# On Computing Techniques for Sombor Index of Some Graphs

Authors: Kiran Naz; Sarfraz Ahmad; Eihab Bashier
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In all types of topological indicators, degree-based indicators play a major role in chemical graph theory. The topological index is a fixed numeric value associated with graph isomerism. Firstly, in 1972, the concept of degree-based index was developed by Gutman and Trinajstic. These degree-based indices are divided into two ways, namely, degree and connection number. These degree-based graph indices are positive-valued for non-regular graphs and zero for regular graphs. In this article, we discussed the degree-based Sombor, reduced Sombor, and average Sombor indices for wheel graph, gear graph, helm graph, flower graph, sunflower graph, and lobster graph.


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## Body

## 1. Introduction

Graph theory has proved to be very useful in modeling key system components with limited components. Image models are used to represent the telephone network, train network, communication problems, traffic network, etc. A graph is a simple way to represent information that connects relationships between objects. Things are represented by nodes and relationships with edges [1].Chemical graph theory is the source of mathematical chemistry that used graph theory in the structure of composite chemical cells. The topological index [2, 3] is part of a chemical graph theory that integrates physicochemical factors such as freezing point, boiling point, melting point, infrared spectrum, electrical parameters, viscosity, and density of substrate chemical graphs.Biological testing of chemical compounds is very expensive. It requires a very large laboratory and advanced equipment to test these compounds. This process is expensive and time consuming. Because of this feature, pharmaceutical companies are keenly interested in finding new ideas or ways to reduce costs. One can reduce costs when there is no need for a laboratory and no need for equipment, but you just need to study a specific chemical structure using topological indicators. Topological indices are of different types such as distance-based topological indices [4], spectrum-based topological indices, and degree-based topological indices.A topological index is a number to describe the graph of a molecule. Topological indices are related to topological distances in a graph or vertex adjacency. Wiener index [5, 6] is the first topological index which is equal to the sum of all distances between the vertices.(1)WG=∑vi,vj⊆VGdvi,vj.Gutman and Trinajestic introduced the first and second Zagreb indices in [7] as(2)M1G=∑vi∈VGdegvi2,M2G=∑vivj∈EGdegvidegvj.The first natural degree-based topological indicator was introduced by Milan Randic. His index was defined as(3)RG=∑vi∼vj1dviGdvjG.LetE be the edge of the graph G, between the vertices vi and vj. Later on, Ernesto Estrada derived a new topological indicator and named it atom bond connectivity index. It is defined as(4)ABCG=∑vi∼vjdviG+dvjG−2dviGdvjG.Furtula et al. derived the modified version of the atom bomb connectivity index, and they named it augmented Zagreb index. It is defined as(5)AZIG=∑vi∼vjdviGdvjGdviG+dvjG−23.The first geometric arithmetic index was proposed by Vukicevic in [8].(6)GAG=∑vi∼vjdviGdvjG1/2dviG+dvjG.In 1980s, Siemion Fajtlowicz introduced a new topological indicator and named it as harmonic index. It is also a degree-based topological indicator.(7)HG=∑vi∼vj2dviG+dvjG,where dviGdvjG and 1/2dviG+dvjG are the geometric and arithmetic means, respectively. The sum connectivity index is a new invention by Nenad Trinajstić and Bo Zhou. It is defined as(8)SCIG=∑vi∼vj1dviG+dvjG,where the product is replaced by sum in Randic index. In this article, we will compute the degree-based indices like SO, SOred, and SOavg. SO, SOred, and SOavg are defined in [9] as(9)SOG=∑ei,j∈EGdegvi2+degvj2,SOredG=∑ei,j∈EGdegvi−12+degvj−12,SOavgG=∑ei,j∈EGdegvi−2mn2+degvj−2mn2.We have discussed wheel graph, gear graph, helm graph, flower graph, sunflower graph, and lobster graph with the help of all these indices. Recently, Sombor index has received a lot of attention within mathematics and chemistry. For example, the chemical function of the Sombor index, especially the ability to speculate and discriminate, has been investigated. The results show that the Sombor index [9] can be used effectively in modeling computer thermodynamic structures and confirming the validity of this new indicator in QSPR analysis.


## 2. Wheel Graph

The wheel graphWn is determined by connecting K1 and Cn. K1 is the graph of order 1 and Cn is the cycle graph as shown in Figure 1. The size of wheel graph [7] is 2n and the order of wheel graph is n+1. Apparently, Wn wheel graph has every node of degree 3 other than the internal node. Internal node has degree n. The wheel graphs are planer graphs.All dual planer graphs are isomorphic to wheel graphs. The chromatic number of wheel graph Wn is 3 if n is odd and 4 if n is even. For n=3, we have calculated three different Sombor indices of W3:(10)SOW3G=618,SOredW3G=122,SOavgW3G=0.Figure 1
Wheel graph.The general representation of the wheel graph for three different Sombor indices is given in Theorems1–3.Theorem 1.
LetWn be a wheel graph of order n+1. Then, the Sombor index of Wn is(11)SOWn=n18+n9+n2.Proof.
SinceWn is a wheel graph of order n+1, where n≥3, by definition,(12)SOWn=∑ei,j∈EWndegvi2+degvj2,SOWn=n18+n9+n2.Theorem 2.
LetWn be a wheel graph of order n+1. Then, the reduced Sombor index of Wn is(13)SOredWn=n8+n4+n−12.Proof.
SinceWn is a wheel graph of order n+1, where n≥3, by definition,(14)SOredWn=∑ei,j∈EWndegvi−12+degvj−12,SOredWn=2n2+nn2−2n+5.Theorem 3.
LetWn be a wheel graph of order n+1. Then, the average Sombor index of Wn is(15)SOavgWn=2n3−4nn+1+n3−4nn+12+n−4nn+12.Proof.
SinceWn is a wheel graph of order n+1, where n≥3, by definition,(16)SOavgWn=∑ei,j∈EWndegvi−2mn2+degvj−2mn2,SOavgWn=2n−n−3n+1+nn−32n2+1n+12.


## 3. Comparison betweenSO, SOred, and SOavg for Wheel Graph

By getting results from Theorems1–3, we will make the comparison [10] between the values of Sombor, reduced Sombor, and average Sombor indices of a wheel graph. Table 1 and Figure 2 represent the numerical and graphical comparison of the three indices.Table 1
Comparison between Sombor, red. Sombor, and avg. Sombor indices.
nSOWnSOredWnSOavgWn325.455816.97060436.970625.73591.3425550.368036.50282.7420665.705149.28164.1841783.008964.07095.65698102.293280.86837.15179123.565399.67178.6629Figure 2
Comparison between Sombor, red. Sombor, and avg. Sombor indices for wheel graph.


## 4. Gear Graph

The gear graphGn, also known as bipartite wheel graph, is determined by adding a new node between each pair of adjacent nodes of rim as shown in Figure 3. The size of gear graph is 3n and the order of gear graph [7] is 2n+1. The gear graphs are a special case of Jahangir graph. The gear graphs are matchsticks and a unit distance graphs. For n=3, we have calculated three different Sombor indices of G3:(17)SOG3=613+92,SOredG3=65+62,SOavgG3=307+927.Figure 3
Gear graph.The general representation of the gear graph for three different Sombor indices is given in Theorems4–6.Theorem 4.
LetGn be a gear graph of order 2n+1. Then, the Sombor index of Gn is(18)SOGn=2n13+n9+n2.Proof.
SinceGn is a gear graph of order 2n+1, where n≥3, by definition,(19)SOGn=∑ei,j∈EGndegvi2+degvj2,SOGn=2n13+n9+n2.Theorem 5.
LetGn be a gear graph of order 2n+1. Then, the reduced Sombor index of Gn is(20)SOredGn=2n5+n4+n−12.Proof.
SinceGn is a gear graph of order 2n+1, where n≥3, by definition,(21)SOredGn=∑ei,j∈EGndegvi−12+degvj−12,SOredGn=2n5+nn2−2n+5.Theorem 6.
LetGn be a gear graph of order 2n+1. Then, the average Sombor index of Gn is(22)SOavgGn=2n2−6n2n+12+3−6n2n+12+n3−6n2n+12+n−6n2n+12.Proof.
SinceGn is a gear graph of order 2n+1, where n≥3, by definition,(23)SOavgGn=∑ei,j∈EGndegvi−2mn2+degvj−2mn2,SOavgGn=2n2−6n2n+12+3−6n2n+12+n3−6n2n+12+n−6n2n+12.


## 5. Comparison betweenSO, SOred, and SOavg for Gear Graph

By getting results from Theorems4–6, we will make the comparison [10] between the values of Sombor, reduced Sombor, and average Sombor indices of a gear graph. The numerical and graphical representation of indices is given in Table 2 and Figure 4, respectively.Table 2
Comparison between Sombor, red. Sombor, and avg. Sombor indices.
nSOGnSOredGnSOavgGn334.361221.90176.1040448.844432.310711.4603565.210344.721419.2124683.515859.143829.07127103.788175.576840.97808126.040994.018054.91149150.2814114.465170.8613Figure 4
Comparison between Sombor, red. Sombor, and avg. Sombor indices for gear graph.


## 6. Helm Graph

The helm graphHn is determined by adding a single edge to every node of the rim as shown in Figure 5. The size of helm graph is 3n and the order of helm graph is 2n+1. The chromatic number of helm graph is 3 if n is even and 4 if n is odd. The helm graph [7] contains three type of vertices,n vertices on outer rim, npendant vertices, and nvertices of degree 4. It is a node prime graph. For n=3, we have calculated three different Sombor indices of H3:(24)SOH3=317+332+15,SOredH3=318+313+6,SOavgH3=32217+31097+3027.Figure 5
Helm graph.The general representation of the helm graph for three different Sombor indices is given in Theorems4–6.Theorem 7.
LetHn be a gear graph of order 2n+1. Then, the Sombor index of Hn is(25)SOHn=n17+n32+n16+n2.Proof.
SinceHn is a helm graph of order 2n+1, where n≥3, by definition,(26)SOHn=∑ei,j∈EHndegvi2+degvj2,SOHn=n17+n32+n16+n2.Theorem 8.
LetHn be a helm graph of order 2n+1. Then, the reduced Sombor index of Hn is(27)SOredHn=3n+n18+n9+n−12.Proof.
SinceHn is a helm graph of order 2n+1, where n≥3, by definition,(28)SOredHn=∑ei,j∈EHndegvi−12+degvj−12,SOredHn=3n+n18+n9+n−12.Theorem 9.
LetHn be a helm graph of order 2n+1. Then, the average Sombor index of Hn is(29)SOavgHn=n1−6n2n+12+4−6n2n+12+2n4−6n2n+1+n4−6n2n+12+n−6n2n+12.Proof.
SinceHn is a helm graph of order 2n+1, where n≥3, by definition,(30)SOavgHn=∑ei,j∈EHndegvi−2mn2+degvj−2mn2,SOavgHn=n1−6n2n+12+4−6n2n+12+2n4−6n2n+1+n4−6n2n+12+n−6n2n+12.


## 7. Comparison betweenSO, SOred, and SOavg for Helm Graph

By getting results from Theorems7–9, we are able to make the comparison [10] between the values of Sombor, reduced Sombor, and average Sombor indices of a helm graph. The numerical and graphical representation of indices is given in Table 3 and Figure 6, respectively.Table 3
Comparison between Sombor, red. Sombor, and avg. Sombor indices.
nSOHnSOredHnSOavgHn344.339932.544616.9065461.747345.941123.6225580.915461.213232.75136101.946478.441644.11837124.895597.655957.59928149.7939118.867373.13899176.6594142.079888.7120Figure 6
Comparison between Sombor, red. Sombor, and avg. Sombor indices for helm graph.


## 8. Flower Graph

The flower graphFln is determined from the helm graph by joining each single node to the apex of helm graph as shown in Figure 7. The size of flower graph is 4n and the order of flower graph [7] is 2n+1. For n=3, we have calculated three different Sombor indices of Fl3:(31)SOFl3=65+122+610+613,SOredFl3=310+92+326+334,SOavgFl3=6297+1227+61067+6857.Figure 7
Flower graph.The general representation of the flower graph for three different Sombor indices is given in Theorems10–12.Theorem 10.
LetFln be a flower graph of order 2n+1. Then, the Sombor index of Fln is(32)SOFln=n20+n32+n4+2n2+n16+2n2.Proof.
SinceFln is a flower graph of order 2n+1, where n≥3, by definition,(33)SOFln=∑ei,j∈EFlndegvi2+degvj2,SOFln=n20+n32+n4+2n2+n16+2n2.Theorem 11.
LetFln be a flower graph of order 2n+1. Then, the reduced Sombor index of Fln is(34)SOredFln=n10+n18+n1+2n−12+n9+2n−12.Proof.
SinceFln is a flower graph of order 2n+1, where n≥3, by definition,(35)SOredFln=∑ei,j∈EFlndegvi−12+degvj−12,SOredFln=n10+n18+n1+2n−12+n9+2n−12.Theorem 12.
LetFln be a flower graph of order 2n+1. Then, the average Sombor index of Fln is(36)SOavgFln=n2−8n2n+12+4−8n2n+12+2n4−8n2n+1+n2−8n2n+12+2n−8n2n+12+n4−8n2n+12+2n−8n2n+12.Proof.
SinceFln is a flower graph of order 2n+1, where n≥3, by definition,(37)SOavgFln=∑ei,j∈EFlndegvi−2mn2+degvj−2mn2,SOavgFln=n2−8n2n+12+4−8n2n+12+2n4−8n2n+1+n2−8n2n+12+2n−8n2n+12+n4−8n2n+12+2n−8n2n+12.


## 9. Comparison betweenSO, SOred, and SOavg for Flower Graph

By getting results from Theorems10–12, we will make the comparison between the values of Sombor, reduced Sombor, and average Sombor indices of a flower graph. The numerical and graphical representation of indices is given in Table 4 and Figure 8, respectively.Table 4
Comparison between Sombor, red. Sombor, and avg. Sombor indices.
nSOFlnSOredFlnSOavgFln370.993955.004723.767504109.277988.367045.68715155.4868129.735775.67616209.6618179.1122113.68147271.8194236.4949159.69058341.9674301.8822213.70039420.1096375.2728275.7095Figure 8
Comparison between Sombor, red. Sombor, and avg. Sombor indices for flower graph.


## 10. Sunflower Graph

The sunflower graphSfn is determined from the flower graph by expanding n single edges to the apex of flower graph as shown in Figure 9. The size of sunflower graph [7] is 5n and the order of flower graph is 3n+1. For n=3, we have calculated three different Sombor indices of Sf3:(38)SOSf3=65+122+385+397+382,SOredSf3=310+92+365+373+24,SOavgSf3=62+637+610.Figure 9
Sunflower graph.The general representation of the sunflower graph for three different Sombor indices is given in Theorems13–15.Theorem 13.
LetSfn be a sunflower graph of order 3n+1. Then, the Sombor index of Sfn is(39)SOSfn=n20+n32+n4+3n2+n16+3n2+n1+3n2.Proof.
SinceSfn is a sunflower graph of order 3n+1, where n≥3, by definition,(40)SOSfn=∑ei,j∈ESfndegvi2+degvj2,SOSfn=n20+n32+n4+3n2+n16+3n2+n1+3n2.Theorem 14.
LetSfn be a sunflower graph of order 3n+1. Then, the reduced Sombor index of Sfn is(41)SOredSfn=n10+n18+n1+3n−12+n9+3n−12+n3n−1.Proof.
SinceSfn is a sunflower graph of order 3n+1, where n≥3, by definition,(42)SOredSfn=∑ei,j∈ESfndegvi−12+degvj−12,SOredSfn=n10+n18+n1+3n−12+n9+3n−12+n3n−1.Theorem 15.
LetSfn be a sunflower graph of order 3n+1. Then, the average Sombor index of Sfn is(43)SOavgFln=n2−10n3n+12+4−10n3n+12+2n4−10n3n+1+n2−10n3n+12+3n−10n3n+12+n4−10n3n+12+3n−10n3n+12+n1−10n3n+12+3n−10n3n+12.Proof.
SinceSfn is a sunflower graph of order 3n+1, where n≥3, by definition,(44)SOavgSfn=∑ei,j∈ESfndegvi−2mn2+degvj−2mn2,SOavgFln=n2−10n3n+12+4−10n3n+12+2n4−10n3n+1+n2−10n3n+12+3n−10n3n+12+n4−10n3n+12+3n−10n3n+12+n1−10n3n+12+3n−10n3n+12.


## 11. Comparison betweenSO, SOred, and SOavg for Sunflower Graph

By getting results from Theorems13–15, we are able to develop the comparison between the values of Sombor, reduced Sombor, and average Sombor indices [10] of a sunflower graph. The numerical and graphical representation of indices is given in Table 5 and Figure 10, respectively.Table 5
Comparison between Sombor, red. Sombor, and avg. Sombor indices.
nSOSfnSOredSfnSOavgSfn397.079096.033563.95564187.9409163.4081119.37585279.0960248.7920192.80836388.2396352.1819284.24397515.3776473.5756393.68038660.5124612.9718521.11669823.6455770.3698666.5528Figure 10
Comparison between Sombor, red. Sombor, and avg. Sombor indices for sunflower graph.


## 12. Lobster Graph

The lobsterLn2,r is a graph formed from a path on n nodes as a backbone, each node in the backbone is adjacent to two different node hands, and each node hand is adjacent to r different node fingers each of which has degree 1. The lobster graph or lobster tree is a tree [1] in which the removal of leaf node leaves a caterpillar graph as shown in Figure 11. The size of lobster graph is 2n2r+3−2 and the order of flower graph is n2r+3. We consider a special case p=2 of a regular lobster graph Lnp,r. For n,r=2, we have calculated three different Sombor indices of Ln2,r:(45)SOL22,2=122+10+810−32+397+382,SOredL22,2=82+16+213−18,SOavgL22,2=86417+347+807−1527.Figure 11
Lobster graph.The general representation of the lobster graph for three different Sombor indices is given in Theorems16–18.Theorem 16.
LetLn2,r be a lobster graph of order n2r+3. Then, the Sombor index of Ln2,r is(46)SOLn2,r=49+r+12+10+2nr1+r+12+n−332+2n−216+r+12.Proof.
SinceLn2,r is a lobster graph of order n2r+3, where n≥2 and r≥2, by definition,(47)SOLn2,r=∑ei,j∈ELn2,rdegvi2+degvj2,SOLn2,r=49+r+12+10+2nr1+r+12+n−332+2n−216+r+12.Theorem 17.
LetLn2,r be a lobster graph of order n2r+3. Then, the reduced Sombor index of Ln2,r is(48)SOredLn2,r=44+r2+2nr2+2n−29+r2+213+n−318.Proof.
SinceLn2,r is a lobster graph of order n2r+3, where n≥2 and r≥2, by definition,(49)SOredLn2,r=∑ei,j∈ELn2,rdegvi−12+degvj−12,SOredLn2,r=44+r2+2nr2+2n−29+r2+213+n−318.Theorem 18.
LetLn2,r be a lobster graph of order n2r+3. Then, the average Sombor index of Ln2,r is(50)SOavgLn2,r=49−2n2r+3−2n2r+32+r+1−2n2r+3−2n2r+32+2n−34−2n2r+3−2n2r+3+23−2n2r+3−2n2r+32+4−2n2r+3−2n2r+32+2nr1−2n2r+3−2n2r+32+r+1−2n2r+3−2n2r+32+2n−216−2n2r+3−2n2r+32+r+1−2n2r+3−2n2r+32.Proof.
SinceLn2,r is a lobster graph of order n2r+3, where n≥2 and r≥2, by definition,(51)SOavgLn2,r=∑ei,j∈ELn2,rdegvi−2mn2+degvj−2mn2,SOavgLn2,r=49−2n2r+3−2n2r+32+r+1−2n2r+3−2n2r+32+2n−34−2n2r+3−2n2r+3+23−2n2r+3−2n2r+32+4−2n2r+3−2n2r+32+2nr1−2n2r+3−2n2r+32+r+1−2n2r+3−2n2r+32+2n−216−2n2r+3−2n2r+32+r+1−2n2r+3−2n2r+32.


## 13. Comparison betweenSO, SOred, and SOavg for Lobster Graph

By getting results from Theorems16–18, we will make the comparison between the values of Sombor, reduced Sombor, and average Sombor indices [10] of a lobster graph. The numerical and graphical representation of indices is given in Table 6 and Figure 12, respectively.Table 6
Comparison between Sombor, red. Sombor, and avg. Sombor indices.
n,rSOLn2,rSOredLn2,rSOavgLn2,r2,373.820453.390758.94463,4168.5065131.0996141.03314,5314.6446256.3182264.40263,274.917949.735978.76854,3157.2388114.8465148.03585,4287.0170223.5849254.57412,246.611930.282242.19003,3115.529684.1186103.49574,4227.7618177.3423197.8057Figure 12
Comparison between Sombor, red. Sombor, and avg. Sombor indices for lobster graph.


## 14. Conclusion

Topological indicator is a mathematical coding of the graphs. We have discussed the particular cases ofSO, SOred, and SOavg to elaborate the wheel graph, gear graph, helm graph, flower graph, sunflower graph, and lobster graph, and then we find the general representation of these graphs. We have also calculated the comparison in both numerical and graphical forms between these three indices for each graph.



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