Nowadays, molecular descriptors are the most significant invariants utilised in molecular modeling, and as a result, they are controlled by statistics, chemo metrics, and chemo informatics. Topological indices (TIs) are receiving more and more attention from chemists and biologists, because of their significance in QSPR (Quantitative Structure Property Relationships) and QSAR (Quantitative Structure Activity Relationships). In this, Johan Zagreb chromatic indices of some chemical network graphs are presented.

Compounds of chemical molecules tend to be represented as graphs, with nodes and edges denoting the atoms and bonds, respectively. In particular, chemical graph theory is applied in mathematical chemistry to investigate and study the structure of complicated chemical networks and basic chemical structures. Numerical parameter called topological indices is used to describe how a graph is structured. In addition to their many other uses, they have contributed significantly to the development of mathematical chemistry as molecular descriptors

In this, we extend the concept of chromatic Zagreb indices _{1}, c_{2}, c_{3}, · · ·, c_{l}}, l ∈ N sufficiently large is a set of distinct colors, a proper vertex coloring of a graph G is a vertex coloring φ: V(G) → C of G such that no two distinct adjacent vertices have the same color. The minimum number of colors in proper vertex coloring of G is called the chromatic number of G and is denoted as χ(G). Unless mentioned otherwise, we follow the convention that, among the colors in the coloring C = {c_{1}, c_{2}, c_{3}, … c_{l}}, l = χ(G), the color c_{1} will be assigned to maximum possible number of vertices in G, then color

c_{2}_{ }will be assigned to maximum possible number of remaining uncolored vertices and proceeding like this, at the final step, the color c_{l} will be given for the remaining uncolored vertices. This convention is called the rainbow neighbourhood convention _{i}) = c_{l}, in other words we say that φ(vi)=l. A maximal proper coloring of a graph G is a Johan coloring denoted J - coloring, if and only if every vertex of G belongs to a rainbow neighbourhood of G. The maximum number of colors in a J -coloring is denoted by J(G). To learn more about topological indices and their uses refer

Python programmes are simple to understand and can generally run in a matter of seconds. Here, we use a python code to find a family of Zagreb Johan indices for a molecular graph and to generate a number of Zagreb indices for the mentioned molecular networks.

_{r}_{ }is

(1) JZI_{1}(RB_{r})=9r^{2}+ 12r–3.

(2) JZI_{2}(RB_{r})=6r^{2}+ 8r−2.

(3) JZI_{3}(RB_{r})=3r^{2}+ 4r−1.

(4) HJZI_{1}(RB_{r})=27r^{2}+ 36r−9.

(5) HJZI_{2}(RB_{r})=12r^{2}+ 16r−4.

(6) AJZI_{2}(RB_{r})=24r^{2}+32r−8.

(7) ReJZI_{1}(RB_{r})=^{2}+6n−

(8) ReJZI_{2}(RB_{r})=2n^{2}+

(9) ReJZI_{3}(RB_{r})=18r^{2}+24r−6.

_{r}. We presume that the vertex setto be

In the graph RB_{r}, we use two colors and the coloring partition given as follows,

Using the above coloring partition, the first Johan Zagreb Index is,

=3r^{2}+4r−1(3)

=9r^{2}+12r−3JZI(RB)=9r+12r−3.

_{r}_{ }is,

(1) JZI_{1}(HB_{r})=9r^{2}+27r−18.

(2) JZI_{2}(HB_{r})=6r^{2}+18r−12.

(3) JZI_{3}(HB_{r})=3r^{2}+9r−6.

(4) JZJI_{1}(HB_{r}) = 27r^{2} + 81r − 54.

(5) HJZI_{2}(HB_{r}) = 12r^{2} + 36r − 24.

(6) AJZI_{2}(HB_{r}) = 24r^{2} + 72r − 48.

(7) ReJZI_{1}(HB_{r})=9r^{2}+ 27r−9.

(8) ReJZI_{2}(HB_{r}) = 2r^{2} + 6r − 4.

(9) ReJZI_{3}(HB_{r})=18r^{2}+54r−36.HBsetto be

In the graph HB_{r}, we use two colors and the coloring partition given as follows,

Using the above coloring partition the first Johan Zagreb Index is,

= 3r^{2}+9r−6(3)

=9r^{2}+27r−18JZI(HB)=9r+27r−18. The proof is Similar for the all other indices.

_{r}_{ }is

(1) JZI_{1}(JB_{r}) = 18rs + 15r + 3s − 12.

(2) JZI_{2}(JB_{r}) =12rs+10r+3s−8.

(3) JZI_{3}(JB_{r})=6rs+5r+s−4.

(4) HJZI_{1}(JB_{r}) = 54rs + 45r + 9s − 36.

(5) HJZI_{2}(JB_{r}) = 24rs + 20r + 4s − 16.

(6) AJZI_{2}(JB_{r})=48rs + 40r+ 8s −32.

(7) ReJZI_{1}(JB_{r})=

(8) ReJZI_{2}(JB_{r})=4rs+

(9) ReJZI_{3}(JB_{r})=36rs+30r+6s−24.

_{r}. We presume that the vertex set to be

In the graph JB_{r}, we use two colors and the coloring partition given as follows,

Using the above coloring partition, the first Johan Zagreb Index is,

=6rs+5r+s−4(3)

=18rs+15r+3s−12JZI(JB)=18rs+15r+3s−12. The proof is Similar for the all other

Indices.

r=int(input(”Enter r Value:”))

s=int(input(”Enter s Value:”))

JZI1JR_{r,s}= 18 ∗r ∗s + 15 ∗r + 3 ∗s − 12

JZI2JR_{r,s}= 12 ∗r ∗s + 10 ∗r + 2 ∗s − 8

JZI3JR_{r,s}= 6 ∗r ∗s + 5 ∗r + s − 4

HJZ1JR_{r,s}= 54 ∗r ∗s + 45 ∗r + 9s − 36

HJZI2JR_{r,s}=24∗r∗s+20 ∗r+4s−16

AJZIJR_{r,s}= 48 ∗r ∗s + 40∗r + 8s – 32

ReJZI1JR_{r,s}=9∗r∗s+=

ReJZI3JR =36 ∗r∗s+30∗r+6∗s−24

print(”ZJI1(JR_{r,s}) =: ”,ZJI1JR_{r,s})

print(”ZJI2(JR_{r,s})=:”,ZJI2JR_{r,s})

print(”ZJI3(JR_{r,s}) =: ”,ZJI3JR_{r,s})

print(”HZJI1(JR_{r,s})=:”,HZJI1JR_{r,s})

print(”HZJI2(JR_{r,s})=:”, HZJI2JR_{r,s})

print(”AZJI(JR_{r,s})=:”,AZJIJR_{r,s})

print(”ReZJI1(JR_{r,s})=:”, ReZJI1JR_{r,s})

print(”ReZJI2(JR_{r,s})=:”, ReZJI2JR_{r,s})

print(”ReZJI3(JR_{r,s})=:”,ReZJI3JR_{r,s})

This study introduces a number of Johan Zagreb Indices that are relevant to chemical graph theory. Although many various chromatic indices have been employed in the literature, Johan Zagreb chromatic indices outperform those that have already been developed. Additionally, it provides the Python Program to calculate these indices. The physico-chemical characteristics of chemical molecules are predicted using them. The creation of new indices will aid in the discovery of drugs more in future study. Interdisciplinary research is quite valuable in this situation, that this explanation would encourage some readers to research these indices in more depth. So hopefully more research can be done in the fascinating field of graph theoretical indices.

This work has been presented in “International conference on Recent Strategies in Mathematics and Statistics" 112 (ICRSMS2022), Organized by the Department of Mathematics of Stella Maris College and of IIT Madras during 19 to 21 May, at Chennai, India The Organizer claims the peer review responsibility.