# On Certain Aspects of Topological Indices

Authors: Tanweer Ul Islam; Zeeshan Saleem Mufti; Aqsa Ameen; Muhammad Nauman Aslam; Ali Tabraiz
Journal: Journal of Mathematics
 (2021)

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## Abstract

A topological index, also known as connectivity index, is a molecular structure descriptor calculated from a molecular graph of a chemical compound which characterizes its topology. Various topological indices are categorized based on their degree, distance, and spectrum. In this study, we calculated and analyzed the degree-based topological indices such as first general Zagreb indexMrG, geometric arithmetic index GAG, harmonic index HG, general version of harmonic index HrG, sum connectivity index λG, general sum connectivity index λrG, forgotten topological index FG, and many more for the Robertson apex graph. Additionally, we calculated the newly developed topological indices such as the AG2G and Sanskruti index for the Robertson apex graph G.


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## Body

## 1. Introduction

The Robertson apex graph (see Figure1) was introduced by Neil Robertson. Robertson’s apex graph is the graph which has 15-vertex set and total number of edges is 32. In this graph, each vertex has the degree 4 except one vertex which has degree 6. The Robertson apex graph is formed by adding an additional apex vertex in the (planar) rhombic dodecahedral graph which is associated to all degree 3 vertex or by contracting two opposite vertices of the tesseract graph. The apex graph discussed here is in the form of m,n unit, where “m” represents the number of rows and “n” represents the number of columns. First, we have found the degree-based topological indices of 1,1 unit, 1,2 unit, and so on, and finally, we have generated the formulas for the degree-based topological indices of m,n unit.Figure 1
Robertson apex graph.A graph is a planar graph which can be drawn without any edge crossing. In an apex graph, the removal of a vertex gives a planar graph. Sometimes an apex graph is also called a virtually planar graph and the apices of the graph are the set of vertices whose removal reduces the graph planar. Apex graph differs from the critical nonplanar graph in which an apex graph requires only that their exists at least one vertex whose removal gives a planar graph although a critical nonplanar graph requires that removal of every vertex make a planar graph.A graph can be recognized by a polynomial, a matrix, a sequence of numbers, or a numeric number which represents the whole graph, and these representations are designed to be uniquely defined for that graph [1–4]. A topological index, which is a graph invariant that does not depend on the classification or graphic demonstration of the graph, is a numerical parameter mathematically derived from the graph. The topological indices of molecular graphs are generally used for establishing correlations between the arrangement of a molecular compound and its physico-chemical properties or biological activity. Topological indices are major tools for analyzing many physico-chemical properties of molecules without performing any testing. Some most significant types of topological indices of graphs are distance-based topological indices, degree-based topological indices, and spectrum-based topological indices. One of the most investigated categories of topological indices used in mathematical chemistry is called degree-based topological indices, which are defined in terms of the degrees of the vertices of a graph. Thus, we can write the definition of such a topological index in the form given as(1)TIG=∑pq∈GFdp,dq,where G = (V(G), E(G)) is a simple, undirected, and connected graph and d(u) denotes the degree of the vertex u. Topological indices also called molecular descriptors are presented to explain the physiochemical properties of molecules [5–7]. For more details on topological indices, see [8–12].Gutman and Das proposed the first general Zagreb index in [13]:(2)MrG=∑p∈VGdpr.Yuan et al. in 2009 defined geometric-arithmetic (GA) index and compared GA index by the well-known Randic index [14]:(3)GAG=∑pq∈EG2dp⋅dqdp+dq.Zhong [15] introduced the harmonic index which was described by(4)HG=∑pq∈EG2dp+dq.Yan et al. [16] introduced the general version of harmonic index which was defined by(5)HrG=∑pq∈EG2dp+dqr.In 2009, Zhou et al. [17] defined the sum-connectivity index λ which was formulated by(6)λG=∑pq∈EG1dp+dq.Few years ago, Yan et al. [16] introduced the general sum connectivity index as follows:(7)λrG=∑pq∈EGdp+dqr.In 2015, Gao et al. [18] reinvestigated the forgotten topological index or (F-index) and it is denoted by F(G):(8)FG=∑pq∈EGdp2+dq2.Arithmetic-geometric index [19] is defined as(9)AG1G=∑pq∈EGdp+dq2dp⋅dq.Multiplicative geometric arithmetic index [20] is stated as follows:(10)GAIIG=∏pq∈EG2dp⋅dqdp+dq.Multiplicative sum connectivity index [21] is defined as(11)SCIIG=∑pq∈EG1dp+dq.TheAG2 index [19] is defined as(12)AG2G=∑pq∈EGSp+Sq2Sp⋅Sq.In 2016, Hosamani [22] formulated the Sanskruti index S(G) of a general graph G. It is defined as(13)SG=∑pq∈EGSp⋅SqSp+Sq−23.


## 2. Results and Discussion

Theorem 1.
LetG be a graph with m≥3 and n≥3. Then,(14)MrG=2×4r3mn+9n+9m+22+5×6rmn+n+m+1+8r2mn+4n+3m+6+2×9rm+1+10rmn+n.Proof.
LetG be a graph and r is a real number. Then, by using Table 1, we have(15)MrG=2×4r3mn+9n+9m+22+5×6rmn+n+m+1+8r2mn+4n+3m+6+2×9rm+1+10rmn+n.Table 1
The vertex partition of graphG based on degree of vertices.
Degree of vertexNo. of vertices46mn + 18n + 18m + 4465mn + 5n + 5m + 582mn + 4n + 3m + 692m + 210mn + nTotal40mn + 72m + 72n + 136Theorem 2.
LetG be a graph with m≥3 edges and n≥3 vertices.(1)
(16)GAG=10mn+26n+48213m+100213+256×4mn+4n+4m+4+232×8mn+24n+16m+40+2710×4mn+4n+473×8mn+8n+8m+8+256×6m+6+1415×6mn+6n.(2)
(17)AG1G=10mn+26n+39m+79+51264mn+4n+4m+4+3428mn+24n+16m+40+720104mn+4n+71238mn+8n+8m+8+51266m+6+415156mn+6n.(3)
(18)HG=1857140n+2771420mn+372041365+180941365m.(4)
(19)HrG=14r×4mn+20n+20m+60+15r×4mn+4n+4m+4+16r×8mn+24n+16m+40+213r×12m+12+17r×4mn+4n+16r×6mn+6n+6m+6+17r×8mn+8n+8m+8+215r×6m+6+18r×6mn+6n.(5)
(20)λG=142×4mn+20n+20m+60+11010×4mn+4n+4m+4+163×8mn+24n+16m+40+11313×12m+12+11414×4mn+4n+1636mn+6n+6m+6+11414×8mn+8n+8m+8+11515×6m+6+32mn+32n.(6)
(21)λrG=8r4mn+20n+20m+60+10r4mn+4n+4m+4+12r8mn+24n+16m+40+13r12m+12+14r4mn+4n+12r6mn+6n+6m+6+14r8mn+8n+8m+8+15r6m+6+16r6mn+6n.(7)
(22)FG=8426+5280n+3488mn+5226m.Proof.
(1) The geometric aritheoremetic index is defined as(23)GAG=∑pq∈EG2dp×dqdp+dq.
This implies that(24)GAG=E4,4×24×44+4+E4,6×24×64+6+E4,8×24×84+8+E4,9×24×94+9+E4,10×24×104+10+E6,6×26×66+6+E6,8×26×86+8+E6,9×26×96+9+E6,10×26×106+10.
By using Table2, we get(25)GAG=4mn+20n+20m+60×24×44+4+4mn+4n+4m+4×24×64+6+8mn+24n+16m+40×24×84+8+12m+12×24×94+9+4mn+4n×24×104+10+6mn+6m+6n+6×26×66+6+8mn+8n+8m+8×26×86+8+6m+6×26×96+9+6mn+6n×26×106+10.
After simplification, we obtain(26)GAG=10mn+26n+48213m+100213+256×4mn+4n+4m+4+232×8mn+24n+16m+40+2710×4mn+4n+473×8mn+8n+8m+8+256×6m+6+1415×6mn+6n.(2) The aritheoremetic-geometric index is defined as(27)AG1G=∑pq∈EGdp+dq2dp×dq.
This implies that(28)AG1G=E4,4×4+424×4+E4,6×4+624×6+E4,8×4+824×8+E4,9×4+924×9+E4,10×4+1024×10+E6,6×6+626×6+E6,8×6+826×8+E6,9×6+926×9+E6,10×6+1026×10.
By using Table2, we obtain(29)AG1G=4mn+20n+20m+60×4+424×4+4mn+4n+4m+4×4+624×6+8mn+24n+16m+40×4+824×8+12m+12×4+924×9+4mn+4n×4+1024×10+6mn+6m+6n+6×6+626×6+8mn+8n+8m+8×6+826×8+6m+6×6+926×9+6mn+6n×6+1026×10.
After simplification, we obtain(30)AG1G=10mn+26n+39m+79+51264mn+4n+4m+4+3428mn+24n+16m+40+720104mn+4n+71238mn+8n+8m+8+51266m+6+415156mn+6n.(3) The harmonic index is defined as(31)HG=∑pq∈EG2dp+dq.
This implies that(32)HG=E4,4×24+4+E4,6×24+6+E4,8×24+8+E4,9×24+9+E4,10×24+10+E6,6×26+6+E6,8×26+8+E6,9×26+9+E6,10×26+10.
By using Table2, we obtain(33)HG=4mn+20n+20m+60×24+4+4mn+4n+4m+4×+24+6+8mn+24n+16m+40×24+8+12m+12×24+9+4mn+4n×24+10+6mn+6m+6n+6×26+6+8mn+8n+8m+8×26+8+6m+6×26+9+6mn+6n×26+10.
After simplification, we obtain(34)HG=1857140n+2771420mn+372041365+180941365m.
Hence, it is proved.(4) The general harmonic index is defined as(35)HrG=∑pq∈EG2dp+dqr.
This implies that(36)HrG=E4,4×24+4r+E4,6×24+6r+E4,8×24+8r+E4,9×24+9r+E4,10×24+10r+E6,6×26+6r+E6,8×26+8r+E6,9×26+9r+E6,10×26+10r.
By using Table2, we obtain(37)HrG=4mn+20n+20m+60×24+4r+4mn+4n+4m+4×+24+6r+8mn+24n+16m+40×24+8r+12m+12×24+9r+4mn+4n×24+10r+6mn+6m+6n+6×26+6r+8mn+8n+8m+8×26+8r+6m+6×26+9r+6mn+6n×26+10r.
After simplification, we obtain(38)HrG=14r×4mn+20n+20m+60+15r×4mn+4n+4m+4+16r×8mn+24n+16m+40+213r×12m+12+17r×4mn+4n+16r×6mn+6n+6m+6+17r×8mn+8n+8m+8+215r×6m+6+18r×6mn+6n.
Hence, it is proved.(5) The sum-connectivity index is defined as(39)λG=∑pq∈EG1dp+dq.
This implies that(40)λG=E4,4×14+4+E4,6×14+6+E4,8×14+8+E4,9×14+9+E4,10×14+10+E6,6×16+6+E6,8×16+8+E6,9×16+9+E6,10×16+10.
By using Table2, we obtain(41)λG=4mn+20n+20m+60×14+4+4mn+4n+4m+4×14+6+8mn+24n+16m+40×14+8+12m+12×14+9+4mn+4n×14+10+6mn+6m+6n+6×16+6+8mn+8n+8m+8×16+8+6m+6×16+9+6mn+6n×16+10.
After simplification, we obtain(42)λG=142×4mn+20n+20m+60+11010×4mn+4n+4m+4+163×8mn+24n+16m+40+11313×12m+12+11414×4mn+4n+1636mn+6n+6m+6+11414×8mn+8n+8m+8+11515×6m+6+32mn+32n,which completes the proof.(6) The general sum-connectivity index is defined as(43)λrG=∑pq∈EGdp+dqr.
This implies that(44)λrG=E4,4×4+4r+E4,6×4+6r+E4,8×4+8r+E4,9×4+9r+E4,10×4+10r+E6,6×6+6r+E6,8×6+8r+E6,9×6+9r+E6,10×6+10r.
By using Table2, we obtain(45)λrG=4mn+20n+20m+60×4+4r+4mn+4n+4m+4×4+6r+8mn+24n+16m+40×4+8r+12m+12×4+9r+4mn+4n×4+10r+6mn+6m+6n+6×6+6r+8mn+8n+8m+8×6+8r+6m+6×6+9r+6mn+6n×6+10r.
After simplification, we obtain(46)λrG=8r4mn+20n+20m+60+10r4mn+4n+4m+4+12r8mn+24n+16m+40+13r12m+12+14r4mn+4n+12r6mn+6n+6m+6+14r8mn+8n+8m+8+15r6m+6+16r6mn+6n.
Hence, it is proved.(7) TheF-index is defined as(47)FG=∑pq∈EGdp2+dq2.
This implies that(48)FG=E4,4×42+42+E4,6×42+d62+E4,8×42+82+E4,9×42+92+E4,10×42+102+E6,6×62+62+E6,8×62+82+E6,9×62+92+E6,10×62+102.
By using Table2, we obtain(49)FG=4mn+20n+20m+60×42+42+4mn+4n+4m+4×42+62+8mn+24n+16m+40×42+82+12m+12×42+92+4mn+4n×42+102+6mn+6m+6n+6×62+62+8mn+8n+8m+8×62+82+6m+6×62+92+6mn+6n×62+102.
After simplification, we obtain(50)FG=8426+5280n+3488mn+5226m.Table 2
The vertex division of degree-based topological indices of (m, n) Robertson apex graph.
dp,dq,wherepq∈EGNo. of edges4,44mn + 20n + 20m + 604,64mn + 4n + 4m + 44,88mn + 24n + 16m + 404,912m + 124,104mn + 4n6,66mn + 6m + 6n + 66,88mn + 8n + 8m + 86,96m + 66,106mn + 6nTotal40mn + 72m + 72n + 136Theorem 3.
LetG be a graph with m≥3 edges and n≥3 vertices.(1)
(51)GAIIG=2+2564mn+4n+4m+4+2328mn+24n+16m+40+121312m+12+27104mn+4n+4738mn+8n+8m+8+2566m+6+14156mn+6n.(2)
(52)SCIIG=1424mn+20n+20m+60+110104mn+4n+4m+4+1638mn+24n+16m+40+1131312m+12+114144mn+4n+1636mn+6n+6m+6+114148mn+8n+8m+8+115156m+6+146mn+6n.Proof.
(1) Multiplicative geometric aritheoremetic index is defined as(53)GAIIG=∏pq∈EG2dp×dqdp+dq.
This implies that(54)GAIIG=∏pq∈E1G2dp×dqdp+dq×∏pq∈E2G2dp×dqdp+dq×∏pq∈E3G2dp×dqdp+dq×∏pq∈E4G2dp×dqdp+dq×∏pq∈E5G2dp×dqdp+dq×∏pq∈E6G2dp×dqdp+dq×∏pq∈E7G2dp×dqdp+dq×∏pq∈E8G2dp×dqdp+dq×∏pq∈E9G2dp×dqdp+dq,GAIIG=2dp×dqdp+dqE1G×2dp×dqdp+dqE2G×2dp×dqdp+dqE3G×2dp×dqdp+dqE4G×2dp×dqdp+dqE5G×2dp×dqdp+dqE6G×2dp×dqdp+dqE7G×2dp×dqdp+dqE8G×2dp×dqdp+dqE9G.
By using Table2, we obtain(55)GAIIG=24×44+44mn+20n+20m+60×24×64+64mn+4n+4m+4×24×84+88mn+24n+16m+40×24×94+912m+12×24×104+104mn+4n×26×66+66mn+6m+6n+6×26×86+88mn+8n+8m+8×26×96+96m+6×26×106+106mn+6n.
After simplification, we obtained that(56)GAIIG=2+2564mn+4n+4m+4+2328mn+24n+16m+40+121312m+12+27104mn+4n+4738mn+8n+8m+8+2566m+6+14156mn+6n.(2) Multiplicative sum connectivity index is defined as(57)SCIIG=∏pq∈EG1dp+dq.
This implies that(58)SCIIG=∏pq∈E1G1dp+dq×∏pq∈E2G1dp+dq×∏pq∈E3G1dp+dq×∏pq∈E4G1dp+dq×∏pq∈E5G1dp+dq×∏pq∈E6G1dp+dq×∏pq∈E7G1dp+dq×∏pq∈E8G1dp+dq×∏pq∈E9G1dp+dq,SCIIG=1dp+dqE1G×1dp+dqE2G×1dp+dqE3G×1dp+dqE4G×1dp+dqE5G×1dp+dqE6G×1dp+dqE7G|×1dp+dqE8G×1dp+dqE9G.
By using Table2, we obtain(59)SCIIG=14+44mn+20n+20m+60×14+64mn+4n+4m+4×14+88mn+24n+16m+40×14+912m+12×14+104mn+4n×16+66mn+6m+6n+6×16+88mn+8n+8m+8×16+96m+6×16+106mn+6n.
After simplification, we obtained that(60)SCIIG=1424mn+20n+20m+60+110104mn+4n+4m+4+1638mn+24n+16m+40+1131312m+12+114144mn+4n+1636mn+6n+6m+6+114148mn+8n+8m+8+115156m+6+146mn+6n.Theorem 4.
(1)
(61)AG2G=2n+2mn+674208m+842+81410m410+82105mn105+8334448m+81722+4217634m+41763+756971394+63493986+1188534+69290m290+7124364m+41218+2215n30+143mn3+417804mn+4n390+191510+5128n+126+8536126m+61806+439244mn+4n−4m−4462+4710924mn+4n546+9522362m+2559+121432mn+2n−2m−2143+2021n21+95m5+61205m205+31105mn210+51216n+86+10310+95+43152+53mn6+5912382+5485170+585151+2656964m+429+1615n15+55330m774924m+441+2621n42+7128n+43.(2)
(62)SG=362506.41n+303544.76mn+7812.50mn+339240.75m+429700.18.Proof.
(1) TheAG2G index is defined as(63)AG2G=∑pq∈EGdp+dq2dp×dq.
This implies that(64)AG2G=E16,20×16+20216×20+E16,24×16+24216×24+E16,25×16+25216×25+E16,29×16+29216×29+E18,20×18+20218×20+E18,24×18+24218×24+E18,25×18+25218×25+E18,30×18+30218×30+E18,41×18+41218×41+E18,42×18+42218×42+E20,25×20+25220×25+E20,30×20+30220×30+E20,34×20+34220×34+E20,41×20+41220×41+E20,42×20+42220×42+E24,34×24+34224×34+E24,36×24+36224×36+E25,34×25+34225×34+E25,36×25+36225×36+E25,42×25+42225×42+E29,34×29+34229×34+E29,36×29+36229×36+E29,40×29+40229×40+E29,42×29+42229×42+E30,36×30+36230×36+E30,40×30+40230×40+E30,52×30+52230×52+E34,41×34+41234×41+E36,41×36+41236×41+E36,42×36+42236×42+E40,41×40+41240×41+E40,42×40+42240×42+E41,42×41+42241×42+E41,43×41+43241×43+E42,42×42+42242×42+E42,43×42+43242×43+E42,44×42+44242×44+E42,52×42+52242×52+E43,52×43+52243×52+E44,52×44+52244×52.
By using Table3, we obtain(65)AG2G=20×16+20216×20+8n+12×16+24216×24+12m+12×16+25216×25+4m+4×16+29216×29+4×18+20218×20+8n+4×18+24218×24+4×18+25218×25+4n×18+30218×30+4×18+41218×41+4n×18+42218×42+4m×20+25220×25+4mn×20+30220×30+8×20+34220×34+4m×20+41220×41+4mn×20+42220×42+8×24+34224×34+16n+8×24+36224×36+8×25+34225×34+8m×25+36225×36+8m+8×25+42225×42+4×29+34229×34+4m+4×29+36229×36+4m×29+40229×40+4m+4×29+42229×42+8n×30+36230×36+8mn×30+40230×40+4mn+4n×30+52230×52+4×34+41234×41+4m+4×36+41236×41+8n×36+42236×42+4m×40+41240×41+8mn×40+42240×42+8m+8×41+42241×42+4m+4×41+43241×43+2mn+2n−2m−2×42+42242×42+6m+6×42+43242×43+4mn+4n−4m−4×42+44242×44+4mn+4n×42+52242×52+2m+2×43+52243×52+2mn+2n−2m−2×44+52244×52.
After simplification, we obtain that(66)AG2G=2n+2mn+674208m+842+81410m410+82105mn105+8334448m+81722+4217634m+41763+756971394+63493986+1188534+69290m290+7124364m+41218+2215n30+143mn3+417804mn+4n390+191510+5128n+126+8536126m+61806+439244mn+4n−4m−4462+4710924mn+4n546+9522362m+2559+121432mn+2n−2m−2143+2021n21+95m5+61205m205+31105mn210+51216n+86+10310+95+43152+53mn6+5912382+5485170+585151+2656964m+429+1615n15+55330m774924m+441+2621n42+7128n+43.(2) The Sanskruti indexSG of a molecular graph G is defined as(67)SG=∑pq∈EGSp⋅SqSp+Sq−23.
This implies that(68)SG=E16,20×16.2016+20−23+E16,24×16.2416+24−23+E16,25×16.2516+25−23+E16,29×16.2916+29−23+E18,20×18.2018+20−23+E18,24×18.2418+24−23+E18,25×18.2518+25−23+E18,30×18.3018+30−23+E18,41×18.4118+41−23+E18,42×18.4218+42−23+E20,25×20.2520+25−23+E20,30×20.3020+30−23+E20,34×20.3420+34−23+E20,41×20.4120+41−23E20,42×20.4220+42−23+E24,34×24.3424+34−23+E24,36×24.3624+36−23+E25,34×25.3425+34−23+E25,36×25.3625+36−23+E25,42×25.4225+42−23+E29,34×29.3429+34−23+E29,36×29.3629+36−23+E29,40×29.4029+40−23+E29,42×29.4229+42−23+E30,36×30.3630+36−23+E30,40×30.4030+40−23+E30,52×30.5230+52−23+E34,41×34.4134+41−23+E36,41×36.4136+41−23+E36,42×36.4236+42−23+E40,41×40.4140+41−23+E40,42×40.4240+42−23+E41,42×41.4241+42−23+E41,43×41.4341+43−23+E42,42×42.4242+42−23+E42,43×42.4342+43−23+E42,44×42.4442+44−23+E42,52×42.5242+52−23+E43,52×43.5243+52−23+E44,52×44.5244+52−23.
By using Table3, we obtain(69)SG=20×16.2016+20−23+8n+12×16.2416+24−23+12m+12×16.2516+25−23+4m+4×16.2916+29−23+4×18.2018+20−23+8n+4×18.2418+24−23+4×18.2518+25−23+4n×18.3018+30−23+4×18.4118+41−23+4n×18.4218+42−23+4m×20.2520+25−23+4mn×20.3020+30−23+8×20.3420+34−23+4m×20.4120+41−23+4mn×20.4220+42−23+8×24.3424+34−23+16n+8×24.3624+36−23+8×25.3425+34−23+8m×25.3625+36−23+8m+8×25.4225+42−23+4×29.3429+34−23+4m+4×29.3629+36−23+4m×29.4029+40−23+4m+4×29.4229+42−23+8n×30.3630+36−23+8mn×30.4030+40−23+4mn+4n×30.5230+52−23+4×34.4134+41−23+4m+4×36.4136+41−23+8n×36.4236+42−23+4m×40.4140+41−23+8mn×40.4240+42−23+8m+8×41.4241+42−23+4m+4×41.4341+43−23+2mn+2n−2m−2×42.4242+42−23+6m+6×42.4342+43−23+4mn+4n−4m−4×42.4442+44−23+4mn+4n×42.5242+52−23+2m+2×43.5243+52−23+2mn+2n−2m−2×44.5244+52−23.
After simplification, we obtain that(70)SG=362506.41n+303544.76mn+7812.50mn+339240.75m+429700.18.Table 3
The line division of neighbors of (m, n) Robertson apex graph.
Sp,Sq,wherepq∈EGNo. of edgesSp,Sq,wherepq∈EGNo. of edges(16, 20)20(16, 24)8n + 12(16, 25)12m + 12(16, 29)4m + 4(18, 20)4(18, 24)8n + 4(18, 25)4(18, 30)4n(18, 41)4(18, 42)4n(20, 25)4m(20, 30)4mn(20, 34)8(20, 41)4m(20, 42)4mn(24, 34)8(24, 36)16n + 8(25, 34)8(25, 36)8m(25, 42)8m + 8(29, 34)4(29, 36)4m + 4(29, 40)4m(29, 42)4m + 4(30, 36)8n(30, 40)8mn(30, 52)4mn + 4n(34, 41)4(36, 41)4m + 4(36, 42)8n(40, 41)4m(40, 42)8mn(41, 42)8m + 8(41, 43)4m + 4(42, 42)2(mn + n − m − 1)(42, 43)6m + 6(42, 44)4(mn + n − m − 1)(42, 52)4mn + 4n(43, 52)2m + 2(44, 52)2(mn + n − m − 1)Total40mn + 72(m + n) + 136


## 3. Conclusion

In this paper, we deal with the general graph and study their topological indices. We compute the first general Zagreb indexMr. By means of edge dividing approaches, we present the exact expression of several important indices, including geometric arithmetic index GA, Multiplicative geometric arithmetic index GAII, harmonic index, general version of Harmonic index HrG, sum connectivity index λG, general sum connectivity index λrG, forgotten topological index FG, and aritheoremetic geometric index AG1G of abovementioned indices for graph G. We have determined and computed the closed formulas of the AG2G and Sanskruti index SG for the Robertson apex graph. These results are novel and significant contributions in graph theory and network science, and they provide a good basis to understand the topology of this graph. Furthermore, we have computed closed result, and we focus on the theoretical analysis of topological indices expression for this structure.In the future, we are interested in designing some incipient architectures/networks and then studying their topological indices, which will be quite auxiliary to understanding their underlying topology.



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*Source: 9913529-2021-05-03.xml*