Chemical graph theory is a branch of mathematics which combines graph theory and chemistry. Chemical reaction network theory is a territory of applied mathematics that endeavors to display the conduct of genuine compound frameworks. It pulled the research community due to its applications in theoretical and organic chemistry since 1960. Additionally, it also increases the interest the mathematicians due to the interesting mathematical structures and problems are involved. The structure of an interconnection network can be represented by a graph. In the network, vertices represent the processor nodes and edges represent the links between the processor nodes. Graph invariants play a vital feature in graph theory and distinguish the structural properties of graphs and networks. In this paper, we determined the newly introduced topological indices namely, first ve -degree Zagreb α index, first ve -degree Zagreb β index, second ve -degree Zagreb index, ve -degree Randic index, ve -degree atom-bond connectivity index, ve -degree geometric-arithmetic index, ve -degree harmonic index and ve -degree sum-connectivity index for honey comb derived network. In the analysis of the quantitative structure property relationships (QSPRs) and the quantitative structureactivity relationships (QSARs), graph invariants are important tools to approximate and predicate the properties of the biological and chemical compounds. Also, we give the numerical and graphical representation of our outcomes.

Honey comb derived network ev-degree topological indices
Introduction

A structural molecular diagram is a basic diagram in the study of structural chemical graph theory where atoms are spoken to by nodes and chemical bonds are spoken to by lines. A diagram is associated if there is an association between any pair of nodes. A network is an associated diagram that has no various lines between two nodes and loop. The number of nodes which are associated with a fixed node v is known as the degree of v and is denoted by dv . The collection of all the adjacent nodes to the node v is referred to open neighborhood of v and can be represented by N(v) . The open neighborhood became the closed neighborhood when we include the node v in the collection and is represented by N[v] . The shortest distance between two vertices u,v ∈V(G) is denoted by d(u,v), and the maximum value of d(u,v) in G is called the diameter of G, denoted as diam(G). For basic definition, see West .

The relation between the (QSPR) and (QSAR) predict the properties and natural exercises of unstudied material. In these materials, the topological indices and some physico-chemical properties are utilized to anticipate bioactivity for chemical compounds . A number represents a topological index in a diagram of a chemical compound, which can be utilized to portray the underlined chemical compound and help to foresee its physio-chemical properties. In 1947 Weiner established the framework of topological index. He was approximated the breaking point of alkanes and presented the Weiner index . In 1975, Milan Randic presented Randic index . In 1998, Bollobas et al.  and Amic et al.  proposed the general Randic index and has been concentrated by both scientist and mathematicians . The Randic index is one of the most important and generally considered and applied topological index. Numerous surveys, papers and books  are composed on graph invariant. For detail of different topological indices, see . Chellali et al.  introduced two novel degree thoughts which they called “ ve -degree and ev -degree”. Horoldagva et al.  contributed to the study related to “ ve -degree and ev -degree”. The new style degree base indices have been applied to already existing indices and found the better results in . It has been found that the ve -degree Zagreb index has more grounded estimate power than the old-style Zagreb index.

The <inline-formula id="ieqn-26"> <!--<alternatives><inline-graphic xlink:href="ieqn-26.tif"/><tex-math id="tex-ieqn-26"><![CDATA[${{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-26"><mml:mrow><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:mrow></mml:math> <!--</alternatives>--></inline-formula>-degree and <inline-formula id="ieqn-27"> <!--<alternatives><inline-graphic xlink:href="ieqn-27.tif"/><tex-math id="tex-ieqn-27"><![CDATA[${{ev}}$]]></tex-math>--><mml:math id="mml-ieqn-27"><mml:mrow><mml:mrow><mml:mi>e</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:mrow></mml:math> <!--</alternatives>--></inline-formula>-degree Based Topological Indices

Chellali et al.  gave the definition of ev -degree of an edge e=uvE which is denoted by dev(e) , and is the cardinality of nodes of the union of the closed neighborhoods of u and v . The ve -degree of the node vV , denoted by dve(v) , and is the cardinality of lines of different lines that are incident to any node from the closed neighborhood of v . Throughout this paper we consider G is a connected graph, e=uvE(G) and vV . For some basic definitions regarding “ ev -degree and ve -degree topological indices” . The topological indices related to ev -degree are: The ev -degree Zagreb index, ev -degree Randic index, The topological indices related to ev -degree are: The first ve -degree Zagreb α index, first ve -degree Zagreb β index, second ve -degree Zagreb index, ve -degree Randic index, ve -degree atom-bond connectivity index, ve -degree geometric-arithmetic index, ve -degree harmonic index and ve -degree sum-connectivity index.

Main Results

In the present section, we determined our computational results for Honey Comb derived network (see Fig. 1), which is a planar graph. The number of nodes and lines in HcDN1(n) are 9n23n+1 and 27n221n+6 respectively.

<inline-formula id="ieqn-59"> <!--<alternatives><inline-graphic xlink:href="ieqn-59.tif"/><tex-math id="tex-ieqn-59"><![CDATA[$HcDN1(n)$]]></tex-math>--><mml:math id="mml-ieqn-59"><mml:mi>H</mml:mi><mml:mi>c</mml:mi><mml:mi>D</mml:mi><mml:mi>N</mml:mi><mml:mn>1</mml:mn><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> <!--</alternatives>--></inline-formula> network with <inline-formula id="ieqn-60"> <!--<alternatives><inline-graphic xlink:href="ieqn-60.tif"/><tex-math id="tex-ieqn-60"><![CDATA[$n = 4$]]></tex-math>--><mml:math id="mml-ieqn-60"><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math> <!--</alternatives>--></inline-formula>

There are five types of lines in HcDN1(n) based on degrees of end nodes of each line. Tab. 1 shows line partition of HcDN1(n) . Tab. 2 represents the number of nodes corresponding their degrees.

Line partition <inline-formula id="ieqn-63"> <!--<alternatives><inline-graphic xlink:href="ieqn-63.tif"/><tex-math id="tex-ieqn-63"><![CDATA[$HcDN1$]]></tex-math>--><mml:math id="mml-ieqn-63"><mml:mi>H</mml:mi><mml:mi>c</mml:mi><mml:mi>D</mml:mi><mml:mi>N</mml:mi><mml:mn>1</mml:mn></mml:math> <!--</alternatives>--></inline-formula>
(d(u),d(v)) Number of lines
(3,3) 6
(3,5) 12(n1)
(3,6) 6n
(5,6) 18(n1)
(6,6) 27n257n+30
Number of nodes with corresponding degrees
d(u) Number of nodes
3 6n
5 6(n1)
6 9n215n+7

In Tab. 3, We partition the lines, based on ev -degree of the HcDN1 . In Tabs. 4 and 5, we partition the nodes, based on ve -degree of HcDN1 .

Line partition of <inline-formula id="ieqn-86"> <!--<alternatives><inline-graphic xlink:href="ieqn-86.tif"/><tex-math id="tex-ieqn-86"><![CDATA[$HcDN1$]]></tex-math>--><mml:math id="mml-ieqn-86"><mml:mi>H</mml:mi><mml:mi>c</mml:mi><mml:mi>D</mml:mi><mml:mi>N</mml:mi><mml:mn>1</mml:mn></mml:math> <!--</alternatives>--></inline-formula>
Number of lines Degree of its end nodes ev -degrees
6 (3,3) 6
12(n1) (3,5) 8
6n (3,6) 9
18(n1) (5,6) 11
27n257n+30 (6,6) 12
Node partition of <inline-formula id="ieqn-103"> <!--<alternatives><inline-graphic xlink:href="ieqn-103.tif"/><tex-math id="tex-ieqn-103"><![CDATA[$HcDN1$]]></tex-math>--><mml:math id="mml-ieqn-103"><mml:mi>H</mml:mi><mml:mi>c</mml:mi><mml:mi>D</mml:mi><mml:mi>N</mml:mi><mml:mn>1</mml:mn></mml:math> <!--</alternatives>--></inline-formula>
Number of nodes ve -degrees
12 12
6(n2) 14
6(n1) 20
6 22
6(n2) 25
6(n1) 29
9n227n+19 30
The <inline-formula id="ieqn-119"> <!--<alternatives><inline-graphic xlink:href="ieqn-119.tif"/><tex-math id="tex-ieqn-119"><![CDATA[$ve$]]></tex-math>--><mml:math id="mml-ieqn-119"><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:math> <!--</alternatives>--></inline-formula>-degree of the end nodes of lines of <inline-formula id="ieqn-120"> <!--<alternatives><inline-graphic xlink:href="ieqn-120.tif"/><tex-math id="tex-ieqn-120"><![CDATA[$HcDN1$]]></tex-math>--><mml:math id="mml-ieqn-120"><mml:mi>H</mml:mi><mml:mi>c</mml:mi><mml:mi>D</mml:mi><mml:mi>N</mml:mi><mml:mn>1</mml:mn></mml:math> <!--</alternatives>--></inline-formula>
Number of lines ve -degrees of its end nodes
6 (12,12)
12 (12,20)
12 (12,22)
12 (20,22)
12 (22,29)
6 (29,29)
6(3n5) (29,30)
6(n1) (20,29)
12(n2) (14,20)
6(n2) (14,25)
12(n2) (20,25)
12(n2) (25,29)
6(n2) (25,30)
27n293n+78 (30,30)
Computing Indices for <inline-formula id="ieqn-150"> <!--<alternatives><inline-graphic xlink:href="ieqn-150.tif"/><tex-math id="tex-ieqn-150"><![CDATA[${{HcDN1}}$]]></tex-math>--><mml:math id="mml-ieqn-150"><mml:mrow><mml:mrow><mml:mi>H</mml:mi><mml:mi>c</mml:mi><mml:mi>D</mml:mi><mml:mi>N</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:mrow></mml:math> <!--</alternatives>--></inline-formula> Formulas

In this section, we will calculate ev -degree and ve -degree based indices of the different types of indices which are given as under;

ev degree Zagreb Index

Now with the help of Tab. 3, we compute the ev -degree based Zagreb index of HcDN1 as:

Mev(HcDN1)=eE(HcDN1)dev(e)2,

Mev(HcDN1)=6×62+12(n1)×82+6n×92+18(n1)×112+(27n257n+30)×122=216+768n768+486n+2178n2178+3888n28208n+4320=3888n24776n+1590.

• The first ve -degree Zagreb α index

Now with the help of Tab. 4, we compute the first ve -degree Zagreb α index of HcDN1 as:

M1αve(HcDN1)=vV(HcDN1)dve(v)2,

M1αve(HcDN1)=12×122+6(n2)×142+6(n1)×202+6×222+6(n2)×252+6(n1)×292+(9n227n+19)×302=1728+1176n2352+2400n2400+2904+3750n7500+5046n5046+8100n224300+17100+5046n5046+8100n224300+17100=8100n211928n+4434.

• The first ve -degree Zagreb β index

Now with the help of Tab. 5, we compute the first ve -degree Zagreb β index of HcDN1 as:

M1βve(HcDN1)=uvE(HcDN1)(dve(u)+dve(v)),

M1βve(HcDN1)=6×24+12×32+12×34+12×42+12×51+6×58+6(3n5)×59+6(n1)×49+12(n2)×34+6(n2)×39+12(n2)×45+12(n2)×54+6(n2)×55+(27n293n+78)×6=144+384+408+504+612+348+1062n1770+294+408n816+234n468+540n1080+648n1296+330n660+1620n25580n+4680=1620n22064n+696.

• The second ve-degree Zagreb index

Now with the help of Tab. 5, we compute the second ve -degree based Zagreb index of HcDN1 as:

M2ve(HcDN1)=uvE(HcDN1)(dve(u)×dve(v)),

M2ve(HcDN1)=6×144+12×240+12×264+12×440+12×638+6×841+6(3n5)×870+6(n1)×580+12(n2)×280+6(n2)×350+12(n2)×500+12(n2)×725+6(n2)×750+(27n293n+78)×900=864+2880+3168+5280+7656+5046+15660n26100+3480n3480+3360n6720+2100n4200+6000n12000+8700n17400+4500n9000+24300n283700n+70200=24300n239900n+16194.

• The ve -degree Randic index

Now with the help of Tab. 5, we compute the ve -degree Randic index of HcDN1 as:

Rve(HcDN1)=uvE(HcDN1)(dve(u)×dve(v))12,

Rve(HcDN1)=6×14412+12×24012+12×26412+12×44012+12×63812+6×84112+6(3n5)×87012+6(n1)×58012+12(n2)×28012+6(n2)×35012+12(n2)×50012+12(n2)×72512+6(n2)×75012+(27n293n+78)×90012

=612+12415+12266+122110+12638+6841+18870n30870+62145n62145+12270n24270+6514n12514+12520n24520+12529n24529+6530n12530+2730n29330n+7830

=910n2+(18870+3145+670+6514+12520+12529+65303110)n+12+315+666+6110+12638+6841308703145127012514245202452912530+135

=0.9n20.0013n+0.122

• The ev -degree Randic index

Now with the help of Tab. 3, we compute the ev -degree based Randic index of HcDN1 as:

Rev(HcDN1)=eE(HcDN1)dev(e)12,

Rev(HcDN1)=6×612+12(n1)×612+6n×912+18(n1)×1112+(27n257n+30)×1212

=2712n2+(62+2+18115712)n+(6621811+153).

=7.794n24.785+1.44.

• The atom-bond connectivity index

Now with the help of Tab. 5, we compute the atom-bond connectivity index of HcDN1 as:

ABCve(HcDN1)=uvE(HcDN1)dve(u)+dve(v)2dve(u)×dve(v),

ABCve(HcDN1)=6×242144+12×322240+12×342264+12×422440+12×512638+6×582841+6(3n5)×592870+6(n1)×492580+12(n2)×+6(n2)×392350+12(n2)×452500+12(n2)×542725+6(n2)×552750+(27n293n+78)×602900

=62212+1230415+482266+24102110+84638+1214841+1857870n3057870+6472145n6472145+482270n962270+637514n1237514+1243520n2443520+2413529n4813529+653530n1253530+275830n2935830n+785830

=95810n2+(1857870+347145+2435+63714+124320+2413529+653530315810)n+222+32+2433+1211+84638+12148413057870347214548351237514244352048135291253530+13585.

=6.854n2+18.92n+1.855.

• The geometric-arithmetic index

Now with the help of Tab. 5, we compute the geometric-arithmetic index of HcDN1 as:

GAve(HcDN1)=uvE(HcDN1)2degve(u)×dve(v)dve(u)+dve(v),

GAve(HcDN1)=6×214424+12×224032+12×226434+12×244042+12×263851+6×284158+6(3n5)×287059+6(n1)×258049+12(n2)×228034+6(n2)×235039+12(n2)×250045+12(n2)×272554+6(n2)×275055+(27n293n+78)×290060

=6+315+246617+81107+863817+684129+3687059n6087059+2414549n2414549+247017n487017+201413n401413+8203n16203+20299n40299+123011n243011+27n293n+78

=(3687059+2414549+247017+201413+8203+20299+12301193)n+6+315+246617+81107+863817+684129608705924145494870174014131620340299243011+78+27n2

=27n221.669n+6.195.

• The harmonic index

Now with the help of Tab. 5, we compute the Harmonic index of HcDN1 as:

Hve(HcDN1)=uvE(HcDN1)2dve(u)+dve(v),

Hve(HcDN1)=6×224+12×232+12×234+12×242+12×251+6×258+6(3n5)×259+6(n1)×249+12(n2)×234+6(n2)×239+12(n2)×245+12(n2)×254+6(n2)×255+(27n293n+78)×260

=12+34+1217+47+817+629+3659n6059+12491249+1217n2417+413n813+815n1615+49n89+1255n2455+910n23110n+135

=0.90n20.036n+0.122.

• The sum-connectivity index

Now with the help of Tab. 5, we compute the Sum-connectivity index of HcDN1 as:

χve(HcDN1)=uvE(HcDN1)(dve(u)+dve(v))12,

χve(HcDN1)=6×2412+12×3212+12×3412+12×4212+12×5112+6×5812+6(3n5)×5912+6(n1)×4912+12(n2)×3412+6(n2)×3912+12(n2)×4512+12(n2)×5412+6(n2)×5512+(27n293n+78)×6012

=36+32+1234+1242+1251+658+1859n3059+67n67+1234n2434+639n1239+45n85+46n86+655n1255+27215n293215n+3915

=27215n2+(1859+67+1234+639+45+46+65593215)n+36+32+1234+1242+1251+6583059672434123985861255+3915

=3.486n21.556n+0.531.

Numerical and Graphical Representation and Discussion

The ve and ev for ten different types od degree base topological descriptors for the HcDN1 are calculated both numerically and graphically. From Fig. 2 it is clearly noted that the behavior of first Zegreb alpha index, first Zagreb beta index and second Zagreb index is almost same in the increasing direction as the value of n increases while ev Zagreb index value has a very rapid increase with the increase value of n . From Fig. 3 it is clearly noted that the behavior of atom bond connectivity index and geometric arithmetic index are almost closely increasing with the increase value of n while ev Randic index value has a very rapid increase with the increase value of n . The numerical representation of HcDN1 is shown in Tabs. 68. The graphical representation of HcDN1 are shown in Figs. 24.

Numerical comparison of <inline-formula id="ieqn-189"> <!--<alternatives><inline-graphic xlink:href="ieqn-189.tif"/><tex-math id="tex-ieqn-189"><![CDATA[${M^{ev}},M_1^{\alpha ve},M_1^{\beta ve}$]]></tex-math>--><mml:math id="mml-ieqn-189"><mml:mrow><mml:msup><mml:mi>M</mml:mi><mml:mrow><mml:mi>e</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo>,</mml:mo><mml:msubsup><mml:mi>M</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mi>α</mml:mi><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msubsup><mml:mo>,</mml:mo><mml:msubsup><mml:mi>M</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mi>β</mml:mi><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msubsup></mml:math> <!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-190"> <!--<alternatives><inline-graphic xlink:href="ieqn-190.tif"/><tex-math id="tex-ieqn-190"><![CDATA[$M_2^{ve}$]]></tex-math>--><mml:math id="mml-ieqn-190"><mml:msubsup><mml:mi>M</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msubsup></mml:math> <!--</alternatives>--></inline-formula>
n Mev M1αve M1βve M2ve
1 702 606 252 594
2 7590 12978 3048 33594
3 22254 41550 9084 115194
4 44694 86322 18360 245394
5 74910 147294 30876 424194
6 112902 224466 46632 651594
7 158670 317838 65628 927594
8 212214 427410 87864 1252194
9 273534 553182 113340 1625394
10 342630 695154 142056 2047194
Graphical comparison of <inline-formula id="ieqn-245"> <!--<alternatives><inline-graphic xlink:href="ieqn-245.tif"/><tex-math id="tex-ieqn-245"><![CDATA[${M^{ev}}$]]></tex-math>--><mml:math id="mml-ieqn-245"><mml:mrow><mml:msup><mml:mi>M</mml:mi><mml:mrow><mml:mi>e</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-246"> <!--<alternatives><inline-graphic xlink:href="ieqn-246.tif"/><tex-math id="tex-ieqn-246"><![CDATA[$M_1^{\alpha ve}$]]></tex-math>--><mml:math id="mml-ieqn-246"><mml:msubsup><mml:mi>M</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mi>α</mml:mi><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msubsup></mml:math> <!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-247"> <!--<alternatives><inline-graphic xlink:href="ieqn-247.tif"/><tex-math id="tex-ieqn-247"><![CDATA[$M_1^{\beta ve}$]]></tex-math>--><mml:math id="mml-ieqn-247"><mml:msubsup><mml:mi>M</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mi>β</mml:mi><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msubsup></mml:math> <!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-248"> <!--<alternatives><inline-graphic xlink:href="ieqn-248.tif"/><tex-math id="tex-ieqn-248"><![CDATA[$M_2^{ve}$]]></tex-math>--><mml:math id="mml-ieqn-248"><mml:msubsup><mml:mi>M</mml:mi><mml:mn>2</mml:mn><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msubsup></mml:math> <!--</alternatives>--></inline-formula> Numerical comparison of <inline-formula id="ieqn-249"> <!--<alternatives><inline-graphic xlink:href="ieqn-249.tif"/><tex-math id="tex-ieqn-249"><![CDATA[${R^{ev}}$]]></tex-math>--><mml:math id="mml-ieqn-249"><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mrow><mml:mi>e</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-250"> <!--<alternatives><inline-graphic xlink:href="ieqn-250.tif"/><tex-math id="tex-ieqn-250"><![CDATA[$AB{C^{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-250"><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-251"> <!--<alternatives><inline-graphic xlink:href="ieqn-251.tif"/><tex-math id="tex-ieqn-251"><![CDATA[$G{A^{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-251"><mml:mi>G</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula>
n Rev ABCve GAve
1 4.449 27.63 11.526
2 23.046 67.113 70.857
3 57.231 120.304 184.188
4 107.004 187.203 351.519
5 172.365 267.81 572.85
6 253.314 362.125 848.181
7 349.851 470.148 1177.512
8 461.976 591.879 1560.843
9 589.689 727.318 1998.174
10 732.99 876.465 2489.505
Graphical comparison of <inline-formula id="ieqn-295"> <!--<alternatives><inline-graphic xlink:href="ieqn-295.tif"/><tex-math id="tex-ieqn-295"><![CDATA[${R^{ev}}$]]></tex-math>--><mml:math id="mml-ieqn-295"><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mrow><mml:mi>e</mml:mi><mml:mi>v</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-296"> <!--<alternatives><inline-graphic xlink:href="ieqn-296.tif"/><tex-math id="tex-ieqn-296"><![CDATA[$AB{C^{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-296"><mml:mi>A</mml:mi><mml:mi>B</mml:mi><mml:mrow><mml:msup><mml:mi>C</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-297"> <!--<alternatives><inline-graphic xlink:href="ieqn-297.tif"/><tex-math id="tex-ieqn-297"><![CDATA[$G{A^{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-297"><mml:mi>G</mml:mi><mml:mrow><mml:msup><mml:mi>A</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula> Numerical comparison of <inline-formula id="ieqn-298"> <!--<alternatives><inline-graphic xlink:href="ieqn-298.tif"/><tex-math id="tex-ieqn-298"><![CDATA[${R^{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-298"><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-299"> <!--<alternatives><inline-graphic xlink:href="ieqn-299.tif"/><tex-math id="tex-ieqn-299"><![CDATA[${H^{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-299"><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-300"> <!--<alternatives><inline-graphic xlink:href="ieqn-300.tif"/><tex-math id="tex-ieqn-300"><![CDATA[${\chi ^{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-300"><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula>
n Rve Hve χve
1 1.02 0.986 2.461
2 3.719 3.65 11.363
3 8.218 8.114 27.237
4 14.517 14.378 50.083
5 22.615 22.442 79.901
6 32.514 32.306 116.691
7 44.213 43.97 160.453
8 57.116 57.434 211.187
9 73.01 72.698 268.893
10 90.109 89.762 333.571
Graphical comparison of <inline-formula id="ieqn-344"> <!--<alternatives><inline-graphic xlink:href="ieqn-344.tif"/><tex-math id="tex-ieqn-344"><![CDATA[${R^{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-344"><mml:mrow><mml:msup><mml:mi>R</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula>, <inline-formula id="ieqn-345"> <!--<alternatives><inline-graphic xlink:href="ieqn-345.tif"/><tex-math id="tex-ieqn-345"><![CDATA[${H^{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-345"><mml:mrow><mml:msup><mml:mi>H</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula> and <inline-formula id="ieqn-346"> <!--<alternatives><inline-graphic xlink:href="ieqn-346.tif"/><tex-math id="tex-ieqn-346"><![CDATA[${\chi ^{ve}}$]]></tex-math>--><mml:math id="mml-ieqn-346"><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mrow><mml:mi>v</mml:mi><mml:mi>e</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> <!--</alternatives>--></inline-formula>
Conclusion

There are many applications of topological descriptors in computer science, networks, agriculture and chemical graph theory etc. These descriptors help in finding the behavior of their structures. We dealt with the honey comb derived network and computed ten different types of topological descriptors which are base on ev and ve degree. We have computed their explicit formulas and then computed their numerical values for different values of n . Further we plotted their graphs for comparison and discussed their behavior. We observe that the values of all descriptors increases with the increase value of n . In the analysis of the quantitative structure property relationships (QSPRs) and the quantitative structureactivity relationships (QSARs), graph invariants are important tools to approximate and predicate the properties of the biological and chemical compounds. In this paper, we study the vertex-edge based topological indices for honey comb derived network.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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