# On Nirmala indices–based entropy measures for the complex structure of
ruthenium bipyridine

H. M. Nagesh1,∗
Muhammad Kamran Siddiqui2
Department of Science and Humanities,
PES University, Bangalore, India.  2Department of Mathematics, COMSATS
University Islamabad,
Lahore Campus, Lahore, Pakistan.  email<EMAIL_ADDRESS>
###### Abstract

A numerical parameter, known as a topological index, is employed to represent
the molecular structure of a compound by considering its graph-theoretical
properties. In the study of quantitative structure-activity relationships
(QSAR) and quantitative structure-property relationships (QSPR), topological
indices are used to predict the physicochemical properties of chemical
compounds. Graph entropies have evolved as information-theoretic tools to
investigate the structural information of a molecular graph. In this study, we
compute the Nirmala index, the first and second inverse Nirmala index of the
complex structure of ruthenium bipyridine, with the help of its M-polynomial.
Furthermore, entropy measures for the complex structure of ruthenium
bipyridine are computed using Shannon’s entropy model. The comparison between
the Nirmala indices and their associated entropy measures is presented through
numerical computation. The correlation between the Nirmala indices and
associated entropy measurements is then examined using a regression model.

###### keywords:

Nirmala index, first inverse Nirmala index, second inverse Nirmala index,
graph entropy, ruthenium bipyridine.

##  1 Introduction

Topological indices are structural invariants based on molecular graphs that
capture the underlying connectivity of molecular networks, and thus these
indices have received considerable attention in recent years owing to their
applications in quantitative structure-activity and quantitative structure-
property relationships (QSPR) relations [1, 2, 3, 4]. Degree-based topological
indices have been studied extensively and they have been employed to predict
the physicochemical properties of molecular structures [5]. Information
entropy metrics can be used to characterize the information complexity of
complex chemical structures, including the complex structure of ruthenium
bipyridine. Shannon first proposed the idea of information entropy to assess
and measure the complexity of data and information transmission, but it has
since been extensively used in many different scientific domains. The analysis
of molecular structure complexity and quantum chemical electron densities [6]
is one of the most significant uses of information entropy. In QSAR and QSPR
investigations, topological indices combined with entropy measures may be a
more effective tool. Information entropy has been discovered to be directly
correlated with the physical features of fullerenes in several kinds of
natural substances, including their formal carbon atom oxidation states and
rotational symmetry numbers [7].

Let $\Upsilon=(V(\Upsilon),E(\Upsilon))$ be an ordered pair of a simple,
connected, and undirected graph with non-empty vertex set $V(\Upsilon)$ and
edge set $E(\Upsilon)$. The total number of edges incident to a vertex $v\in
V(\Upsilon)$ is known as $degree$ of the vertex $v$ and is denoted as
$d_{\Upsilon}(v)$. Let $e=pq$ represent an edge of the graph $\Upsilon$, where
$p$ and $q$ are the end vertices of the edge $e$. The topological index based
on degrees, defined on the edge set $E(\Upsilon)$ of a graph $\Upsilon$ [8] is
given by

$I(\Upsilon)=\displaystyle\sum_{uv\in
E(\Upsilon)}f(d_{\Upsilon}(u),d_{\Upsilon}(v)),$

where $f(x,y)$ is a non-negative and symmetric function that depends on the
mathematical formulation of the topological index.

Several topological indices have been reported in the literature and have
shown benefits in several fields, including drug development, biology,
chemistry, computer science, and physics. Proposed by H. Wiener in 1947, the
Wiener index was the first and most studied topological index. One noteworthy
application is the prediction of paraffin boiling temperatures [9]. Another
well-known degree-based topological statistic is the connectedness index,
sometimes known as the _Randić index_ and first introduced by Milan Randić in
1975. Its importance in making drugs is generally recognized [10]. For
additional information on topological indices and their applications, readers
are referred to [11, 12].

Numerous efforts have been undertaken to improve the category of degree-based
topological indices by the addition of new indices.

Kulli in [13] introduced a novel degree-based topological index of a molecular
graph $\Upsilon$, which is called the _Nirmala index_ as follows.

$N(\Upsilon)=\displaystyle\sum_{uv\in
E(\Upsilon)}\sqrt{d_{\Upsilon}(u)+d_{\Upsilon}(v)}$ (1)

Later in 2021, Kulli [14] introduced the notion of the first inverse Nirmala
index $IN_{1}(\Upsilon)$ and second inverse Nirmala index $IN_{2}(\Upsilon)$
of a molecular graph $\Upsilon$ as follows.

$IN_{1}(\Upsilon)=\displaystyle\sum_{uv\in
E(\Upsilon)}\sqrt{\frac{1}{d_{\Upsilon}(u)}+\frac{1}{d_{\Upsilon}(v)}}=\displaystyle\sum_{uv\in
E(\Upsilon)}\left(\frac{1}{d_{\Upsilon}(u)}+\frac{1}{d_{\Upsilon}(v)}\right)^{\frac{1}{2}}$
(2) $IN_{2}(\Upsilon)=\displaystyle\sum_{uv\in
E(\Upsilon)}\frac{1}{\sqrt{\frac{1}{d_{\Upsilon}(u)}+\frac{1}{d_{\Upsilon}(v)}}}=\displaystyle\sum_{uv\in
E(\Upsilon)}\left(\frac{1}{d_{\Upsilon}(u)}+\frac{1}{d_{\Upsilon}(v)}\right)^{-\frac{1}{2}}$
(3)

In the past, several topological indices were computed utilizing their
conventional mathematical formulation. There are several attempts to look into
a compact method that can recover many topological indices of a particular
class. In this regard, the concept of a general polynomial was developed,
whose values of the necessary topological indices at a given point are
produced by its derivatives, integrals, or a combination of both. For
instance, the distance-based topological indices are recovered by the Hosoya
polynomial [15], whereas the NM-polynomial generates the neighborhood degree
sum-based topological indices [16].

The M-polynomial was introduced by Deutsch and Klažar in [17] to find the
degree-based topological indices.

Definition 1.1 ([17]) The M-polynomial of a graph $\Upsilon$ is defined as:

$M(\Upsilon;x,y)=\displaystyle\sum_{\delta\leq i\leq
j\leq\Delta}m_{i,j}(\Upsilon)x^{i}y^{j}$,

where $\delta=min\\{d_{\Upsilon}(u)|u\in V(\Upsilon)\\}$,
$\Delta=max\\{d_{\Upsilon}(u)|u\in V(\Upsilon)\\}$, and $m_{ij}$ is the number
of edges $uv\in E(\Upsilon)$ such that
$d_{\Upsilon}(u)=i,d_{\Upsilon}(v)=j\,(i,j\geq 1)$.

The M-polynomial-based derivation formulae to compute the different Nirmala
indices are listed in Table 1.

Table 1. Relationship between the M-polynomial and Nirmala indices for a graph
$\Upsilon$.

Sl. No | Topological Index | $f(x,y)$ | Derivation from $M(\Upsilon;x,y)$
---|---|---|---
1 | Nirmala index (N) | $\sqrt{(}x+y)$ | $D_{x}^{\frac{1}{2}}J(M(\Upsilon;x,y))|_{x=1}$
2 | First inverse Nirmala index $(IN_{1})$ | $\sqrt{(}\frac{x+y}{xy})$ | $D_{x}^{\frac{1}{2}}JS_{y}^{\frac{1}{2}}S_{x}^{\frac{1}{2}}(M(\Upsilon;x,y))|_{x=1}$
3 | Second inverse Nirmala index $(IN_{2})$ | $\sqrt{(}\frac{xy}{x+y})$ | $S_{x}^{\frac{1}{2}}JD_{y}^{\frac{1}{2}}D_{x}^{\frac{1}{2}}(M(\Upsilon;x,y))|_{x=1}$

Here,
$D_{x}^{\frac{1}{2}}(h(x,y))=\sqrt{x\cdot\frac{\partial(h(x,y))}{\partial
x}}\cdot\sqrt{h(x,y)}$;
$D_{y}^{\frac{1}{2}}(h(x,y))=\sqrt{y\cdot\frac{\partial(h(x,y))}{\partial
y}}\cdot\sqrt{h(x,y)}$;
$S_{x}^{\frac{1}{2}}(h(x,y))=\sqrt{\displaystyle\int_{0}^{x}\frac{h(t,y)}{t}}dt\cdot\sqrt{h(x,y)}$;
$S_{y}^{\frac{1}{2}}(h(x,y))=\sqrt{\displaystyle\int_{0}^{y}\frac{h(x,t)}{t}}dt\cdot\sqrt{h(x,y)}$;
and $J(h(x,y))=h(x,x)$ are the operators.

We refer readers to [18, 19, 20, 21, 22, 23, 24] for additional information on
degree-based topological indices utilizing the M-polynomial.

Shannon [25] first proposed the concept of entropy, which is a probability
distribution-based measure of the uncertainty or unpredictability of the
information in a system. Next, entropy was used to assess the structural
information of networks, graphs, and chemical structures. In the past few
years, graph entropies have gained increased utility across various fields,
including mathematics, computer science, biology, chemistry, sociology, and
ecology. There are various forms of graph entropy measurements, including
extrinsic and intrinsic measures. These measures relate to the probability
distribution concerning graph invariants, such as edges and vertices.
Interested readers are referred to [26, 27, 28, 29] for additional details on
degree-based graph entropy metrics and their applications.

###  1.1 Entropy of a graph in terms of vertex degree

Let $\Upsilon$ be a simple and connected graph with order $p$ and size $q$.
Then, the graph entropy of a graph $\Upsilon$ as given in [25] is defined as
follows.

$ENT_{\omega}(\Upsilon)=-\displaystyle\sum_{i=1}^{p}\frac{\omega(s_{i})}{\displaystyle\sum_{j=1}^{p}\omega(s_{j})}log\left(\frac{\omega(s_{i})}{\displaystyle\sum_{j=1}^{p}\omega(s_{j})}\right)$
(4)

Here, $\omega$ is a meaningful information function and $s_{i}\in V(\Upsilon)$
for every $i\in\left[1,2,\ldots,p\right]$.

Let $\omega(s_{i})=d_{\Upsilon}(s_{i})$. Then equation (4) reduces to

$\displaystyle ENT_{\omega}(\Upsilon)=$
$\displaystyle-\displaystyle\sum_{i=1}^{p}\frac{d_{\Upsilon}(s_{i})}{\displaystyle\sum_{j=1}^{p}d_{\Upsilon}(s_{j})}log\left(\frac{d_{\Upsilon}(s_{i})}{\displaystyle\sum_{j=1}^{p}d_{\Upsilon}(s_{j})}\right)$
$\displaystyle=$ $\displaystyle
log\left(\displaystyle\sum_{j=1}^{p}d_{\Upsilon}(s_{j})\right)-\frac{1}{\displaystyle\sum_{j=1}^{p}d_{\Upsilon}(s_{j})}\displaystyle\sum_{i=1}^{p}d_{\Upsilon}(s_{i})log(d_{\Upsilon}(s_{i}))$

By the fundamental theorem of graph theory,
$\displaystyle\sum_{i=1}^{p}d_{\Upsilon}(s_{i})=2q$.
Hence,

$\displaystyle ENT_{\omega}(\Upsilon)=$ $\displaystyle
log(2q)-\frac{1}{2q}\displaystyle\sum_{i=1}^{p}d_{\Upsilon}(s_{i})log(d_{\Upsilon}(s_{i}))$
(5)

###  1.2 Entropy of a graph in terms of edge-weight

In 2014, Chen et al. [30] proposed the concept of the entropy of edge-weighted
graphs as follows.

Let $\Upsilon=(V(\Upsilon),E(\Upsilon),\omega(st))$ be an edge-weight graph,
where $V(\Upsilon)$ is a set of vertices, $E(\Upsilon)$ is a set of edges; and
$\omega(st)$ denotes the weight of an edge $st\in E(\Upsilon)$. Then the
entropy of a graph in terms of edge weight is defined as,

$\displaystyle ENT_{\omega}(\Upsilon)=$
$\displaystyle-\displaystyle\sum_{s^{{}^{\prime}}t^{{}^{\prime}}\in
E(\Upsilon)}\frac{\omega(s^{{}^{\prime}}t^{{}^{\prime}})}{\displaystyle\sum_{st\in
E(\Upsilon)}\omega(st)}log\left(\frac{\omega(s^{{}^{\prime}}t^{{}^{\prime}})}{\displaystyle\sum_{st\in
E(\Upsilon)}\omega(st)}\right)$ $\displaystyle=$
$\displaystyle-\displaystyle\sum_{s^{{}^{\prime}}t^{{}^{\prime}}\in
E(\Upsilon)}\frac{\omega(s^{{}^{\prime}}t^{{}^{\prime}})}{\displaystyle\sum_{st\in
E(\Upsilon)}\omega(st)}\left[log(\omega(s^{{}^{\prime}}t^{{}^{\prime}}))-log\left(\displaystyle\sum_{st\in
E(\Upsilon)}\omega(st)\right)\right]$ $\displaystyle=$ $\displaystyle
log\left(\displaystyle\sum_{st\in
E(\Upsilon)}\omega(st)\right)-\displaystyle\sum_{s^{{}^{\prime}}t^{{}^{\prime}}\in
E(\Upsilon)}\frac{\omega(s^{{}^{\prime}}t^{{}^{\prime}})}{\displaystyle\sum_{st\in
E(\Upsilon)}\omega(st)}log(\omega(s^{{}^{\prime}}t^{{}^{\prime}}))$

Hence,

$\displaystyle ENT_{\omega}(\Upsilon)=$ $\displaystyle
log\left(\displaystyle\sum_{st\in
E(\Upsilon)}\omega(st)\right)-\frac{1}{\left(\displaystyle\sum_{st\in
E(\Upsilon)}\omega(st)\right)}\displaystyle\sum_{s^{{}^{\prime}}t^{{}^{\prime}}\in
E(\Upsilon)}\omega(s^{{}^{\prime}}t^{{}^{\prime}})log(\omega(s^{{}^{\prime}}t^{{}^{\prime}}))$
(6)

In 2023, Virendra Kumar et al. [31] introduced the notion of the Nirmala
indices-based entropy by considering the meaningful information function
$\omega$ as a function associated with the definitions of the Nirmala indices
as given in equations (1-3).

Nirmala entropy: Let $\omega(st)=\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}$. Then
from the definition of the Nirmala index as given in Equation (1), we have

$\displaystyle\sum_{st\in E(\Upsilon)}\omega(st)=\displaystyle\sum_{st\in
E(\Upsilon)}\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}=N(\Upsilon)$

Hence, using equation (6), the Nirmala entropy of a graph $\Upsilon$ is given
by

$ENT_{N}(\Upsilon)=log(N(\Upsilon))-\frac{1}{N(\Upsilon)}\displaystyle\sum_{st\in
E(\Upsilon)}\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}\times
log(\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)})$ (7)

First inverse Nirmala entropy: Let
$\omega(st)=\sqrt{\frac{1}{d_{\Upsilon}(s)}+\frac{1}{d_{\Upsilon}(t)}}$. Then
from the definition of the first inverse Nirmala index as given in Equation
(2), we have,

$\displaystyle\sum_{st\in E(\Upsilon)}\omega(st)=\displaystyle\sum_{st\in
E(\Upsilon)}\sqrt{\frac{1}{d_{\Upsilon}(s)}+\frac{1}{d_{\Upsilon}(t)}}=IN_{1}(\Upsilon)$

Hence, using equation (6), the first inverse Nirmala entropy of a graph
$\Upsilon$ is given by

$ENT_{IN_{1}}(\Upsilon)=log(IN_{1}(\Upsilon))-\frac{1}{IN_{1}(\Upsilon)}\displaystyle\sum_{st\in
E(\Upsilon)}\sqrt{\frac{1}{d_{\Upsilon}(s)}+\frac{1}{d_{\Upsilon}(t)}}\times
log\left(\sqrt{\frac{1}{d_{\Upsilon}(s)}+\frac{1}{d_{\Upsilon}(t)}}\right)$
(8)

Second inverse Nirmala entropy: Let
$\omega(st)=\frac{\sqrt{d_{\Upsilon}(s)\cdot
d_{\Upsilon}(t)}}{\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}}$. Then from the
definition of the second inverse Nirmala index as given in Equation (3), we
have,

$\displaystyle\sum_{st\in E(\Upsilon)}\omega(st)=\displaystyle\sum_{st\in
E(\Upsilon)}\frac{\sqrt{d_{\Upsilon}(s)\cdot
d_{\Upsilon}(t)}}{\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}}=IN_{2}(\Upsilon)$

Hence, using equation (6), the second inverse Nirmala entropy of a graph
$\Upsilon$ is given by

$ENT_{IN_{2}}(\Upsilon)=log(IN_{2}(\Upsilon))-\frac{1}{IN_{2}(\Upsilon)}\displaystyle\sum_{st\in
E(\Upsilon)}\frac{\sqrt{d_{\Upsilon}(s)\cdot
d_{\Upsilon}(t)}}{\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}}\times
log\left(\frac{\sqrt{d_{\Upsilon}(s)\cdot
d_{\Upsilon}(t)}}{\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}}\right)$ (9)

###  1.3 Methodology

In this paper, the complicated structural details of ruthenium bipyridine are
obtained. Further, we use Shannon’s entropy measures by employing novel
information functions derived from various Nirmala indices definitions. We
conduct a comprehensive mathematical and computational exploration of these
measures within the complex structure of ruthenium bipyridine. The rest of the
paper is organized as follows: Section 2 discusses the 2D and 3D structure of
ruthenium bipyridine. In Section 3, we compute the Nirmala indices of
ruthenium bipyridine using its M-polynomial, which allows us to calculate the
Nirmala indices-based entropy measures of ruthenium bipyridine $RB_{n}$.
Section 4 compares the Nirmala indices and their associated entropy measures
through numerical data. Section 5 deals with a regression model to illustrate
how the estimated Nirmala indices and associated entropy values fit the curve.
Lastly, a discussion and a conclusion are presented in sections 6 and 7,
respectively.

##  2 Complex structure of ruthenium bipyridine

Ruthenium (II) tris (bipyridine) is one of the compounds that has had the
largest impact on the advancement of chemistry. The solvation of ruthenium
dyes [32] in various solvents and electrolytes has received a lot of interest,
particularly in Dye-Sensitive Solar Cells (DSSCs). The emission and absorption
spectra of ruthenium tris (bipyridine) complexes [33] adorned with various
electron-rich and redox-active amine substituents exhibit reversible changes
in the deep red to near-infrared range when low electrochemical potentials are
applied.

Multiple targets and pathways are involved in the anticancer actions of
ruthenium complexes. Comparing the ruthenium complexes to other metal-based
molecules under study, they demonstrated impressive anticancer activity.
Because of their great selectivity and low toxicity, organometallic ruthenium
(II) complexes have been thoroughly investigated for possible biological
applications. Several ruthenium compounds that effectively increased
selectivity to various processes by strictly controlling the coordination
sphere have been the subject of numerous investigations [34, 35]. For a
ruthenium core, a range of molecular structures can be readily constructed
because of ruthenium’s versatility in oxidation states and its easy
interaction with ligands. The readers are referred to [36] for further
information and an analysis of the properties and uses of ruthenium.

This paper aims to support scientists in their exploration of ruthenium
bipyridine’s physical properties. Since ruthenium is among the rarest metals,
conducting trials would be too expensive to influence the price. Therefore,
cost management can be achieved by mathematical chemistry. It is far more time
and money-consuming for a chemist to do lab tests to investigate the
properties of the chemical than it is for a mathematician to research the
chemical’s structure, which is free and allows for quicker discovery of the
qualities. Figure 1 shows the 2D and 3D structure of complex ruthenium
bipyridine [37].

Figure 1. 2D and 3D structure of complex ruthenium bipyridine.

###  2.1 Complex structure of ruthenium bipyridine graph

By incorporating triphenylamine into the coordination molecule of ruthenium
tris bipyridine, the ruthenium bipyridine graph’s complex structure is formed,
with each atom—both metal and organic ligand—representing the graph’s vertices
and bonds acting as its edges. The molecular graph of the complex structure of
ruthenium bipyridine $RB_{n}$ is shown in Figure 2, where $n$ is the number of
triphenylamine polymers around the ruthenium tris bipyridine complexes. The
order and size of $RB_{n}$ are

$|V(RB_{n})=6\left[6(2^{n})+6(2^{n}-1)\right]+\left[6(2^{n-2})+6(2^{n-2}-1)\right]+1$
and

$|E(RB_{n})=6\left[6(2^{n+1})+6(2^{n+1}-1)\right]+3\left[6(2^{n})+6(2^{n}-1)\right]+9$,
respectively.

Figure 2. The complex structure of ruthenium bipyridine graph $RB_{n}$.

Motivated by the studies described in [31, 37], we aim to calculate the
Nirmala index, as well as the first and second inverse Nirmala index, for the
complex structure of ruthenium bipyridine $RB_{n}$ using M-polynomial.
Furthermore, entropy measures for the complex structure of ruthenium
bipyridine are computed using Shannon’s entropy model.

##  3 Main results

In this section, we first find the M-polynomial of ruthenium bipyridine
$RB_{n}$. Then, we compute the Nirmala index, and first and second inverse
Nirmala indices using its M-polynomial. For the complex structure of ruthenium
bipyridine graph $RB_{n}$, the size of $RB_{n}$ is

$|E(RB_{n})|=6\left[6(2^{n+1})+6(2^{n+1}-1)\right]+3\left[6(2^{n})+6(2^{n}-1)\right]+9$.

The edge set partitions of the complex structure of ruthenium bipyridine graph
$RB_{n}$ are shown in Table 2.

Sl. No | Edge set | $(d_{\Upsilon}(r),d_{\Upsilon}(s))$ | Number of repetitions
---|---|---|---
1 | $E_{1}$ | (2,2) | $18\cdot 2^{n+2}-18$
2 | $E_{2}$ | (2,3) | $36\cdot 2^{n+1}-24$
3 | $E_{3}$ | (3,3) | $9\cdot 2^{n+2}-9$
4 | $E_{4}$ | (3,6) | $6$

Table 2. Edge set partition of $RB_{n}$ according to degrees of end vertices
of an edge.

It is easy to observe that

$|E(RB_{n})|=\left[6(2^{n+1})+6(2^{n+1}-1)\right]+3\left[6(2^{n})+6(2^{n}-1)\right]+9=72\cdot
2^{n+1}+36\cdot 2^{n}-45.$ $\displaystyle\sum_{i=1}^{4}|E_{i}|=27\cdot
2^{n+2}+36\cdot 2^{n+1}-45.$

Using the laws of exponents, one can easily prove that

$72\cdot 2^{n+1}+36\cdot 2^{n}-45=180\cdot 2^{n}-45.$ $27\cdot 2^{n+2}+36\cdot
2^{n+1}-45=180\cdot 2^{n}-45.$

Therefore, $|E(RB_{n})|=\displaystyle\sum_{i=1}^{4}|E_{i}|$.

###  3.1 Nirmala indices of $RB_{n}$

We now find the M-polynomial of $RB_{n}$ as follows:

###### Theorem 3.1

Let $\Upsilon=RB_{n}$ be the molecular graph for the complex structure of
ruthenium bipyridine. Then the M-polynomial of $\Upsilon$ is
$M(\Upsilon;x,y)=(18\cdot 2^{n+2}-18)x^{2}y^{2}+(36\cdot
2^{n+1}-24)x^{2}y^{3}+(9\cdot 2^{n+2}-9)x^{3}y^{3}+6x^{3}y^{6}$.

Proof. Let $\Upsilon=RB_{n}$ be the molecular graph for the complex structure
of ruthenium bipyridine.
From Table 2, $\displaystyle\sum_{i=1}^{4}|E_{i}|=27\cdot 2^{n+2}+36\cdot
2^{n+1}-45$.

Since each vertex of $\Upsilon$ is of degree either $2$ or $3$ or $6$, the
partitions of edge set $E(\Upsilon)$ are:
$E_{1}(\Upsilon):=\\{e=ab\in
E(\Upsilon):d_{\Upsilon}(a)=2,d_{\Upsilon}(b)=2\\}$;
$E_{2}(\Upsilon)):=\\{e=ab\in
E(\Upsilon):d_{\Upsilon}(a)=2,d_{\Upsilon}(b)=3\\}$;
$E_{3}(\Upsilon):=\\{e=ab\in
E(\Upsilon):d_{\Upsilon}(a)=3,d_{\Upsilon}(b)=3\\}$;
$E_{4}(\Upsilon):=\\{e=ab\in
E(\Upsilon):d_{\Upsilon}(a)=3,d_{\Upsilon}(b)=6\\}$.

Clearly, $|E_{1}(\Upsilon)|=18\cdot 2^{n+2}-18;|E_{2}(\Upsilon)|=36\cdot
2^{n+1}-24;|E_{3}(\Upsilon)|=9\cdot 2^{n+2}-9;\\\ |E_{4}(\Upsilon)|=6$.
Therefore,

$\displaystyle M(\Upsilon;x,y)=$ $\displaystyle\displaystyle\sum_{\delta\leq
i\leq
j\leq\Delta}m_{i,j}(\Upsilon)x^{i}y^{j}=m_{22}(\Upsilon)x^{2}y^{2}+m_{23}(\Upsilon)x^{2}y^{3}+m_{33}(\Upsilon)x^{3}y^{6}+m_{36}(\Upsilon)x^{3}y^{6}$
$\displaystyle=$ $\displaystyle(18\cdot 2^{n+2}-18)x^{2}y^{2}+(36\cdot
2^{n+1}-24)x^{2}y^{3}+(9\cdot 2^{n+2}-9)x^{3}y^{3}+6x^{3}y^{6}.$

Now we evaluate the Nirmala indices of $RB_{n}$ with the help of its
M-polynomial.

###### Theorem 3.2

Let $\Upsilon=RB_{n}$ be the molecular graph for the complex structure of
ruthenium bipyridine. Then the Nirmala indices of $\Upsilon$ are:

a. $N(\Upsilon)=(36\cdot 2^{n+2}-36)+\sqrt{5}(36\cdot
2^{n+1}-24)+\sqrt{6}(9\cdot 2^{n+2}-9)+18$,

b. $IN_{1}(\Upsilon)=(18\cdot 2^{n+2}-18)+\frac{\sqrt{5}}{\sqrt{6}}(36\cdot
2^{n+1}-24)+\frac{\sqrt{6}}{3}(9\cdot 2^{n+2}-9)+\frac{6}{\sqrt{2}}$,

c. $IN_{2}(\Upsilon)=(18\cdot 2^{n+2}-18)+\frac{\sqrt{6}}{\sqrt{5}}(36\cdot
2^{n+1}-24)+\frac{27}{\sqrt{6}}(2^{n+2}-1)+6\sqrt{2}$.

Proof. Let $\Upsilon=RB_{n}$ be the molecular graph for the complex structure
of ruthenium bipyridine. From Theorem 3.1, the M-polynomial of $\Upsilon$ is

$M(\Upsilon;x,y)=(18\cdot 2^{n+2}-18)x^{2}y^{2}+(36\cdot
2^{n+1}-24)x^{2}y^{3}+(9\cdot 2^{n+2}-9)x^{3}y^{3}+6x^{3}y^{6}$.

From Table 2, we have

$\displaystyle(i)\,D_{x}^{\frac{1}{2}}J(M(\Upsilon;x,y))$
$\displaystyle=D_{x}^{\frac{1}{2}}J\left[(18\cdot
2^{n+2}-18)x^{2}y^{2}+(36\cdot 2^{n+1}-24)x^{2}y^{3}+(9\cdot
2^{n+2}-9)x^{3}y^{3}+6x^{3}y^{6}\right]$
$\displaystyle=D_{x}^{\frac{1}{2}}\left[(18\cdot 2^{n+2}-18)x^{4}+(36\cdot
2^{n+1}-24)x^{5}+(9\cdot 2^{n+2}-9)x^{6}+6x^{9}\right]$
$\displaystyle=2(18\cdot 2^{n+2}-18)x^{4}+\sqrt{5}(36\cdot
2^{n+1}-24)x^{5}+\sqrt{6}(9\cdot 2^{n+2}-9)x^{6}+18x^{9}$
$\displaystyle=(36\cdot 2^{n+2}-36)x^{4}+\sqrt{5}(36\cdot
2^{n+1}-24)x^{5}+\sqrt{6}(9\cdot 2^{n+2}-9)x^{6}+18x^{9}.$
$\displaystyle(ii)\,D_{x}^{\frac{1}{2}}JS_{y}^{\frac{1}{2}}S_{x}^{\frac{1}{2}}(M(\Upsilon;x,y))$
$\displaystyle=$ $\displaystyle
D_{x}^{\frac{1}{2}}JS_{y}^{\frac{1}{2}}S_{x}^{\frac{1}{2}}\left[(18\cdot
2^{n+2}-18)x^{2}y^{2}+(36\cdot 2^{n+1}-24)x^{2}y^{3}+(9\cdot
2^{n+2}-9)x^{3}y^{3}+6x^{3}y^{6}\right]$ $\displaystyle=$ $\displaystyle
D_{x}^{\frac{1}{2}}JS_{y}^{\frac{1}{2}}\left[\frac{1}{\sqrt{2}}(18\cdot
2^{n+2}-18)x^{2}y^{2}+\frac{1}{\sqrt{2}}(36\cdot
2^{n+1}-24)x^{2}y^{3}+\frac{1}{\sqrt{3}}(9\cdot
2^{n+2}-9)x^{3}y^{3}+\frac{1}{\sqrt{2}}(6x^{3}y^{6})\right]$
$\displaystyle=D_{x}^{\frac{1}{2}}J\left[\frac{1}{2}(18\cdot
2^{n+2}-18)x^{2}y^{2}+\frac{1}{\sqrt{6}}(36\cdot
2^{n+1}-24)x^{2}y^{3}+\frac{1}{3}(9\cdot
2^{n+2}-9)x^{3}y^{3}+\frac{1}{\sqrt{18}}(6x^{3}y^{6})\right]$
$\displaystyle=D_{x}^{\frac{1}{2}}\left[\frac{1}{2}(18\cdot
2^{n+2}-18)x^{4}+\frac{1}{\sqrt{6}}(36\cdot
2^{n+1}-24)x^{5}+\frac{1}{3}(9\cdot
2^{n+2}-9)x^{6}+\frac{1}{\sqrt{18}}(6x^{9})\right]$ $\displaystyle=(18\cdot
2^{n+2}-18)x^{4}+\frac{\sqrt{5}}{\sqrt{6}}(36\cdot
2^{n+1}-24)x^{5}+\frac{\sqrt{6}}{3}(9\cdot
2^{n+2}-9)x^{6}+\frac{6}{\sqrt{2}}x^{9}.$

$\displaystyle(iii)\,$ $\displaystyle
S_{x}^{\frac{1}{2}}JD_{y}^{\frac{1}{2}}D_{x}^{\frac{1}{2}}(M(\Upsilon;x,y))$
$\displaystyle=S_{x}^{\frac{1}{2}}JD_{y}^{\frac{1}{2}}D_{x}^{\frac{1}{2}}\left[(18\cdot
2^{n+2}-18)x^{2}y^{2}+(36\cdot 2^{n+1}-24)x^{2}y^{3}+(9\cdot
2^{n+2}-9)x^{3}y^{3}+6x^{3}y^{6}\right]$
$\displaystyle=S_{x}^{\frac{1}{2}}JD_{y}^{\frac{1}{2}}\left[\sqrt{2}(18\cdot
2^{n+2}-18)x^{2}y^{2}+\sqrt{2}(36\cdot 2^{n+1}-24)x^{2}y^{3}+\sqrt{3}(9\cdot
2^{n+2}-9)x^{3}y^{3}+6\sqrt{3}x^{3}y^{6}\right]$
$\displaystyle=S_{x}^{\frac{1}{2}}J\left[2(18\cdot
2^{n+2}-18)x^{2}y^{2}+\sqrt{6}(36\cdot 2^{n+1}-24)x^{2}y^{3}+3(9\cdot
2^{n+2}-9)x^{3}y^{3}+6\sqrt{18}x^{3}y^{6}\right]$
$\displaystyle=S_{x}^{\frac{1}{2}}\left[(36\cdot
2^{n+2}-36)x^{4}+\sqrt{6}(36\cdot 2^{n+1}-24)x^{5}+(27\cdot
2^{n+2}-27)x^{6}+6\sqrt{18}x^{9}\right]$ $\displaystyle=\frac{1}{2}(36\cdot
2^{n+2}-36)x^{4}+\frac{\sqrt{6}}{\sqrt{5}}(36\cdot
2^{n+1}-24)x^{5}+\frac{1}{\sqrt{6}}(27\cdot
2^{n+2}-27)x^{6}+\frac{6}{\sqrt{9}}\sqrt{18}x^{9}$ $\displaystyle=(18\cdot
2^{n+2}-18)x^{4}+\frac{\sqrt{6}}{\sqrt{5}}(36\cdot
2^{n+1}-24)x^{5}+\frac{27}{\sqrt{6}}(2^{n+2}-1)x^{6}+6\sqrt{2}x^{9}.$

Hence, the Nirmala indices of $\Upsilon$ are given by

$\displaystyle(a)\,N(\Upsilon)=$ $\displaystyle
D_{x}^{\frac{1}{2}}J(M(\Upsilon;x,y))|_{x=1}$ $\displaystyle=(36\cdot
2^{n+2}-36)+\sqrt{5}(36\cdot 2^{n+1}-24)+\sqrt{6}(9\cdot 2^{n+2}-9)+18.$
$\displaystyle(b)\,IN_{1}(\Upsilon)=$ $\displaystyle
D_{x}^{\frac{1}{2}}JS_{y}^{\frac{1}{2}}S_{x}^{\frac{1}{2}}(M(\Upsilon;x,y))|_{x=1}$
$\displaystyle=(18\cdot 2^{n+2}-18)+\frac{\sqrt{5}}{\sqrt{6}}(36\cdot
2^{n+1}-24)+\frac{\sqrt{6}}{3}(9\cdot 2^{n+2}-9)+\frac{6}{\sqrt{2}}.$
$\displaystyle(c)\,IN_{2}(\Upsilon)=$ $\displaystyle
S_{x}^{\frac{1}{2}}JD_{y}^{\frac{1}{2}}D_{x}^{\frac{1}{2}}(M(\Upsilon;x,y))|_{x=1}$
$\displaystyle=(18\cdot 2^{n+2}-18)+\frac{\sqrt{6}}{\sqrt{5}}(36\cdot
2^{n+1}-24)+\frac{27}{\sqrt{6}}(2^{n+2}-1)+6\sqrt{2}.$

###  3.2 Entropy measures of the complex structure of ruthenium bipyridine
$RB_{n}$

We proceed with the computation of graph entropy metrics for the complex
structure of ruthenium bipyridine $RB_{n}$ using Shannon’s entropy. Initially,
we assess the degree-based graph entropy expression. Subsequently, we
determine the mathematical formulations for entropy measures based on Nirmala
indices, utilizing the previously derived expressions of the Nirmala indices.

Nirmala entropy of $RB_{n}$:
From Theorem 3.2, the Nirmala index of $\Upsilon$ is given by

$N(\Upsilon)=(36\cdot 2^{n+2}-36)+\sqrt{5}(36\cdot 2^{n+1}-24)+\sqrt{6}(9\cdot
2^{n+2}-9)+18.$

From Table 2 and Equation (7), we have

$\displaystyle ENT_{N}(\Upsilon)=$ $\displaystyle
log(N(\Upsilon))-\frac{1}{N(\Upsilon)}\displaystyle\sum_{st\in
E(\Upsilon)}\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}\times
log(\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)})$ $\displaystyle=$ $\displaystyle
log(N(\Upsilon))-\frac{1}{N(\Upsilon)}\left[\displaystyle\sum_{i=1}^{4}\displaystyle\sum_{st\in
E_{i}(\Upsilon)}\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}\times
log(\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)})\right]$
$\displaystyle=log(N(\Upsilon))-\frac{1}{N(\Upsilon)}\left[(18\cdot
2^{n+2}-18)\cdot\sqrt{2+2}\cdot log(\sqrt{2+2})\right]$
$\displaystyle-\frac{1}{N(\Upsilon)}\left[(36\cdot
2^{n+1}-24)\cdot\sqrt{2+3}\cdot log(\sqrt{2+3})\right]$ $\displaystyle-$
$\displaystyle\frac{1}{N(\Upsilon)}\left[(9\cdot
2^{n+2}-9)\cdot\sqrt{3+3}\cdot log(\sqrt{3+3})+(6\cdot\sqrt{3+6}\cdot
log(\sqrt{3+6})\right]$
$\displaystyle=log(N(\Upsilon))-\frac{1}{N(\Upsilon)}\left[(18\cdot
2^{n+2}-18)\cdot 2\cdot log(\sqrt{2+2})\right]$
$\displaystyle-\frac{1}{N(\Upsilon)}\left[(36\cdot
2^{n+1}-24)\cdot\sqrt{5}\cdot log(\sqrt{5})\right]$ $\displaystyle-$
$\displaystyle\frac{1}{N(\Upsilon)}\left[(9\cdot 2^{n+2}-9)\cdot\sqrt{6}\cdot
log(\sqrt{6})+6\cdot 3\cdot log(3)\right]$
$\displaystyle=log(N(\Upsilon))-\frac{1}{N(\Upsilon)}\left[(18\cdot
2^{n+2}-18)\cdot 2\cdot log(\sqrt{2+2})\right]$
$\displaystyle-\frac{1}{N(\Upsilon)}\left[(36\cdot
2^{n+1}-24)\cdot\sqrt{5}\cdot log(\sqrt{5})\right]$ $\displaystyle-$
$\displaystyle\frac{1}{N(\Upsilon)}\left[(9\cdot 2^{n+2}-9)\cdot\sqrt{6}\cdot
log(\sqrt{6})+18\cdot log(3)\right]$

Finally, we get the desired formulation of the Nirmala entropy for $RB_{n}$ by
substituting the value of $N(\Upsilon)$ into the previous expression.

First inverse Nirmala entropy of $RB_{n}$:
From Theorem 3.2, the first inverse Nirmala index of $\Upsilon$ is given by

$\displaystyle IN_{1}(\Upsilon)=$ $\displaystyle(18\cdot
2^{n+2}-18)+\frac{\sqrt{5}}{\sqrt{6}}(36\cdot
2^{n+1}-24)+\frac{\sqrt{6}}{3}(9\cdot 2^{n+2}-9)+\frac{6}{\sqrt{2}}.$

From Table 2 and Equation (8), we have

$\displaystyle ENT_{IN_{1}}(\Upsilon)=$ $\displaystyle
log(IN_{1}(\Upsilon))-\frac{1}{IN_{1}(\Upsilon)}\displaystyle\sum_{st\in
E(\Upsilon)}\sqrt{\frac{1}{d_{\Upsilon}(s)}+\frac{1}{d_{\Upsilon}(t)}}\times
log\left(\sqrt{\frac{1}{d_{\Upsilon}(s)}+\frac{1}{d_{\Upsilon}(t)}}\right)$
$\displaystyle=$ $\displaystyle
log(IN_{1}(\Upsilon))-\frac{1}{IN_{1}(\Upsilon)}\left[\displaystyle\sum_{i=1}^{4}\displaystyle\sum_{st\in
E_{i}(\Upsilon)}\sqrt{\frac{1}{d_{\Upsilon}(s)}+\frac{1}{d_{\Upsilon}(t)}}\times
log\left(\sqrt{\frac{1}{d_{\Upsilon}(s)}+\frac{1}{d_{\Upsilon}(t)}}\right)\right]$
$\displaystyle=$ $\displaystyle
log(IN_{1}(\Upsilon))-\frac{1}{IN_{1}(\Upsilon)}\left[(18\cdot
2^{n+2}-18)\cdot\sqrt{\frac{1}{2}+\frac{1}{2}}\cdot
log\left(\sqrt{\frac{1}{2}+\frac{1}{2}}\right)\right]$ $\displaystyle-$
$\displaystyle\frac{1}{IN_{1}(\Upsilon)}\left[(36\cdot
2^{n+1}-24)\cdot\sqrt{\frac{1}{2}+\frac{1}{3}}\cdot
log\left(\sqrt{\frac{1}{2}+\frac{1}{3}}\right)\right]$ $\displaystyle-$
$\displaystyle\frac{1}{IN_{1}(\Upsilon)}\left[(9\cdot
2^{n+2}-9)\cdot\sqrt{\frac{1}{3}+\frac{1}{3}}\cdot
log\left(\sqrt{\frac{1}{3}+\frac{1}{3}}\right)\right]$ $\displaystyle-$
$\displaystyle\frac{1}{IN_{1}(\Upsilon)}\left[6\cdot\sqrt{\frac{1}{3}+\frac{1}{6}}\cdot
log\left(\sqrt{\frac{1}{3}+\frac{1}{6}}\right)\right]$

Since $log(1)=0$,

$\displaystyle ENT_{IN_{1}}(\Upsilon)=$ $\displaystyle
log(IN_{1}(\Upsilon))-\frac{1}{IN_{1}(\Upsilon)}\left[(36\cdot
2^{n+1}-24)\cdot\sqrt{\frac{5}{6}}\cdot log\left(\frac{5}{6}\right)\right]$
$\displaystyle-$ $\displaystyle\frac{1}{IN_{1}(\Upsilon)}\left[(9\cdot
2^{n+2}-9)\cdot\sqrt{\frac{2}{3}}\cdot
log\left(\sqrt{\frac{2}{3}}\right)+6\cdot\sqrt{\frac{1}{2}}\cdot
log\left(\sqrt{\frac{1}{2}}\right)\right]$

Finally, by substituting the value of $IN_{1}(\Upsilon)$ into the preceding
expression, we obtain the desired formulation of the first inverse Nirmala
entropy for $RB_{n}$.

Second inverse Nirmala entropy of $RB_{n}$:
From Theorem 3.2, the second inverse Nirmala index of $\Upsilon$ is given by

$\displaystyle IN_{2}(\Upsilon)=$ $\displaystyle(18\cdot
2^{n+2}-18)+\frac{\sqrt{6}}{\sqrt{5}}(36\cdot
2^{n+1}-24)+\frac{27}{\sqrt{6}}(2^{n+2}-1)+6\sqrt{2}.$

From Table 3 and Equation (9), we have

$\displaystyle ENT_{IN_{2}}(\Upsilon)=$ $\displaystyle
log(IN_{2}(\Upsilon))-\frac{1}{IN_{2}(\Upsilon)}\displaystyle\sum_{st\in
E(\Upsilon)}\frac{\sqrt{d_{\Upsilon}(s)\cdot
d_{\Upsilon}(t)}}{\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}}\times
log\left(\frac{\sqrt{d_{\Upsilon}(s)\cdot
d_{\Upsilon}(t)}}{\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}}\right)$
$\displaystyle=$ $\displaystyle
log(IN_{2}(\Upsilon))-\frac{1}{IN_{2}(\Upsilon)}\left[\displaystyle\sum_{i=1}^{4}\displaystyle\sum_{st\in
E_{i}(\Upsilon)}\frac{\sqrt{d_{\Upsilon}(s)\cdot
d_{\Upsilon}(t)}}{\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}}\times
log\left(\frac{\sqrt{d_{\Upsilon}(s)\cdot
d_{\Upsilon}(t)}}{\sqrt{d_{\Upsilon}(s)+d_{\Upsilon}(t)}}\right)\right]$
$\displaystyle=$ $\displaystyle
log(IN_{2}(\Upsilon))-\frac{1}{IN_{2}(\Upsilon)}\left[(18\cdot
2^{n+2}-18)\cdot\frac{\sqrt{4}}{\sqrt{4}}\cdot
log\left(\frac{\sqrt{4}}{\sqrt{4}}\right)\right]$
$\displaystyle-\frac{1}{IN_{2}(\Upsilon)}\left[(36\cdot
2^{n+1}-24)\cdot\frac{\sqrt{6}}{\sqrt{5}}\cdot
log\left(\frac{\sqrt{6}}{\sqrt{5}}\right)\right]$ $\displaystyle-$
$\displaystyle\frac{1}{IN_{2}(\Upsilon)}\left[(9\cdot
2^{n+2}-9)\cdot\frac{\sqrt{9}}{\sqrt{6}}\cdot
log\left(\frac{\sqrt{9}}{\sqrt{6}}\right)+6\cdot\frac{\sqrt{18}}{\sqrt{9}}\cdot
log\left(\frac{\sqrt{18}}{\sqrt{9}}\right)\right]$

Since $log(1)=0$,

$\displaystyle ENT_{IN_{2}}(\Upsilon)=$ $\displaystyle
log(IN_{2}(\Upsilon))-\frac{1}{IN_{2}(\Upsilon)}\left[(36\cdot
2^{n+1}-24)\cdot\frac{\sqrt{6}}{\sqrt{5}}\cdot
log\left(\frac{\sqrt{6}}{\sqrt{5}}\right)\right]$ $\displaystyle-$
$\displaystyle\frac{1}{IN_{2}(\Upsilon)}\left[(9\cdot
2^{n+2}-9)\cdot\frac{3}{\sqrt{6}}\cdot
log\left(\frac{3}{\sqrt{6}}\right)+6\cdot\sqrt{2}\cdot
log\left(\sqrt{2}\right)\right]$

The second inverse Nirmala entropy for $RB_{n}$ can finally be expressed as
desired by substituting the value of $IN_{2}(\Upsilon)$ in the previous
expression.

##  4 Comparison through numerical demonstrations

Numerous scientific fields, such as computer science, information theory,
chemistry, biological therapies, and pharmacology, frequently use graph
entropy measurements. To accurately quantify these molecular properties,
scientists working in these disciplines rely on numerical calculation
representation. This section uses numerical computation to compare the Nirmala
indices and the accompanying entropy measures. The numerical computation of
the Nirmala indices and accompanying entropy measures for $RB_{n}$, where
$1\leq n\leq 25$ is presented in Table 3.
Table 3. Calculated values of the Nirmala indices and their associated entropy
measures of $RB_{n}$, where $1\leq n\leq 25$.

[n] |  N |  $IN_{1}$ |  $IN_{2}$ |  $ENT_{N}$ |  $ENT_{IN_{1}}$ |  $ENT_{IN_{2}}$
---|---|---|---|---|---|---
[1] |  692.64 |  291.22 |  343.09 |  5.7484 |  5.7292 |  5.7489
[2] |  1479.00 |  625.46 |  733.02 |  6.5111 |  6.5117 |  6.5114
[3] |  3051.71 |  1293.94 |  1512.90 |  7.2373 |  7.2377 |  7.2375
[4] |  6197.14 |  2630.91 |  3072.60 |  7.9466 |  7.9469 |  7.9468
[5] |  12488 |  5304.84 |  6192 |  8.6477 |  8.6480 |  8.6479
[6] |  25069.71 |  10652.70 |  12430.79 |  9.3448 |  9.3451 |  9.3450
[7] |  50233.14 |  21348.41 |  24908.40 |  10.0400 |  10.0402 |  10.0401
[8] |  100559.99 |  42739.85 |  49863.66 |  10.7341 |  10.7344 |  10.7342
[9] |  201213.69 |  85500.72 |  99774.15 |  11.4277 |  11.4280 |  11.4279
[10] |  402521.09 |  171088.46 |  199595.14 |  12.1211 |  12.1214 |  12.1213
[11] |  805135.90 |  342219.94 |  399237.11 |  12.8144 |  12.8147 |  12.8145
[12] |  1610365.52 |  684482.89 |  798521.06 |  13.5076 |  13.5079 |  13.5077
[13] |  3220824.76 |  1369008.80 |  1597088.95 |  14.2008 |  14.2011 |  14.2009
[14] |  6441743.24 |  2738060.63 |  3194224.72 |  14.8940 |  14.8942 |  14.8941
[15] |  12883580.20 |  5476164.27 |  6388496.28 |  15.5871 |  15.5874 |  15.5872
[16] |  25767254.11 |  10952371.56 |  12777039.40 |  16.2803 |  16.2805 |  16.2804
[17] |  51534601.93 |  21904786.14 |  25554125.63 |  16.9734 |  16.9737 |  16.9735
[18] |  103069297.57 |  43809615.30 |  51108298.10 |  17.6666 |  17.6668 |  17.6667
[19] |  206138688.86 |  87619273.63 |  102216643.02 |  18.3597 |  18.3600 |  18.3598
[20] |  412277471.44 |  175238590.27 |  204433332.88 |  19.0529 |  19.0531 |  19.0530
[21] |  82455036.60 |  350477223.57 |  408866712.60 |  19.7460 |  19.7463 |  19.7461
[22] |  1649110166.91 |  700954490.15 |  817733472.03 |  20.4391 |  20.4394 |  20.4393
[23] |  3298220427.54 |  1401909023.33 |  1635466990.89 |  21.1323 |  21.1326 |  21.1324
[24] |  6596440948.79 |  2803818089.67 |  3270934028.62 |  21.8254 |  21.8257 |  21.8256
[25] |  13192881991.29 |  5607636222.37 |  6541868104.06 |  22.5186 |  22.5189 |  22.5187

From Table 3, the following two remarks are possible.
Remark 1: The Nirmala indices and associated entropy measures of the complex
structure of ruthenium bipyridine increase as the values of $n$ increase.
Remark 2: For complex structure of ruthenium bipyridine graph
$\Upsilon=RB_{n}$, we have the following inequality relationships:

$IN_{1}(\Upsilon)<IN_{2}(\Upsilon)<N(\Upsilon)$

$ENT_{N}(\Upsilon)\approx ENT_{IN_{1}}(\Upsilon)\approx
ENT_{IN_{2}}(\Upsilon)$.

###  4.1 The logarithmic regression model

To investigate the relationship between the dependent variable and one or more
predictor variables in our dataset, we apply logarithmic regression analysis.
Logarithmic regression is a nonlinear technique that modifies the dependent or
predictor variables using logarithmic functions. The following is the equation
for the logarithmic regression model.

$y=a*log(x)+b,$

where the response variable is $y$ and the predictor variable is $x$. The
regression coefficients $a$ and $b$ represent the relation between $x$ and
$y$.

Here, for $1\leq n\leq 25$, we examine the link between the Nirmala indices
and entropy metrics of the complex structure of ruthenium bipyridine $RB_{n}$
using a logarithmic regression analysis. The study utilized many statistical
measurements, such as the squared correlation coefficient ($R^{2}$), the sum
of square error (SSE), adjusted squared correlation coefficient (Adj. R-sq),
root mean square error (RMSE), and squared correlation coefficient ($R^{2}$).
A low RMSE value (nearer to 0) implies that the model performs well, whereas a
larger $R^{2}$ value (near 1) suggests that the regression line fits the data
better. In this instance, obtaining a larger $R^{2}$ value is our main goal.

The statistics of curve fitting of the Nirmala indices versus Nirmala entropy
measures for the complex structure of ruthenium bipyridine $RB_{n}$ using the
logarithmic regression are shown in Table 4.

Model |  $R^{2}$ |  SSE |  Adj. R-sq |  RMSE
---|---|---|---|---
$ENT_{N}=1.0129*log(N)-0.8842$ |  0.9920 |  4.9922 |  0.9917 |  0.4658
$ENT_{IN_{1}}=1.0002*log(IN_{1})+0.0683$ |  1 |  0.000242 |  1 |  0.0032
$ENT_{IN_{2}}=1.0002*log(IN_{2})+0.0854$ |  1 |  0.000316 |  1 |  0.0011

Table 4. Statistics of curve fitting of the Nirmala indices vs. Nirmala
entropy measures of the complex structure of ruthenium bipyridine $RB_{n}$.

Figure 3. Curve fitting plots for the Nirmala indices vs. Nirmala entropy
measures of the complex structure of ruthenium bipyridine $RB_{n}$.

##  5 Discussion

Researchers can gain a better understanding of the molecular properties and
behavior of chemical and biological systems by using topological indices,
which are numerical representations of molecular groups in graph theory.
Accurate calculation of numerical indices gives scholars relevant information
that enhances their knowledge of the topic. In this work, we study the so-
called degree-based topological indices, that is, Nirmala indices, of the
complex structure of ruthenium bipyridine $RB_{n}$. Table 3 shows that as $n$
increases, the Nirmala indices and related entropy measures of $RB_{n}$ also
increase. Entropy metrics assess the uncertainty or information content of a
dataset to assist estimate its complexity and distribution. Data analysis,
thermodynamics, and information theory heavily rely on these concepts.
Accurate numerical entropy calculations provide academics with valuable
insights, enhancing their comprehension of the network under investigation.
Considering its advantages, it validates our focus on the edge weight entropy
of $RB_{n}$. Table 3 illustrates that entropy measurements of the complex
structure of ruthenium bipyridine $RB_{n}$ rise with increasing $n$.

To account for non-linearity in data and produce predictions, the social
sciences, biology, and economics employ logarithmic regression, which is a
non-linear regression technique. The study’s statistical measures, such as the
squared correlation coefficient ($R^{2}$), root mean square error (RMSE),
adjusted squared correlation coefficient (Adj. R-sq), and the sum of square
errors (SSE), are shown in Table 4. A regression line that fits the data
better is indicated by a greater $R^{2}$ value, which is closer to 1. The
Nirmala indices and the related entropy measure values of $RB_{n}$ appear to
fit the curve well, as shown in Figure 3.

##  6 Conclusion

This research has made use of the definitions of the Nirmala indices and
entropy measures derived from them. A mathematical formulation of the Nirmala
indices of the complex structure of ruthenium bipyridine $RB_{n}$ has been
achieved. The entropy measures of $RB_{n}$ based on the Nirmala indices have
been analyzed using its M-polynomial. Table 3 shows that when $n$ increases,
the Nirmala indices and related entropy measurements of the complex structure
of ruthenium bipyridine $RB_{n}$ also increase. Additionally, the inequality
relationships shown below are also accurate.

$IN_{1}(\Upsilon)<IN_{2}(\Upsilon)<N(\Upsilon)$

$ENT_{N}(\Upsilon)\approx ENT_{IN_{1}}(\Upsilon)\approx
ENT_{IN_{2}}(\Upsilon)$.

The topology and structural characteristics of the complex structure of
ruthenium bipyridine $RB_{n}$ will be examined with the help of the study’s
results.

## Funding

No funding is available for this study.

## Author contributions

All authors contributed equally.

## Data Availability Statement

This manuscript has no associated data.

## Declarations

Conflict of interest The authors declare that they have no known competing
financial interests or personal relationships that could have appeared to
influence the work reported in this paper.

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