On Computation of Edge Degree-Based Banhatti Indices of a Certain Molecular Network

Department of General Education, Anhui Xinhua University, Hefei 230088, China Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan Department of Mathematics and Statistics, Institute of Southern Punjab, Multan, Pakistan School of Mathematics and Physics, Anhui Jianzhu University, Hefei, China


Introduction
Topological indices are graph invariants associated with numbers that describe the properties of the graph. In chemical graph theory, topological indices play a vital role to explore the structures of different graphs. In 1947, Harold Wiener gave the idea of topological indices [1]. After that, he published a series of papers describe the relation between wiener index and physicochemical properties of carbon-based compounds [2,3] in 1947 and [4,5] in 1948. e analysis of topological indices has great importance in nanotechnology and theoretical chemistry. e irregularity of graph was discussed [6] in 1997. In the last decade of the 20 th century, a large number of topological indices were introduced that were related to the Wiener index. In the second decade of the 21 st century, irregularity topological indices were computed for different chemical structures. In [7], it was shown that Randic and modified Zagreb indices are in oneto-one correspondence for all acyclic molecules which consist of no more than 100 atoms. In [8], the new notion of total irregularity was introduced, and the authors determined the graphs with maximum total irregularity. In [9][10][11], the total irregularity of graphs was discussed under the graph operations. In [12], the total irregularity of graphs was discussed to study QSPR. An Indian mathematician Kulli in 2016 [13] introduced some new Banhatti indices such as K Banhatti indices, modified Banhatti indices and, hyper K Banhatti indices. In the last decade, irregular, distance-and degree-based topological indices became hot topics for research in chemical graph theory. Many researchers computed these indices for different chemical graphs to study their biochemical properties. In [14] [20], Chu et al. studied the irregular indices for metal organic frameworks and certain 2D lattices. e Zagreb connection index is computed for silicate, hexagonal, honeycomb, and oxide networks in [21] in 2021. In [22], Rao et al. studied some degree-based topological indices of a caboxy-terminated dendritic macromolecule. In [23], the authors computed the face index for Boron triangular nanotubes and for quadrilateral sections cut from a regular hexagonal lattice. In [24], Hussain et al. computed topological indices for new classes of Benes network.
Let G(V, E) be a graph where V is a set of vertices and E is a set of edges. A cardinality of edges associated with a vertex is called the degree of the vertex. Here, we use a special term of e � st as an edge of G where the vertex s and vertex t are linked together by edge e. Let d G (e) denote the degree of an edge e in G, which is defined by For more details, refer the work of Kulli [25]. e first and second K Banhatti indices were introduced by Kulli in [13] as (1) e first and second K hyper Banhatti index of G were introduced by Kulli in [26] defined as e first and second modified Banhatti indices of G were introduced by Kulli in [27] as , e harmonic K-Banhatti index of a graph G was introduced by Kulli in [27] as Let G be a graph of water-soluble dendritic unimolecular polyether micelle. It has 38(2 n ) − 4 number of vertices and 42(2 n ) − 5 number of edges where n is the number of growth of the graph. e graph has 4(2 n ) number of vertices having degree 1, 22(2 n ) − 2 vertices having degree 2 and 12(2 n ) − 2 vertices having degree 3. e graph has 4(2 n ) number of edges having degree (1, 3), 8(2 n ) + 2 edges having degree (2, 2), 28(2 n ) − 8 edges having degree (2, 3), and (2, 3) number of edges having degree (3, 3). In Figure 1, the graph G is given for n � 4. Dendritic unimolecular micelles play an important role in drug delivery systems. Unimolecular micelles have a unique property of uniform size and high stability. Also, they have attracted increasing attention due to their high functionality in various applications.
In the next section, we will compute the Banhatti indices for the water-soluble dendritic unimolecular polyether micelle. Table 1 shows the partition of the edge set for the molecular graph G of water-soluble dendritic unimolecular polyether micelle.

Theorem 1.
Let G be the molecular graph of water-soluble dendritic unimolecular polyether micelle; then, the first K Banhatti index of G is Proof. By using Table 1 and the definition of the first K Banhatti index, we have   Figure 1: Graph of water-soluble unimolecular polyether micelle for growth four.

Theorem 2. Let G be the molecular graph of water-soluble dendritic unimolecular polyether micelle; then, the second K Banhatti index of G is
Proof. To compute the second K Banhatti index, we will use Table 1.
Proof. e edge partition given in Table 1 and the definition of the first K hyper Banhatti index give Proof. e result follows by using the values from Table 1 and the definition of the second K hyper Banhatti index.
Proof. By using the definition of the first modified Banhatti index and Table 1, we have , □ Theorem 6. Let G be the molecular graph of water-soluble dendritic unimolecular polyether micelle; then, the second modified Banhatti index of G is Proof. e second modified Banhatti index can be computed by using Table 1 as , , Proof. e result can be obtained as follows by using Table 1 and the definition of the harmonic Banhatti index: � 4 2 n 2 (1 + 2) +(3 + 2) + 8 2 n + 2 2 (2 + 2) +(2 + 2) +

Graphical Analysis and Conclusions
is section actually provides the summary of this article. Table 2 gives the comparison for the said topological indices of the graph. We can see that mB 2 (G) gives the least values for different growths of the graph whereas HB 2 (G) gives largest values. In Table 2, we can check the values for some test values of parameter n. Also, the graphical comparison is presented in Figure 2.

Data Availability
No data were used to support for this research.

Conflicts of Interest
e authors declare no conflicts of interest.