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

Authors: Jiang-Hua Tang; Muhammad Abid; Kashif Ali; Asfand Fahad; Muhammad Anwar Chaudhry; Muhammad Imran Qureshi; Jia-Bao Liu
Journal: Journal of Mathematics
 (2021)

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## Abstract

Chemical graph theory deals with the basic properties of a molecular graph. In graph theory, we correlate molecular descriptors to the properties of molecular structures. Here, we compute some Banhatti molecular descriptors for water-soluble dendritic unimolecular polyether micelle. Our results prove to be very significant to understand the behaviour of water-soluble dendritic unimolecular polyether micelle as a drug-delivery agent.


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## Body

## 1. 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. The analysis of topological indices has great importance in nanotechnology and theoretical chemistry. The irregularity of graph was discussed [6] in 1997. In the last decade of the 20th century, a large number of topological indices were introduced that were related to the Wiener index. In the second decade of the 21st century, irregularity topological indices were computed for different chemical structures. In [7], it was shown that Randic and modified Zagreb indices are in one-to-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 [9–11], 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], Zheng et al. computed some eccentricity-based topological indices and polynomials of Poly(EThylene Amido Amine) (PETAA) dendrimers. In [15], Ye et al. worked on the Zagreb connection number index of nanotubes and regular hexagonal lattice. In [16], Fahad et al. studied the topological descriptors of Poly Propyl Ether Imine (PETIM) dendrimers. In [17], Qureshi studied the Zagreb connection index of drug-related chemical structures. In [18], Zhang et al. worked on a newly defined topological index named face index for silicon carbides. In [19], Luo et al. computed lower bounds on the entire Zagreb indices of trees. In [20], Chu et al. studied the irregular indices for metal organic frameworks and certain 2D lattices. The 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.LetGV,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 dGe denote the degree of an edge e in G, which is defined by dGe=dGs+dGt−2 with e=st. For more details, refer the work of Kulli [25].The first and secondK Banhatti indices were introduced by Kulli in [13] as(1)B1G=∑e=stϵEGdGs+dGe,B2G=∑e=stϵEGdGs∗dGe.The first and secondK hyper Banhatti index of G were introduced by Kulli in [26] defined as(2)HB1G=∑e=stϵEGdGs+dGe2,HB2G=∑e=stϵEGdGs∗dGe2.The first and second modified Banhatti indices ofG were introduced by Kulli in [27] as(3)mB1G=∑e=stϵEG1dGs+dGe+dGt+dGe,mB2G=∑e=stϵEG2dGs+dGe+dGt+dGe.The harmonicK-Banhatti index of a graph G was introduced by Kulli in [27] as(4)HbG=∑e=stϵEG2dGs+dGe+dGt+dGe.LetG be a graph of water-soluble dendritic unimolecular polyether micelle. It has 382n−4 number of vertices and 422n−5 number of edges where n is the number of growth of the graph. The graph has 42n number of vertices having degree 1, 222n−2 vertices having degree 2 and 122n−2 vertices having degree 3. The graph has 42n number of edges having degree 1,3, 82n+2 edges having degree 2,2, 282n−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.Figure 1
Graph of water-soluble unimolecular polyether micelle for growth four.In the next section, we will compute the Banhatti indices for the water-soluble dendritic unimolecular polyether micelle.


## 2. Main Results

Table1 shows the partition of the edge set for the molecular graph G of water-soluble dendritic unimolecular polyether micelle.Table 1
Edge partition of water-soluble dendritic unimolecular polyether micelle.
dGs,dGt, where stϵEGdGeNumber of edges1,3242n2,2282n+22,33282n−83,3422n+1Theorem 1.
LetG be the molecular graph of water-soluble dendritic unimolecular polyether micelle; then, the first K Banhatti index of G is(5)B1G=4322n−58.Proof.
By using Table1 and the definition of the first K Banhatti index, we have(6)B1G=∑e=stϵEGdGs+dGe,=42n1+2+3+2+82n+22+2+2+2+282n−82+3+3+3+22n+13+4+3+4,=322n+642n+16+3082n−88+282n+14,=4322n−58.Theorem 2.
LetG be the molecular graph of water-soluble dendritic unimolecular polyether micelle; then, the second K Banhatti index of G is(7)B2G=5642n−80.Proof.
To compute the secondK Banhatti index, we will use Table 1.(8)B2G=∑e=stϵEGdGs∗dGe,=42n1∗2+3∗2+82n+22∗2+2∗2+282n−82∗3+3∗3+22n+13∗4+3∗4,=322n+642n+16+4202n−120+482n+24,=5642n−80.Theorem 3.
LetG be the molecular graph of water-soluble dendritic unimolecular polyether micelle; then, the first K hyper Banhatti index of G is(9)HB1G=22962n−326.Proof.
The edge partition given in Table1 and the definition of the first K hyper Banhatti index give(10)HB1G=∑e=stϵEGdGs+dGe2,=42n1+22+3+22+82n+22+22+2+22+282n−82+32+3+32+22n+13+42+3+42,=1362n+2562n+64+17082n−488+1962n+98,=22962n−326.Theorem 4.
LetG be the molecular graph of water-soluble dendritic unimolecular polyether micelle; then, the second K hyper Banhatti index of G is(11)HB2G=42682n−584.Proof.
The result follows by using the values from Table1 and the definition of the second K hyper Banhatti index.(12)HB2G=∑e=stϵEGdGs∗dGe2,=42n1∗22+3∗22+82n+22∗22+2∗22+282n−82∗32+3∗32+22n+13∗42+3∗42,=1602n+2562n+64+32762n−936+5762n+288,=42682n−584.Theorem 5.
LetG be the molecular graph of water-soluble dendritic unimolecular polyether micelle; then, the first modified Banhatti index of G is(13)mB1G=4.18842n−0.4059.Proof.
By using the definition of the first modified Banhatti index and Table1, we have(14)mB1G=∑e=stϵEG1dGs+dGe+dGt+dGe,=42n11+2+3+2+82n+212+2+2+2+282n−812+3+3+3+22n+113+4+3+4,=2n2+2n+14+282n11−811+2n7+114,=4.18842n−0.4059.Theorem 6.
LetG be the molecular graph of water-soluble dendritic unimolecular polyether micelle; then, the second modified Banhatti index of G is(15)mB2G=3.452n−0.2416.Proof.
The second modified Banhatti index can be computed by using Table1 as(16)mB2G=∑e=stϵEG2dGs+dGe+dGt+dGe,=42n11×2+3×2+82n+222×2+2×2+282n−822×3+3×3+22n+123×4+3×4,=2n2+2n+14+282n15−815+2n12+124,=3.452n−0.2416.Theorem 7.
LetG be the molecular graph of water-soluble dendritic unimolecular polyether micelle; then, the harmonic Banhatti index of G is(17)HbG=8.37662n−0.8116.Proof.
The result can be obtained as follows by using Table1 and the definition of the harmonic Banhatti index:(18)HbG=∑e=stϵEG2dGs+dGe+dGt+dGe,=42n21+2+3+2+82n+222+2+2+2+282n−822+3+3+3+22n+123+4+3+4,=2n+22n+12+562n11−1611+22n7+214,=8.37662n−0.8116.


## 3. Graphical Analysis and Conclusions

This section actually provides the summary of this article. Table2 gives the comparison for the said topological indices of the graph. We can see that mB2G gives the least values for different growths of the graph whereas HB2G 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.Table 2
Banhatti indices of water-soluble dendritic unimolecular polyether micelle.
Banhatti indicesn=1n=2n=3n=4n=5B1G80616703398685413766B2G104821764432894417968HB1G42668858180423641073146HB2G7952164883356067704135992mB1G7.970916.347733.101366.6085133.6229mB2G6.658413.558427.358454.9584110.1584HbG15.941632.694866.2012133.214267.2396Figure 2
Comparison graph of Banhatti indices of water-soluble unimolecular polyether micelle for growth four.



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