On Computing Techniques for Sombor Index of Some Graphs

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.


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, tra c network, etc. A graph is a simple way to represent information that connects relationships between objects. ings 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. e 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.
is process is expensive and time consuming. Because of this feature, pharmaceutical companies are keenly interested in nding 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 speci c chemical structure using topological indicators. Topological indices are of di erent types such as distance-based topological indices [4], spectrum-based topological indices, and degreebased 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 rst topological index which is equal to the sum of all distances between the vertices.
Gutman and Trinajestic introduced the rst and second Zagreb indices in [7] as e rst natural degree-based topological indicator was introduced by Milan Randic. His index was de ned as Let E be the edge of the graph G, between the vertices v i and v j . Later on, Ernesto Estrada derived a new topological indicator and named it atom bond connectivity index. It is defined as Furtula et al. derived the modified version of the atom bomb connectivity index, and they named it augmented Zagreb index. It is defined as e first geometric arithmetic index was proposed by Vukicevic in [8].
In 1980s, Siemion Fajtlowicz introduced a new topological indicator and named it as harmonic index. It is also a degree-based topological indicator. where are the geometric and arithmetic means, respectively. e sum connectivity index is a new invention by Nenad Trinajstić and Bo Zhou. It is defined as where the product is replaced by sum in Randic index. In this article, we will compute the degree-based indices like SO, SO red , and SO avg . SO, SO red , and SO avg are defined in [9] as 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. e 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.

Wheel Graph
e wheel graph W n is determined by connecting K 1 and C n . K 1 is the graph of order 1 and C n is the cycle graph as shown in Figure 1. e size of wheel graph [7] is 2n and the order of wheel graph is n + 1. Apparently, W n wheel graph has every node of degree 3 other than the internal node. Internal node has degree n. e wheel graphs are planer graphs.All dual planer graphs are isomorphic to wheel graphs. e chromatic number of wheel graph W n is 3 if n is odd and 4 if n is even. For n � 3, we have calculated three different Sombor indices of W 3 : e general representation of the wheel graph for three different Sombor indices is given in eorems 1-3.

Theorem 1. Let
Wn be a wheel graph of order n + 1. en, the Sombor index of Wn is Proof. Since Wn is a wheel graph of order n + 1, where n ≥ 3, by definition, □ Theorem 2. Let Wn be a wheel graph of order n + 1. en, the reduced Sombor index of Wn is Proof. Since Wn is a wheel graph of order n + 1, where n ≥ 3, by definition, □ Theorem 3. Let Wn be a wheel graph of order n + 1. en, the average Sombor index of Wn is Proof. Since Wn is a wheel graph of order n + 1, where n ≥ 3, by definition,

Comparison between SO, SO red , and SO avg for Wheel Graph
By getting results from eorems 1-3, 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.

Gear Graph
e gear graph G n , 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. e size of gear graph is 3n and the order of gear graph [7] is 2n + 1. e gear graphs are a special case of Jahangir graph. e gear graphs are matchsticks and a unit distance graphs. For n � 3, we have calculated three different Sombor indices of G 3 : e general representation of the gear graph for three different Sombor indices is given in eorems 4-6.

Theorem 4. Let G n be a gear graph of order
Proof. Since G n is a gear graph of order 2n + 1, where n ≥ 3, by definition, □ Theorem 5. Let Gn be a gear graph of order 2n + 1. en, the reduced Sombor index of Gn is Proof. Since Gn is a gear graph of order 2n + 1, where n ≥ 3, by definition, Mathematical Problems in Engineering 3 (22) Proof. Since Gn is a gear graph of order 2n + 1, where n ≥ 3, by definition,

Comparison between SO, SO red , and SO avg for Gear Graph
By getting results from eorems 4-6, we will make the comparison [10] between the values of Sombor, reduced Sombor, and average Sombor indices of a gear graph. e numerical and graphical representation of indices is given in Table 2 and Figure 4, respectively.

Helm Graph
e helm graph H n is determined by adding a single edge to every node of the rim as shown in Figure 5. e size of helm graph is 3n and the order of helm graph is 2n + 1. e chromatic number of helm graph is 3 if n is even and 4 if n is odd. e 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 H 3 : (24) e general representation of the helm graph for three different Sombor indices is given in eorems 4-6.
Proof. Since Hn is a helm graph of order 2n + 1, where n ≥ 3, by definition,

Comparison between SO, SO red , and SO avg for Helm Graph
By getting results from eorems 7-9, we are able to make the comparison [10] between the values of Sombor, reduced Sombor, and average Sombor indices of a helm graph. e numerical and graphical representation of indices is given in Table 3 and Figure 6, respectively.

Flower Graph
e flower graph Fl n is determined from the helm graph by joining each single node to the apex of helm graph as shown in Figure 7. e 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 Fl 3 : e general representation of the flower graph for three different Sombor indices is given in eorems 10-12.     2n (36)

Mathematical Problems in Engineering
Proof. Since Fl n is a flower graph of order 2n + 1, where n ≥ 3, by definition,

Comparison between SO, SO red , and SO avg for Flower Graph
By getting results from eorems 10-12, we will make the comparison between the values of Sombor, reduced Sombor, and average Sombor indices of a flower graph. e numerical and graphical representation of indices is given in Table 4 and Figure 8, respectively.

Sunflower Graph
e sunflower graph Sf n is determined from the flower graph by expanding n single edges to the apex of flower graph as shown in Figure 9. e 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 Sf 3 : e general representation of the sunflower graph for three different Sombor indices is given in eorems 13-15.
Theorem 13. Let Sf n be a sunflower graph of order 3n + 1.
en, the Sombor index of Sf n is SO Sf n � n �� 20 Proof. Since Sf n is a sunflower graph of order 3n + 1, where n ≥ 3, by definition, Proof. Since Sf n is a sunflower graph of order 3n + 1, where n ≥ 3, by definition,

Comparison between SO, SO red , and SO avg for Sunflower Graph
By getting results from eorems 13-15, we are able to develop the comparison between the values of Sombor, reduced Sombor, and average Sombor indices [10] of a sunflower graph. e numerical and graphical representation of indices is given in Table 5 and Figure 10, respectively.

Lobster Graph
e lobster L n (2, 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. e 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. e size of lobster graph is 2n(2r + 3) − 2 and the order of flower graph is n(2r + 3). We consider a special case p � 2 of a regular lobster graph L n (p, r). For n, r � 2, we have calculated three different Sombor indices of L n (2, r): e general representation of the lobster graph for three different Sombor indices is given in eorems 16-18.

Comparison between SO, SO red , and SO avg for Lobster Graph
By getting results from eorems 16-18, we will make the comparison between the values of Sombor, reduced Sombor, and average Sombor indices [10] of a lobster graph. e numerical and graphical representation of indices is given in Table 6 and Figure 12, respectively.

Conclusion
Topological indicator is a mathematical coding of the graphs. We have discussed the particular cases of SO, SO red , and SO avg 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.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.