3n−1n,IRDIFHCn=9.9996n−1,IRLHCn=4.8660n−1,IRLUHCn=6n−1,IRLFHCn=4.8990n−1,IRLAHCn=4.8000n−1,IRD1HCn=8.3172n−1,IRGAHCn=0.2448n−1.Proof.
By using the edge partition based on degrees of end vertices of each edge of the honeycomb networkHCn given in Table 9, we have the following computations for the irregularity indices of the honeycomb network HCn:(18)VARHCn=M1HCnn−2mn2=3n18n−106n2−29n2−3n6n22=36n2n−136n4=n−1n2,ALHCn=∑uv∈Edu−dv=2−26+2−312n−1+3−39n2−15n+6=12n−1,IR1HCn=FHCn−2mnM1HCn=162n2−114n−29n2−3n54n2−306n2=30n−1,IR2HCn=M2HCnm−2mn=327n2−21n+23n3n−1−23n3n−16n2=27n2−21n+2n3n−1−3n−1n,IRFHCn=FHCn−2M2HCn=162n2−114n−281n2−63n+6=12n−1,IRFWHCn=IRFHCnM2HCn=12n−181n2−63n+6=14n−127n2−21n+2,IRAHCn=∑uv∈Edu−1/2−dv−1/22=12−1226+12−13212n−1+13−13292−15n+6=0.2016n−1,IRBHCn=∑uv∈Edu1/2−dv1/22=2−226+2−3212n−12+3−329n2−15n+6=1.2120n−1,IRCHCn=∑uv∈Edudvm−2mn=46+12n−126+9n2−15n+693n3n−1−23n3n−16n2=4+46n−1+9n2−15n+6n3n−1−3n−1n,IRDIFHCn=∑uv∈Edudv−dvdu=622−22+12n−1223−32+9n2−15n+633−33=9.9996n−1,IRLHCn=∑uv∈Elndu−lndv=6ln2−ln2+12n−12ln2−ln3+9n2−15n+6ln3−ln3=4.8660n−1,IRLUHCn=∑uv∈Edu−dvmindu,dv=62−22+12n−122−32+9n2−15n+63−33=6n−1,IRLFHCn=∑uv∈Edu−dvdudv=62−24+12n−123−26+9n2−15n+63−39=4.8990n−1,IRLAHCn=2∑uv∈Edu−dvdu+dv=262−24+212n−123−25+29n2−15n+63−36=4.8000n−1,IRD1HCn=∑uv∈Eln1+du−dv=ln1+2−26+ln1+2−312n−12+ln1+3−39n2−15n+6=8.3172n−1,IRGAHCn=∑uv∈Elndu+dv2dudv=6ln2+224+12n−12ln2+326+9n2−15n+6ln3+329=0.2448n−1.
Specific values for the irregularities of the honeycomb network for different values of involved parameters are given in Table10.Table 10
Values of irregularity indices forHCn.
Irregularity indicesn = 1n = 2n = 3n = 4n = 5VARHCn0.00000.25000.22220.18750.1600ALHCn012243648IR1HCn0306090120IR2HCn0.00000.10770.08710.07040.0586IRFHCn012243648IRFWHCn0.00000.05880.04400.03430.0280IRAHCn0.00000.20160.40320.60480.8064IRBHCn0.00001.21202.42403.63604.8480IRCHCn0.00000.07980.06650.05440.0456IRDIFHCn0.00009.999619.999229.998839.9984IRLHCn0.00004.86609.732014.598019.4640IRLUHCn06121824IRLFHCn0.00004.89909.798014.697019.5960IRLAHCn0.00004.80009.600014.400019.2000IRD1HCn0.00008.317216.634424.951633.2688IRGAHCn0.00000.24480.24960.73440.9792


## 3.1. Silicate Network

Silicates are the most popular, significant, and most complex mineral class. TetrahedronSiO4 is the basic chemical unit of silicates. By fusing metal oxides or metal carbonates with sand, silicates are produced. The fact is that all silicates contain tetrahedron of SiO4. For chemistry, the corner vertices of tetrahedron SiO4 reflect oxygen ions, and the silicon ion is represented by the middle vertex. We name the corner vertices as oxygen nodes in graph theory and the middle vertex as the silicon node. Although the tetrahedron is arranged linearly, chain silicates are produced (refer Figure 1).Figure 1
Structure of the silicate network.The total number of vertices and edges inSLn is 15n2+3n and 36n2, respectively.Theorem 1.
The irregularity indices for the silicate networkSLn for n>1 are(6)VARSLn=54n2+16n−225n+12,ALSLn=3n36n+6,IR1SLn=9072n3−3690n2+486n5n+1,IR2SLn=324n−9012n−24n5n+1,IRFSLn=3n54n+18,IRFWSLn=54n+18324n−90,IRASLn=0.1716n3n+1,IRBSLn=3.0883n3n+1,IRCSLn=30.7279n−3.72796n−24n5n+1,IRDIFSLn=27n2+9n,IRLSLn=0.693118n2+6n,IRLUSLn=6n3n+1,IRLFSLn=4.2426n3n+1,IRLASLn=4n3n+1,IRD1SLn=8.3178n3n+1,IRGASLn=0.3533n3n+1.Proof.
By using the edge partition based on degrees of end vertices of each edge of the silicate networkSLn given in Table 1, we compute the irregularity indices of the silicate network SLn, and the computations are given as follows:(7)VARSLn=M1SLnn−2mn2=3n126n−223n5n+1−236n23n5n+12=54n2+16n−225n+12,ALSLn=∑uv∈Edu−dv=3−36n+3−6182+6n+6−618n2−12n=3n36n+6,IR1SLn=FSLn−2mnM1SLn=2106n2−486n−236n2378n2−66n3n5n+1=9072n3−3690n2+486n5n+1,IR2SLn=M2SLnm−2mn=3n324n−9036n2−236n23n5n+1=324n−9012n−24n5n+1,IRFSLn=FSLn−2M2SLn=2106n2−486n−2972n2−270n=3n54n+18,IRFWSLn=IRFSLnM2SLn=3n54n+183n324n−90=54n+18324n−90,IRASLn=∑uv∈Edu−1/2−dv−1/22=13−1326n+13−16218n2+6n+16−162182−12n=0.1716n3n+1,IRBSLn=∑uv∈Edu1/2−dv1/22=3−326n+3−6218n2+6n+6−6218n2−6n=3.0883n3n+1,IRCSLn=∑uv∈Edudvm−2mn=96n+18n2+6n18+18n2−12n3636n2−72n23n5n+1=30.7279n−3.72796n−24n5n+1,IRDIFSLn=∑uv∈Edudv−dvdu=6n33−33+18n2+6n36−63+18n2−12n66−66=27n2+9n,IRLSLn=∑uv∈Elndu−lndv=6nln3−ln3+18n2+6nln3−ln6+18n2−12nln6−ln6=0.693118n2+6n,IRLUSLn=∑uv∈Edu−dvmindu,dv=6n3−33+6n3n+13−63+18n2−12n6−66=6n3n+1.
Specific values of irregularity indices ofSLn for different values of parameters are given in Table 2.Table 1
Partition of the edge set ofSLn which is based on degrees of end vertices.
du,dv3,33,66,6Number of edges6n18n2+6n18n2−12nTable 2
Values of irregularity indices forSLn.
Irregularity indicesn = 1n = 2n = 3n = 4n = 5VARSLn1.33331.86782.00002.05442.0828ALSLn126468102618002790IR1SLn9785344.363613324.500024929.142940160.7692IR2SLn0.41590.45820.44970.44110.4344IRFSLn216756162028084320IRFWSLn0.30770.22580.20410.19400.1882IRASLn0.68642.40245.14808.923213.7280IRBSLn12.353243.236292.6490160.5916247.6640IRCSLn0.50000.44700.41420.39450.3816IRDIFSLn36126270468720IRLSLn16.634458.2204124.7580216.2472332.6880IRLUSLn2484180312480IRLFSLn16.970459.3964127.2780220.6152339.4080IRLASLn1656120208320IRD1SLn33.2712116.4492249.5340432.5256665.4240IRGASLn1.41324.946210.599018.371628.2640(8)IRLFSLn=∑uv∈Edu−dvdudv=6n3−39+18n2+6n6−318+18n2−12n6−636=4.2426n3n+1,IRLASLn=2∑uv∈Edu−dvdu+dv=26n3−36+218n2+6n6−39+218n2−12n6−612=4n3n+1,IRD1SLn=∑uv∈Eln1+du−dv=ln1+3−36n+ln1+3−618n2+6n+ln1+6−618n2−12n=8.3178n3n+1,IRGASLn=∑uv∈Elndu+dv2dudv=6nln3+329+18n2+6nln3+6218+18n2−12nln6+6236=0.3533n3n+1.


## 3.2. Chain Silicate Network

Chain silicate network is obtained when tetrahedra are organized in a sequence. Ann-dimensional chain silicate network is represented by CSn, and it is generated by sequentially organizing n tetrahedra. An n-dimensional chain silicate network is shown in Figure 2.Figure 2
Structure of the chain silicate network.The edge partition of the chain silicate network is given in Table3.Table 3
Edge partition ofCSn.
du,dv, where uvϵG3,33,66,6Number of edgesn+422n−1n−2The total number of vertices and edges inCSn is 3n+1 and 6n, respectively.Theorem 2.
The irregularity indices for the chain silicate networkCSn for n>1 are(9)VARCSn=18n2−183n+12,ALCSn=12n−6,IR1CSn=162n2−13n+1,IR2CSn=913n−86n−12n3n+1,IRFCSn=36n−18,IRFWCSn=4n−213n−8,IRACSn=0.028574n−2,IRBCSn=0.51474n−2,IRCCSn=3n+4+324n−2+6n−26n−12n3n+1,IRDIFCSn=6n−3,IRLCSn=0.69314n−2,IRLUCSn=4n−2,IRLFCSn=22n−1,IRLACSn=432n−1,IRD1CSn=4n−2ln4,IRGACSn=2n−1ln32.Proof.
By using the edge partition based on degrees of end vertices of each edge of the chain silicate networkCSn given in Table 3, the computations for irregularities indices are given as follows:(10)VARCSn=M1CSnn−2mn2=183n−13n+1−26n3n+12=18n2−183n+12,ALCSn=∑uv∈Edu−dv=3−3n+4+3−64n−2+6−6n−2=12n−6,IR1CSn=FCSn−2mnM1CSn=270n−162−6n254n−183n+1=162n2−1623n+1,IR2CSn=M2CSnm−2mn=913n−86n−26n3n+1=913n−86n−12n3n+1,IRFCSn=FCSn−2M2CSn=270n−162−1813n−8=36n−18,IRFWCSn=IRFCSnM2CSn=182n−1913n−8=4n−213n−8,IRACSn=∑uv∈Edu−1/2−dv−1/22=13−132n+4+13−1624n−1+16−162n−2=0.028574n−2,IRBCSn=∑uv∈Edu1/2−dv1/22=3−32n+4+3−624n−26−62n−2=0.51474n−2,IRCCSn=∑uv∈Edudvm−2mn=9n+4+4n−218+n−2366n−12n3n+1=3n+4+324n−2+6n−26n−12n3n+1,IRDIFCSn=∑uv∈Edudv−dvdu=n+433−33+4n−236−63+n−266−66=6n−3,IRLCSn=∑uv∈Elndu−lndv=n+4ln3−ln3+4n−2ln3−ln6+n−2ln6−ln6=0.69314n−2,IRLUCSn=∑uv∈Edu−dvmindu,dv=n+43−33+4n−23−63+n−26−66=4n−2,IRLFCSn=∑uv∈Edu−dvdudv=n+43−39+4n−26−318+n−26−636=22n−1,IRLACSn=2∑uv∈Edu−dvdu+dv=2n+43−36+24n−26−39+2n−26−612=2n−143,IRD1CSn=∑uv∈Eln1+du−dv=ln1+3−3n+4+ln1+3−64n−2+ln1+6−6n−2=4n−2ln4,IRGACSn=∑uv∈Elndu+dv2dudv=n+4ln3+329+4n−2ln3+6218+n−2ln6+6236=2n−1ln32.
Specific values of irregularity indices for different values of involved parameters for the chain silicon network are given in Table4.Table 4
Values of irregularity indices forCSn.
Irregularity indicesn = 1n = 2n = 3n = 4n = 5VARCSn0.00001.10201.44001.59761.6875ALCSn618304254IR1CSn0.000069.4286129.6000186.9231243.0000IR2CSn−0.26140.24570.33370.36690.3852IRFCSn185490126162IRFWCSn0.40000.33330.32260.31820.3158IRACSn0.05720.17160.28600.40030.5147IRBCSn1.02943.08825.14707.20589.2646IRCCSn−0.08580.19270.25700.28260.2956IRDIFCSn39152127IRLCSn1.38624.15866.93109.703412.4758IRLUCSn26101418IRLFCSn1.41424.24267.07119.899512.7279IRLACSn1.33334.00006.66679.333312.0000IRD1CSn2.77268.317813.862919.408124.9533IRGACSn0.75202.25613.76025.26436.7683


## 3.3. Oxide Network

Oxide networks play a crucial role in the analysis of silicate networks. If we detach silicone vertices out of a silicate network, we get an oxide networkOXn that is referred to as an n-dimensional oxide network as shown in Figure 3.Figure 3
Structure of the oxide network.There are two types of edge partition in the oxide networkOXn centered on degrees of end vertices. Table 5 shows the edge partition for OXn.Table 5
Edge partition for the oxide network.
du,dv, where uvϵG2,44,4Number of edges12n18n2−12nThe total number of vertices and edges inOXn is 9n2+3n and 18n2, respectively.Theorem 3.
The irregularity indices of oxide networksOXn for n>1 are(11)VAROXn=72n23n−19n2+3n2,ALOXn=72n,IR1OXn=144n3n−13n+1,IR2OXn=163n−13n−12n3n+1,IRFOXn=48n,IRFWOXn=7243n−1,IRAOXn=0.5136n,IRBOXn=4.1177n,IRCOXn=222+6n−43n−12n3n+1,IRDIFOXn=18n,IRLOXn=8.3177n,IRLUOXn=12n,IRLFOXn=12n2,IRLAOXn=8n,IRD1OXn=13.1833n,IRGAOXn=0.7068n.Proof.
By using the edge partition based on degrees of end vertices of each edge of the oxide networkOXn given in Table 5, we have the following computations for irregularities of the oxide network OXn:(12)VAROXn=M1OXnn−2mn2=24n6n−13n3n+1−2.18n23n3n+12=72n23n−19n2+3n2,ALOXn=∑uv∈Edu−dv=2−412n+4−418n2−12n=72n,IR1OXn=FOXn−2mnM1OXn=576n2−144n−18n2224n6n−19n2+3n=144n3n−13n+1,IR2OXn=M2OXnm−2mn=96n3n−118n2−218n23n3n+1=163n−13n−12n3n+1,IRFOXn=FOXn−2M2OXn=576n2−144n−2288n2−96n=48n,IRFWOXn=IRFOXnM2OXn=28n96n3n−1=7243n−1,IRAOXn=∑uv∈Edu−1/2−dv−1/22=12−14212n+14−14218n2−12n=0.5136n,IRBOXn=∑uv∈Edu1/2−dv1/22=2−4212n+4−4218n2−12n=4.1177n,IRCOXn=∑uv∈Edudvm−2mn=222+6n−43n−12n3n+1,IRDIFOXn=∑uv∈Edudv−dvdu=12n24−42+18n244−44=18n.
Specific values of irregularity indices for different values of involved parameters for the oxide network are given in Table6.Table 6
Values of irregularity indices forOXn.
Irregularity indicesn = 1n = 2n = 3n = 4n = 5VAROXn1.00000.81630.64000.52070.4375ALOXn72144216288360IR1OXn72205.7143345.6000487.3846630.0000IR2OXn0.26603.87447.713711.626515.5718IRFOXn4896144192240IRFWOXn0.14580.05830.03650.02650.0208IRAOXn0.51361.02721.54082.05442.5680IRBOXn4.11778.235412.353116.470820.5885IRCOXn0.21890.18090.13960.11240.0938IRDIFOXn1836547290IRLOXn8.317716.635424.953133.270841.5885IRLUOXn1224364860IRLFOXn8.485316.970625.455833.941142.4264IRLAOXn816243240IRD1OXn13.183326.366639.549952.733265.9165IRGAOXn0.70681.41362.12042.82723.5340(13)IRLOXn=∑uv∈Elndu−lndv=12nln2−ln4+18n2ln4−ln4=8.3177n,IRLUOXn=∑uv∈Edu−dvmindu,dv=12n2−42+18n24−44=12n,IRLFOXn=∑uv∈Edu−dvdudv=12n4−28+18n24−416=12n2,IRLAOXn=2∑uv∈Edu−dvdu+dv=212n4−26+218n24−48=8n,IRD1OXn=∑uv∈Eln1+du−dv=ln1+4−212n+ln1+4−418n2=13.1833n,IRGAOXn=∑uv∈Elndu+dv2dudv=12nln4+228+18n2ln4+4216=0.7068n.


## 3.4. Hexagonal Network

It is well known that there are three normal plane tilings with the same regular polygon type as triangle, hexagon-shaped, and square structures. Triangular tiling is used in the design of hexagonal networks. Ann-dimensional hexagonal network is commonly referred to as HXn, where n is each side’s number of hexagonal vertices. A hexagonal network HXn is shown in Figure 4.Figure 4
Structure of the hexagonal network.Table7 shows the edge partition of the hexagonal network HXn. The total number of vertices and edges in hexagonal networks HXn is 3n2−3n+1 and 9n2−15n+6, respectively.Table 7
Edge partition for the hexagonal network.
du,dv, where uvϵG3,43,64,44,66,6Number of edges1266n−312n−292−33n+30Theorem 4.
The irregularity indices of the hexagonal networkHXn for n>1 are(14)VARHXn=72n3+243n2+15n−303n2−3n+12,ALHXn=64n−3,IR1HXn=720n3−2142n2+1878n−4623n2−3n+1,IR2HXn=108n2−268n+1563n2−5n+2−18n2−30n+123n2−3n+1,IRFHXn=68n−5,IRFWHXn=28n−5108n2−268n+156,IRAHXn=0.1008n−0.0418,IRBHXn=2.4240n−0.8981,IRCHXn=412+182+8n−24+4n−824+2n2−33n+303n2−5n+2−18n2−30n+123n2−3n+1,IRDIFHXn=10n−4,IRLHXn=4.8656n+2.1201,IRLUHXn=6n−2,IRLFHXn=4.8990n−20.0913,IRLAHXn=168n−7635,IRDIHXn=13.1833n−9.7311,IRGAHXn=0.2449n−0.0126.Proof.
By using the edge partition based on degrees of end vertices of each edge of the hexagonal networkHXn given in Table 7, we have the following computations for irregularities of the hexagonal network HXn:(15)VARHXn=M1HXnn−2mn2=108n2−228n+1143n2−3n+1−29n2−15n+63n2−3n+12=72n3+243n2+15n−303n2−3n+12,ALHXn=∑uv∈Edu−dv=3−412+3−66+4−46n−18+4−612n−24+6−69n2−33n+30=64n−3,IR1HXn=FHXn−2mnM1HXn=648n2−1560n+906−29n2−15n+6108n2−228n+1143n2−3n+1=720n3−2142n2+1878n−4623n2−3n+1,IR2HXn=M2HXnm−2mn=324n2−804n+4689n2−15n+6−29n2−15n+63n2−3n+1=108n2−268n+1563n2−5n+2−18n2−30n+123n2−3n+1,IRFHXn=FHXn−2M2HXn=648n2−1560n+906−2324n2−804n+468=68n−5,IRFWHXn=IRFHXnM2HXn=68n−5324n2−804n+468=28n−5108n2−268n+156,IRAHXn=∑uv∈Edu−1/2−dv−1/22=13−14212+13−1626+14−1426n−18+14−16212n−24+16−1629n2−33n+30=0.1008n−0.0418,IRBHXn=∑uv∈Edu1/2−dv1/22=3−4212+3−626+4−6212n−24=2.4240n−0.8981,IRCHXn=∑uv∈Edudvm−2mn=1212+618+6n−1816+12n−2424+9n2−33n+3069n2−15n+6−29n2−15n+63n2−3n+1=412+182+8n−24+4n−824+29n2−33n+303n2−5n+2−18n2−30n+123n2−3n+1,IRDIFHXn=∑uv∈Edudv−dvdu=1234−43+636−63+12n−2446−64=10n−4,IRLHXn=∑uv∈Elndu−lndv=12ln3−ln4+6ln3−ln6+12n−24ln4−ln6=4.8656n+2.1201,IRLUHXn=∑uv∈Edu−dvmindu,dv=123−43+63−63+12n−244−64=6n−2,IRLFHXn=∑uv∈Edu−dvdudv=123−412+66−318+12n−246−424=4.8990n−20.0913,IRLAHXn=2∑uv∈Edu−dvdu+dv=2124−37+266−39+212n−246−410=168n−7635.
Specific values of irregularities of the hexagonal network for different values of involved parameters are given in Table8.Table 8
Values of irregularity indices forHXn.
Irregularity indicesn = 1n = 2n = 3n = 4n = 5VARHXn30031.591811.48486.22794.0633ALHXn6305478102IR1HXn−6.000069.4286280.7368509.6757743.9016IR2HXn∞0.17700.38960.33770.2846IRFHXn1866114162210IRFWHXn−1.50000.42310.11730.06650.0462IRAHXn0.05900.15980.26060.36140.4622IRBHXn1.52593.94996.37398.797911.2219IRCHXn∞0.15680.28870.25290.2146IRDIFHXn616263646IRLHXn2.74557.611112.476717.342322.2079IRLUHXn410162228IRLFHXn2.80777.706712.605717.504722.4037IRLAHXn2.62867.428612.228617.028621.8286IRD1HXn3.452216.635529.818843.002156.1854IRGAHXn0.23230.47720.72210.96701.2119(16)IRD1HXn=∑uv∈Eln1+du−dv=ln1+3−412+ln1+3−66+ln1+4−612n−24=13.1833n−9.7311,IRGAHXn=∑uv∈Elndu+dv2dudv=12ln3+4212+6ln3+6218+12n−24ln4+6224=0.2449n−0.0126.


## 3.5. Honeycomb Network

Honeycomb networks are commonly used as an illustration of benzoid hydrocarbons in chemistry in digital effects, cell phone transmitters, and image analysis. If we use dynamically hexagonal tiling in a specific pattern, honeycomb networks will be formed. Ann-dimensional honeycomb network is referred to as HCn, where n is the number of hexagons between the hexagon core and the boundary. Honeycomb network HCn is built from HCn−1 by adding a hexagon sheet across the boundary of HCn−1 (refer Figure 5).Figure 5
Structure of the honeycomb network.Table9 shows the edge partition of the honeycomb network HCn. The total number of vertices and edges in the honeycomb network HCn is 6n2 and 9n2−3n.Table 9
Edge partition of the honeycomb network.
du,dv, where uvϵG2,22,33,3Number of edges612n−19n2−15n+6Theorem 5.
The irregularity indices of the honeycomb networkHCn for n>1 are(17)VARHCn=n−1n2,ALHCn=12n−1,IR1HCn=30n−1,IR2HCn=27n2−21n+2n3n−1−3n−1n,IRFHCn=12n−1,IRFWHCn=14n−127n2−21n+2,IRAHCn=0.2016n−1,IRBHCn=1.2120n−1,IRCHCn=4+46n−1+9n2−15n+6n3n−1−3n−1n,IRDIFHCn=9.9996n−1,IRLHCn=4.8660n−1,IRLUHCn=6n−1,IRLFHCn=4.8990n−1,IRLAHCn=4.8000n−1,IRD1HCn=8.3172n−1,IRGAHCn=0.2448n−1.Proof.
By using the edge partition based on degrees of end vertices of each edge of the honeycomb networkHCn given in Table 9, we have the following computations for the irregularity indices of the honeycomb network HCn:(18)VARHCn=M1HCnn−2mn2=3n18n−106n2−29n2−3n6n22=36n2n−136n4=n−1n2,ALHCn=∑uv∈Edu−dv=2−26+2−312n−1+3−39n2−15n+6=12n−1,IR1HCn=FHCn−2mnM1HCn=162n2−114n−29n2−3n54n2−306n2=30n−1,IR2HCn=M2HCnm−2mn=327n2−21n+23n3n−1−23n3n−16n2=27n2−21n+2n3n−1−3n−1n,IRFHCn=FHCn−2M2HCn=162n2−114n−281n2−63n+6=12n−1,IRFWHCn=IRFHCnM2HCn=12n−181n2−63n+6=14n−127n2−21n+2,IRAHCn=∑uv∈Edu−1/2−dv−1/22=12−1226+12−13212n−1+13−13292−15n+6=0.2016n−1,IRBHCn=∑uv∈Edu1/2−dv1/22=2−226+2−3212n−12+3−329n2−15n+6=1.2120n−1,IRCHCn=∑uv∈Edudvm−2mn=46+12n−126+9n2−15n+693n3n−1−23n3n−16n2=4+46n−1+9n2−15n+6n3n−1−3n−1n,IRDIFHCn=∑uv∈Edudv−dvdu=622−22+12n−1223−32+9n2−15n+633−33=9.9996n−1,IRLHCn=∑uv∈Elndu−lndv=6ln2−ln2+12n−12ln2−ln3+9n2−15n+6ln3−ln3=4.8660n−1,IRLUHCn=∑uv∈Edu−dvmindu,dv=62−22+12n−122−32+9n2−15n+63−33=6n−1,IRLFHCn=∑uv∈Edu−dvdudv=62−24+12n−123−26+9n2−15n+63−39=4.8990n−1,IRLAHCn=2∑uv∈Edu−dvdu+dv=262−24+212n−123−25+29n2−15n+63−36=4.8000n−1,IRD1HCn=∑uv∈Eln1+du−dv=ln1+2−26+ln1+2−312n−12+ln1+3−39n2−15n+6=8.3172n−1,IRGAHCn=∑uv∈Elndu+dv2dudv=6ln2+224+12n−12ln2+326+9n2−15n+6ln3+329=0.2448n−1.
Specific values for the irregularities of the honeycomb network for different values of involved parameters are given in Table10.Table 10
Values of irregularity indices forHCn.
Irregularity indicesn = 1n = 2n = 3n = 4n = 5VARHCn0.00000.25000.22220.18750.1600ALHCn012243648IR1HCn0306090120IR2HCn0.00000.10770.08710.07040.0586IRFHCn012243648IRFWHCn0.00000.05880.04400.03430.0280IRAHCn0.00000.20160.40320.60480.8064IRBHCn0.00001.21202.42403.63604.8480IRCHCn0.00000.07980.06650.05440.0456IRDIFHCn0.00009.999619.999229.998839.9984IRLHCn0.00004.86609.732014.598019.4640IRLUHCn06121824IRLFHCn0.00004.89909.798014.697019.5960IRLAHCn0.00004.80009.600014.400019.2000IRD1HCn0.00008.317216.634424.951633.2688IRGAHCn0.00000.24480.24960.73440.9792


## 4. Graphical Comparison

In this section, we will compare the results of irregularity indices of certain networks in graphical form. Different colors have been used to represent the behavior of irregularity indices for certain networks in the form of graphical lines. And these graphs have been generated by putting the values ofn along the X-axis in the results of irregularity indices along the Y-axis. Tables 1–5 represent such values of results which are obtained after putting values of n, and these values demonstrated in the tables will help us to generate graphical results.In Figures6–21, five different colors of graphical lines have been used. Red color represents irregular behavior in the silicate network SLn, and blue color represents irregular behavior in the chain silicate network CSn. Similarly, other colors which are green, purple, and black represent irregular behavior in the oxide network OXn, hexagonal network HXn, and honeycomb network HCn, respectively. In Figures 6–21, values of n are plotted along the X-axis and behavior of all selected irregularity indices along the Y-axis.Figure 6
VAR ofSLn,CSn, OXn, HXn, and HCn.Figure 7
AL ofSLn,CSn,OXn,HXn, and HCn.Figure 8
IR1 ofSLn,CSn,OXn,HXn, and HCn.Figure 9
IR2 ofSLn,CSn,OXn,HXn, and HCn.Figure 10
IRF ofSLn,CSn,OXn,HXn, and HCn.Figure 11
IRFW ofSLn,CSn,OXn,HXn, and HCn.Figure 12
IRA ofSLn,CSn,OXn,HXn, and HCn.Figure 13
IRB ofSLn,CSn,OXn,HXn, and HCn.Figure 14
IRC ofSLn,CSn,OXn,HXn, and HCn.Figure 15
IRDIF ofSLn,CSn,OXn,HXn, and HCn.Figure 16
IRL ofSLn,CSn,OXn,HXn, and HCn.Figure 17
IRUL ofSLn,CSn,OXn,HXn, and HCn.Figure 18
IRLF ofSLn,CSn,OXn,HXn, and HCn.Figure 19
IRLA ofSLn,CSn,OXn,HXn, and HCn.Figure 20
IRD1 ofSLn,CSn,OXn,HXn, and HCn.Figure 21
IRGA ofSLn,CSn,OXn,HXn, and HCn.


## 5. Concluding Remarks

In this paper, we have computed irregularity indices of certain networks such as silicate network, chain silicate network, oxide network, hexagonal network, and honeycomb network. These results are valuable and helpful to understand deep irregular behavior of certain networks. These results are also useful for researchers to understand how these networks can be constructed with different irregular properties [16–20].



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