The Minimal Molecular Tree for the Exponential Randić Index

Abstract

Topological indices are numerical parameters that provide a way to quantify the structural features of molecules using their graph representations. In chemical graph theory, these indices have been effectively employed to predict various physico-chemical properties of molecules. Among these, the Randić index stands out as a classical and widely used molecular descriptor in chemistry and pharmacology. The Randić index $R\left(G\right)$ for a given graph G is defined as $R\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}$, where $d\left(v_i\right)$ represents the degree of vertex $v_i$ and $E\left(G\right)$ is the set of edges in the graph G. Given the Randić index’s strong discrimination ability in describing molecular structures, a variant known as the exponential Randić index was recently introduced. The exponential Randić index $ER\left(G\right)$ for a graph G is defined as $ER\left(G\right)={\displaystyle \sum _{v_i{v}_j\in E\left(G\right)}}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}$. This paper further explores and fully characterizes the minimal molecular trees in relation to the exponential Randić index. Moreover, the chemical relevance of the exponential Randić index is also investigated.

1. Introduction

In theoretical organic chemistry, understanding the relationship between molecular structures and their physico-chemical properties is of fundamental importance. Graph theory provides a powerful framework for this, where molecules are modeled as graphs, with atoms represented by vertices and chemical bonds by edges. To quantify the structural characteristics of organic molecules and predict their specific physico-chemical properties, several graph-based indices have been developed. Among these, the Randić index is one of the most prominent and widely used. Introduced by Milan Randić in 1975, the Randić index (R) [1] quantifies molecular branching by considering the degrees of a graph’s vertices. Given a graph G with a vertex set $V\left(G\right)=\{v_1,v_2,\dots ,v_n\}$ and an edge set $E\left(G\right)$, the Randić index is defined as:

$$\begin{array}{c}\hfill {\displaystyle R\left(G\right)=\sum _{v_i{v}_j\in E\left(G\right)}\frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}},}\end{array}$$

where $d\left(v_i\right)$ represents the degree of a vertex $v_i$ in G. Randić observed that this index correlates remarkably well with various physico-chemical properties of alkanes, making it a valuable tool in chemical research. Since its introduction, the Randić index has found numerous applications in both chemistry and pharmacology, particularly in the study of molecular structures.

Building upon the original Randić index, researchers later introduced the generalized Randić index to broaden its applicability. The generalized Randić index $\left(R_{\alpha }\right)$ is defined as:

$$\begin{array}{c}\hfill {\displaystyle R_{\alpha }\left(G\right)=\sum _{v_i{v}_j\in E\left(G\right)}(d\left(v_i\right)d\left(v_j\right){)}^{\alpha },}\end{array}$$

where $\alpha$ is a real number that can be adjusted to modify the index’s sensitivity to different structural features. This generalization provides a more flexible approach, allowing for the exploration of new correlations between molecular structure and properties across a wider range of chemical compounds.

Numerous researchers have thoroughly examined the mathematical properties of both R and $R_{\alpha }$ for various class of graphs. Bollobás and Erdos [2] demonstrated that, among all connected graphs, the star graph achieves the lowest value for R. Caporossi et al. [3] found that, among all trees, the path graph has the highest value with respect to R. Hu and Li [4,5] further identified the trees with the smallest and highest values for $R_{\alpha }$. For additional findings on R and $R_{\alpha }$, see references [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. In 2019, a significant development was made when degree-based indices were enhanced to their exponential forms, increasing their discriminative power [21]. Following this advancement, numerous studies explored the implications and applications of these exponential indices [22,23,24,25,26,27,28,29]. This paper focuses specifically on the exponential Randić index $\left(ER\right)$ introduced by Rada [21]. The exponential Randić index ($ER$) for a graph G is defined as:

$$\begin{array}{c}\hfill ER\left(G\right)=\sum _{v_i{v}_j\in E\left(G\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}.\end{array}$$

Let us consider a function:

$$F(x,y)=e^{{\displaystyle \frac{1}{\sqrt{xy}}}}.$$

Then, we have:

$$ER\left(G\right)=\sum _{v_i{v}_j\in E\left(G\right)}F(d\left(v_i\right),d\left(v_j\right)).$$

Cruz et al. [30] showed that, among all trees, the path graph attains the highest value for $ER$. Cruz and Rada [23] also established that the star graph achieves the lowest value of $ER$ among all trees. Further work by Qiu et al. [31] examined the extremal values of $ER$ for quasi-tree graphs and described the specific graphs that reached these values. Lin and Zhu [32] extended this research to unicyclic graphs, determining their extreme values of $ER$ and identifying the graphs that achieved these extreme values.

Identifying extremal graphs across various classes plays a crucial role in mathematical chemistry. A particularly important class is that of molecular trees’ acyclic connected graphs with a maximum degree of 4. Zhong [33] investigated the extremal values of the harmonic index for molecular trees and characterized those that achieved these values. Gutman et al. [34] determined the extreme values of R for molecular trees and described the specific trees attaining these values. Zhou and Trinajstíc [35] identified the molecular trees with the lowest, second-lowest, and third-lowest sum-connectivity indices. Deng et al. [36] analyzed the molecular trees to determine which ones have the lowest and highest values of the Sombor index. In [37], the authors characterized the extremal molecular trees for the exponential augmented Zagreb index. For more results in this direction, see references [38,39,40,41,42,43,44].

The classical Randić index is useful for predicting molecular properties, but it may not always distinguish between similar structures. To address this, the exponential Randić index ($ER$) was introduced. This study aims to enhance the understanding of the $ER$ index by identifying molecular trees with lowest $ER$ values, contributing to the broader knowledge of extremal graph structures. Additionally, we explore the chemical relevance of the $ER$ index to demonstrate its potential in predicting important molecular properties, which could enhance its application in chemistry and pharmacology.

In this paper, we characterize the molecular trees with lowest $ER$ values and investigate the chemical relevance of the $ER$ index.

2. Main Result

This section discusses the minimum value of $ER$ for molecular trees and identifies the extremal trees. For this purpose, we define the following notations.

Let T be a molecular tree with n vertices. The number of vertices in T with degree j is denoted by $n_j$, while $m_{ij}$ represents the number of edges in T that connect vertices of degrees i and j. The neighborhood of a vertex $v_j$ in T is indicated by $N_T\left(v_j\right)$. Let ${\mathcal{C}}_n$ represent the set of all molecular trees with n vertices, and ${\mathcal{C}}_n^{\mathrm{MIN}}$ denote the subset of molecular trees within ${\mathcal{C}}_n$ that achieve the lowest $ER$ index.

The values of $F(d\left(v_i\right),d\left(v_j\right))$ for various edges $v_i{v}_j\in E\left(T\right)$ in the molecular tree T are as follows:

$$F(d\left(v_i\right),d\left(v_j\right))=\left\{\begin{array}{cc}{\displaystyle e^{\frac{1}{\sqrt{2}}}\approx 2.0281}\hfill & \mathrm{for}\left(d\left(v_i\right),d\left(v_j\right)\right)=(1,2),\hfill \\ {\displaystyle e^{\frac{1}{\sqrt{3}}}\approx 1.7813}\hfill & \mathrm{for}\left(d\left(v_i\right),d\left(v_j\right)\right)=(1,3),\hfill \\ {\displaystyle e^{\frac{1}{\sqrt{4}}}\approx 1.6487}\hfill & \mathrm{for}\left(d\left(v_i\right),d\left(v_j\right)\right)=(1,4),\hfill \\ {\displaystyle e^{\frac{1}{\sqrt{4}}}\approx 1.6487}\hfill & \mathrm{for}\left(d\left(v_i\right),d\left(v_j\right)\right)=(2,2),\hfill \\ {\displaystyle e^{\frac{1}{\sqrt{6}}}\approx 1.5042}\hfill & \mathrm{for}\left(d\left(v_i\right),d\left(v_j\right)\right)=(2,3),\hfill \\ {\displaystyle e^{\frac{1}{\sqrt{8}}}\approx 1.4241}\hfill & \mathrm{for}\left(d\left(v_i\right),d\left(v_j\right)\right)=(2,4),\hfill \\ {\displaystyle e^{\frac{1}{\sqrt{9}}}\approx 1.3956}\hfill & \mathrm{for}\left(d\left(v_i\right),d\left(v_j\right)\right)=(3,3),\hfill \\ {\displaystyle e^{\frac{1}{\sqrt{12}}}\approx 1.3347}\hfill & \mathrm{for}\left(d\left(v_i\right),d\left(v_j\right)\right)=(3,4),\hfill \\ {\displaystyle e^{\frac{1}{\sqrt{16}}}\approx 1.2840}\hfill & \mathrm{for}\left(d\left(v_i\right),d\left(v_j\right)\right)=(4,4).\hfill \end{array}\right.$$

(1)

We begin with the following result.

Lemma 1. 

Let $n\ge 4$. If $T\in {\mathcal{C}}_n^{\mathit{MIN}}$, then $m_{22}=0$.

Proof. 

Assume to the contrary that $T_1\in {\mathcal{C}}_n^{\mathrm{MIN}}$ with $m_{22}>0$. Therefore, there exists an edge $v_1{v}_2\in E\left(T_1\right)$ with $d\left(v_1\right)=2$ and $d\left(v_2\right)=2$. Suppose $v_1{u}_1\in E\left(T_1\right)$, $u_1\ne v_2$, and $v_2{u}_2\in E\left(T_1\right)$, $u_2\ne v_1$. Without loss of generality, we can assume that $d\left(u_1\right)\le d\left(u_2\right)$. Now we transform the tree $T_1$ into a new tree $T_1^{\prime }$ (see, Figure 1) by deleting the edge $u_1{v}_1$ and adding the edge $u_1{v}_2$, that is, $T_1^{\prime }:=T_1-u_1{v}_1+u_1{v}_2$.

Figure 1. Two molecular trees $T_1$ and $T_1^{\prime }$.

By (1), we obtain:

$$\begin{array}{cc}\hfill ER\left(T_1\right)-ER\left(T_1^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_1\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_1^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =\sum _{j=1}^2\left(e^{{\displaystyle \frac{1}{\sqrt{2d\left(u_j\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_j\right)}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{4}}}}-e^{{\displaystyle \frac{1}{\sqrt{3}}}}\right).\hfill \end{array}$$

(2)

Table 1 displays the values of $ER\left(T_1\right)-ER\left(T_1^{\prime }\right)$ for various combinations of $d\left(u_1\right)$ and $d\left(u_2\right)$.

Table 1. Approximate values for the expression on the right side of Equation (2).

$d\left(u_1\right)$	1	1	1	1	2	2	2	3	3	4
$d\left(u_2\right)$	1	2	3	4	2	3	4	3	4	4
$ER\left(T_1\right)-ER\left(T_1^{\prime }\right)$	0.361	0.2587	0.2228	0.2036	0.1564	0.1205	0.1013	0.0846	0.0654	0.0462

From (2), we obtain $ER\left(T_1\right)-ER\left(T_1^{\prime }\right)>0$, by Table 1. This contradicts $T_1\in {\mathcal{C}}_n^{\mathrm{MIN}}$, which completes the proof. □

Lemma 2. 

Let $n\ge 5$. If $T\in {\mathcal{C}}_n^{\mathit{MIN}}$, then $m_{23}=0$.

Proof. 

Assume to the contrary that $T_2\in {\mathcal{C}}_n^{\mathrm{MIN}}$ such that $v_1{v}_2\in E\left(T_2\right)$ with $d\left(v_1\right)=2$ and $d\left(v_2\right)=3$. Suppose $N_{T_2}\left(v_1\right)=\{v_2,u_1\}$ and $N_{T_2}\left(v_2\right)=\{v_1,u_2,u_3\}$. By Lemma 1, we have $m_{22}=0$, and hence, $d\left(u_1\right)\ne 2$ as $d\left(v_1\right)=2$ and $u_1{v}_1\in E\left(T\right)$. Thus, we have $d\left(u_1\right)\in \{1,3,4\}$. Without loss of generality, we can assume that $d\left(u_2\right)\le d\left(u_3\right)$. Now we transform $T_2$ to $T_2^{\prime }$, where $T_2^{\prime }:=T_2-u_1{v}_1+u_1{v}_2$, as illustrated in Figure 2. By (1), we obtain:

$$\begin{array}{cc}\hfill ER\left(T_2\right)-ER\left(T_2^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_2\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_2^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =\sum _{j=2}^3\left(e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_j\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{4d\left(u_j\right)}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{2d\left(u_1\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{4d\left(u_1\right)}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{6}}}}-e^{{\displaystyle \frac{1}{\sqrt{4}}}}\right).\hfill \end{array}$$

(3)

Figure 2. Two molecular trees $T_2$ and $T_2^{\prime }$.

Table 2 presents the values of $ER\left(T_2\right)-ER\left(T_2^{\prime }\right)$ for all possible combinations of $d\left(u_1\right)$, $d\left(u_2\right)$, and $d\left(u_3\right)$.

Table 2. Approximate values for the expression on the right side of (3).

$d\left(u_2\right)$	1	1	1	1	2	2	2	3	3	4
$d\left(u_3\right)$	1	2	3	4	2	3	4	3	4	4
$ER\left(T_2\right)-ER\left(T_2^{\prime }\right)$ ($d\left(u_1\right)=1$)	0.5001	0.4476	0.4284	0.4182	0.3951	0.3759	0.3657	0.3567	0.3465	0.3363
$ER\left(T_2\right)-ER\left(T_2^{\prime }\right)$ ($d\left(u_1\right)=3$)	0.2902	0.2377	0.2185	0.2083	0.1852	0.166	0.1558	0.1468	0.1366	0.1264
$ER\left(T_2\right)-ER\left(T_2^{\prime }\right)$ ($d\left(u_1\right)=4$)	0.2608	0.2083	0.1891	0.1789	0.1558	0.1366	0.1264	0.1174	0.1072	0.097

From (3) with Table 2, we obtain that $ER\left(T_2\right)-ER\left(T_2^{\prime }\right)>0$, which contradicts our assumption. Thus, the proof is finished. □

Lemma 3. 

Let $n\ge 6$. If $T\in {\mathcal{C}}_n^{\mathit{MIN}}$, then $m_{33}=0$.

Proof. 

Assume to the contrary that $T_3\in {\mathcal{C}}_n^{\mathrm{MIN}}$ with $m_{33}>0$. So there exists an edge $v_1{v}_2\in E\left(T_3\right)$, where degree of $v_1$ is 3 and degree of $v_2$ is 3. Suppose the neighbors of $v_1$, other than $v_2$, are $u_1$ and $u_3$ and the neighbors of $v_2$, other than $v_1$, are $u_2$ and $u_4$. Based on Lemma 2, we can say that $d\left(u_j\right)\ne 2$ for $1\le j\le 4$. Without loss of generality, we can assume that $d\left(u_3\right)\le d\left(u_1\right)$ and $d\left(u_4\right)\le d\left(u_2\right)$. We consider a set:

$$S=\left\{(i,j,k,\ell )\in {\mathbb{N}}^4|1\le k\le i\le 4,1\le \ell \le j\le 4,i\ne 2\ne k,j\ne 2\ne \ell \right\}.$$

Now we consider the following four cases.

• Case 1. $\left(d\left(u_1\right),d\left(u_2\right),d\left(u_3\right),d\left(u_4\right)\right)\in S\backslash \left\{(1,3,1,3),(1,4,1,3),(1,4,1,4)\right\}$.

We transform the tree $T_3$ into a new tree $T_3^{\prime }:=T_3-u_3{v}_1+u_3{v}_2$, as shown in Figure 3.

Figure 3. Two molecular trees $T_3$ and $T_3^{\prime }$.

Table 3 lists the values of $ER\left(T_3\right)-ER\left(T_3^{\prime }\right)$ for all possible combinations of $d\left(u_1\right)$, $d\left(u_2\right)$, $d\left(u_3\right)$, and $d\left(u_4\right)$.

Table 3. Approximate values for the expression on the right side of (4).

$d\left(u_2\right)$	1	3	4	3	4	4
$d\left(u_4\right)$	1	1	1	3	3	4
$ER\left(T_3\right)-ER(T_3^{\prime }$) ($d\left(u_1\right)=1,d\left(u_3\right)=1$)	0.1225	0.0508	0.0406	-	-	-
$ER\left(T_3\right)-ER(T_3^{\prime }$) ($d\left(u_1\right)=3,d\left(u_3\right)=1$)	0.2607	0.189	0.1788	0.1173	0.1071	0.0969
$ER\left(T_3\right)-ER(T_3^{\prime }$) ($d\left(u_1\right)=4,d\left(u_3\right)=1$)	0.2799	0.2082	0.198	0.1365	0.1263	0.1161
$ER\left(T_3\right)-ER(T_3^{\prime }$) ($d\left(u_1\right)=3,d\left(u_3\right)=3$)	0.189	0.1173	0.1071	0.0456	0.0354	0.0252
$ER\left(T_3\right)-ER(T_3^{\prime }$) ($d\left(u_1\right)=4,d\left(u_3\right)=3$)	0.2082	0.1365	0.1263	0.0648	0.0546	0.0444
$ER\left(T_3\right)-ER(T_3^{\prime }$) ($d\left(u_1\right)=4,d\left(u_3\right)=4$)	0.198	0.1263	0.1161	0.0546	0.0444	0.0342

By (1), we obtain:

$$\begin{array}{cc}\hfill ER\left(T_3\right)-ER\left(T_3^{\prime }\right)=& \sum _{v_i{v}_j\in E\left(T_3\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_3^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill =& \left(e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_1\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{2d\left(u_1\right)}}}}\right)+\sum _{j=2}^4\left(e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_j\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{4d\left(u_j\right)}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{9}}}}-e^{{\displaystyle \frac{1}{\sqrt{8}}}}\right).\hfill \end{array}$$

(4)

From the above with Table 3, we obtain $ER\left(T_3\right)-ER\left(T_3^{\prime }\right)>0$, which leads to a contradiction.

• Case 2. $\left(d\left(u_1\right),d\left(u_2\right),d\left(u_3\right),d\left(u_4\right)\right)=(1,3,1,3)$. The tree $T_3$ is shown as $T_4$ in Figure 4, with $N_{T_4}\left(u_2\right)\backslash \left\{v_2\right\}=\{u_7,u_8\}$ and $N_{T_4}\left(u_4\right)\backslash \left\{v_2\right\}=\{u_5,u_6\}$. According to Lemma 2, it is evident that $d\left(u_j\right)\ne 2$ for $j=5,6,7,8$. Without loss of generality, we can assume that $d\left(u_5\right)\le d\left(u_6\right)$ and $d\left(u_7\right)\le d\left(u_8\right)$. Now we construct a tree $T_4^{\prime }:=T_4-u_2{u}_7-u_4{u}_5+u_5{v}_1+u_7{v}_2$ from $T_4$, as depicted in Figure 4.
  

  Figure 4. Two molecular trees $T_4$ and $T_4^{\prime }$.

Using (1), we obtain:

$$\begin{array}{cc}\hfill & ER\left(T_4\right)-ER\left(T_4^{\prime }\right)=\sum _{v_i{v}_j\in E\left(T_4\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_4^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =\left(e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_6\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{2d\left(u_6\right)}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_5\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{4d\left(u_5\right)}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_8\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{2d\left(u_8\right)}}}}\right)\hfill \\ \hfill & +\left(e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_7\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{4d\left(u_7\right)}}}}\right)+2\left(e^{{\displaystyle \frac{1}{\sqrt{9}}}}-e^{{\displaystyle \frac{1}{\sqrt{8}}}}\right)+2e^{{\displaystyle \frac{1}{\sqrt{3}}}}-2e^{{\displaystyle \frac{1}{\sqrt{4}}}}+e^{{\displaystyle \frac{1}{\sqrt{9}}}}-e^{{\displaystyle \frac{1}{\sqrt{16}}}}.\hfill \end{array}$$

(5)

Table 4 highlights the values of $ER\left(T_4\right)-ER\left(T_4^{\prime }\right)$ for all possible combinations of $d\left(u_5\right)$, $d\left(u_6\right)$, $d\left(u_7\right)$, and $d\left(u_8\right)$.

Table 4. Approximate values for the expression given on the right side of (5).

$d\left(u_6\right)$	1	3	4	3	4	4
$d\left(u_5\right)$	1	1	1	3	3	4
$ER\left(T_4\right)-ER(T_4^{\prime }$) ($d\left(u_8\right)=1,d\left(u_7\right)=1$)	0.0914	0.2296	0.2488	0.1579	0.1771	0.1669
$ER\left(T_4\right)-ER(T_4^{\prime }$) ($d\left(u_8\right)=3,d\left(u_7\right)=1$)	0.2296	0.3678	0.387	0.2961	0.3153	0.3051
$ER\left(T_4\right)-ER(T_4^{\prime }$) ($d\left(u_8\right)=4,d\left(u_7\right)=1$)	0.2488	0.387	0.4062	0.3153	0.3345	0.3243
$ER\left(T_4\right)-ER(T_4^{\prime }$) ($d\left(u_8\right)=4,d\left(u_7\right)=3$)	0.1771	0.3153	0.3345	0.2436	0.2628	0.2526
$ER\left(T_4\right)-ER(T_4^{\prime }$) ($d\left(u_8\right)=3,d\left(u_7\right)=3$)	0.1579	0.2961	0.3153	0.2244	0.2436	0.2334
$ER\left(T_4\right)-ER(T_4^{\prime }$) ($d\left(u_8\right)=4,d\left(u_7\right)=4$)	0.1669	0.3051	0.3243	0.2334	0.2526	0.2424

Using Table 4 in (5), we obtain $ER\left(T_4\right)-ER\left(T_4^{\prime }\right)>0$, which is a contradiction.

• Case 3. $\left(d\left(u_1\right),d\left(u_2\right),d\left(u_3\right),d\left(u_4\right)\right)=(1,4,1,3)$.

The tree $T_3$ is illustrated as $T_5$ in Figure 5, with $N_{T_5}\left(u_2\right)\backslash \left\{v_2\right\}=\{u_7,u_8,u_9\}$ and $N_{T_5}\left(u_4\right)\backslash \left\{v_2\right\}=\{u_5,u_6\}$. By Lemma 2, it is evident that $d\left(u_5\right)\ne 2$ and $d\left(u_6\right)\ne 2$. Without loss of generality, we can assume that $d\left(u_5\right)\le d\left(u_6\right)$. Now we transform the tree $T_5$ into a new tree $T_5^{\prime }:=T_5-u_4{u}_5+u_5{v}_1$, as shown in Figure 5. Using (1), we obtain:

$$\begin{array}{cc}\hfill ER\left(T_5\right)-ER\left(T_5^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_5\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_5^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =\left(e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_6\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{2d\left(u_6\right)}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_5\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{4d\left(u_5\right)}}}}\right)+2e^{{\displaystyle \frac{1}{\sqrt{3}}}}-2e^{{\displaystyle \frac{1}{\sqrt{4}}}}\hfill \\ \hfill & +e^{{\displaystyle \frac{1}{\sqrt{9}}}}-e^{{\displaystyle \frac{1}{\sqrt{12}}}}+e^{{\displaystyle \frac{1}{\sqrt{9}}}}-e^{{\displaystyle \frac{1}{\sqrt{6}}}}.\hfill \end{array}$$

(6)

Figure 5. Two molecular trees $T_5$ and $T_5^{\prime }$.

Table 5 presents the values of $ER\left(T_5\right)-ER\left(T_5^{\prime }\right)$ for different pairs of $d\left(u_5\right)$ and $d\left(u_6\right)$.

Table 5. Approximate values for the expression given on the right side of Equation (6).

$d\left(u_6\right)$	1	3	4	3	4	4
$d\left(u_5\right)$	1	1	1	3	3	4
$ER\left(T_5\right)-ER\left(T_5^{\prime }\right)$	0.1033	0.2415	0.2607	0.1698	0.189	0.1788

Using Table 5 in (6), we obtain $ER\left(T_5\right)-ER\left(T_5^{\prime }\right)>0$. Thus, we get a contradiction.

• Case 4. $\left(d\left(u_1\right),d\left(u_2\right),d\left(u_3\right),d\left(u_4\right)\right)=(1,4,1,4)$. The tree $T_3$ is displayed as $T_6$ in Figure 6, with $N_{T_6}\left(u_2\right)\backslash \left\{v_2\right\}=\{u_8,u_9,u_{10}\}$ and $N_{T_6}\left(u_4\right)\backslash \left\{v_2\right\}=\{u_5,u_6,u_7\}$. Now we transform the tree $T_6$ into a new tree $T_6^{\prime }$, where $T_6^{\prime }:=T_6-u_4{u}_5-u_4{u}_6+u_5{v}_1+u_6{v}_2$, as shown in Figure 6.
  

  Figure 6. Two molecular trees $T_6$ and $T_6^{\prime }$.

Thus, from relation (1), we get:

$$\begin{array}{cc}\hfill ER\left(T_6\right)-ER\left(T_6^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_6\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_6^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =\left(e^{{\displaystyle \frac{1}{\sqrt{4d\left(u_7\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{2d\left(u_7\right)}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{9}}}}-e^{{\displaystyle \frac{1}{\sqrt{16}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{12}}}}-e^{{\displaystyle \frac{1}{\sqrt{8}}}}\right)\hfill \\ \hfill & +\left(e^{{\displaystyle \frac{1}{\sqrt{12}}}}-e^{{\displaystyle \frac{1}{\sqrt{16}}}}\right)+2e^{{\displaystyle \frac{1}{\sqrt{3}}}}-2e^{{\displaystyle \frac{1}{\sqrt{4}}}}.\hfill \end{array}$$

(7)

The values of $ER\left(T_6\right)-ER\left(T_6^{\prime }\right)$ for four cases of $d\left(u_7\right)$ are displayed in Table 6.

Table 6. Approximate values for the expression on the right side of (7).

$d\left(u_7\right)$	1	2	3	4
$ER\left(T_6\right)-ER\left(T_6^{\prime }\right)$	−0.0413	0.1135	0.1686	0.198

Using Table 6 in (7), we obtain $ER\left(T_6\right)-ER\left(T_6^{\prime }\right)>0$ for $d\left(u_7\right)=2,3,4$, leading to a contradiction. Let us consider $d\left(u_7\right)=1$. In this instance, we display the tree $T_6$ as $T_7$ in Figure 7. Now we transform the tree $T_7$ into a tree $T_7^{\prime }$, where $T_7^{\prime }:=T_7-u_4{u}_5-u_4{u}_6-u_2{u}_8-u_2{u}_9-u_2{u}_{10}+u_5{v}_1+u_6{v}_2+u_7{u}_8+u_7{u}_9+u_7{u}_{10}$, as illustrated in Figure 7. Using (1), we obtain:

$$\begin{array}{cc}\hfill ER\left(T_7\right)-ER\left(T_7^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_7\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_7^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =\left(e^{{\displaystyle \frac{1}{\sqrt{9}}}}-e^{{\displaystyle \frac{1}{\sqrt{16}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{12}}}}-e^{{\displaystyle \frac{1}{\sqrt{8}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{4}}}}-e^{{\displaystyle \frac{1}{\sqrt{8}}}}\right)\hfill \\ \hfill & +\left(e^{{\displaystyle \frac{1}{\sqrt{12}}}}-e^{{\displaystyle \frac{1}{\sqrt{4}}}}\right)+2e^{{\displaystyle \frac{1}{\sqrt{3}}}}-2e^{{\displaystyle \frac{1}{\sqrt{4}}}}.\hfill \end{array}$$

Figure 7. Two molecular trees $T_7$ and $T_7^{\prime }$.

From the above, we obtain $ER\left(T_7\right)-ER\left(T_7^{\prime }\right)>0$, which contradicts our assumption. Thus, the proof is finished. □

Lemma 4. 

Let $n\ge 11$. If $T\in {\mathcal{C}}_n^{\mathit{MIN}}$, then $m_{13}=0$.

Proof. 

Assume to the contrary that $T_8\in {\mathcal{C}}_n^{\mathrm{MIN}}$ such that $v_1{v}_2\in E\left(T\right)$ with $d\left(v_1\right)=3$ and $d\left(v_2\right)=1$. Suppose that $N_{T_8}\left(v_1\right)\backslash \left\{v_2\right\}=\{u_1,u_2\}$. Since $n\ge 11$, we have $\left(d\left(u_1\right),d\left(u_2\right)\right)\ne (1,1)$. According to Lemmas 2 and 3, $d\left(u_j\right)\notin \{2,3\}$ for $j=1,2$. Without loss of generality, we can assume that $d\left(u_2\right)\le d\left(u_1\right)$. Since $T_8$ is a molecular tree, using the above results, we obtain $\left(d\left(u_1\right),d\left(u_2\right)\right)=(4,1)$ or $\left(d\left(u_1\right),d\left(u_2\right)\right)=(4,4)$.

• Case 1. $\left(d\left(u_1\right),d\left(u_2\right)\right)=(4,1)$. The tree $T_8$ is shown as $T_9$ in Figure 8, with $N_{T_9}\left(u_1\right)\backslash$$\left\{v_1\right\}=\{u_3,u_4,u_5\}$. Without loss of generality, we can assume that $d\left(u_5\right)\ge d\left(u_4\right)\ge d\left(u_3\right)$. Now we transform the tree $T_9$ into a new tree $T_9^{\prime }$ (see, Figure 8) by deleting the edges $u_2{v}_1$, $u_1{u}_3$, $u_1{u}_4$ and adding the edges $u_3{v}_1$, $u_4{v}_1$, $u_1{u}_2$.
  

  Figure 8. Two molecular trees $T_9$ and $T_9^{\prime }$.

Thus, applying (1), we obtain:

$$\begin{array}{cc}\hfill ER\left(T_9\right)-ER\left(T_9^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_9\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_9^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =\left(e^{{\displaystyle \frac{1}{\sqrt{4d\left(u_5\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{3d\left(u_5\right)}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{3}}}}-e^{{\displaystyle \frac{1}{\sqrt{4}}}}\right).\hfill \end{array}$$

(8)

If $d\left(u_5\right)=1$, then $d\left(u_4\right)=d\left(u_3\right)=1$, and hence, $n=7$, a contradiction as $n\ge 11$. Otherwise, $d\left(u_5\right)\ge 2$. The values of $ER\left(T_9\right)-ER\left(T_9^{\prime }\right)$ for three cases of $d\left(u_5\right)$ are shown in Table 7.

Table 7. Approximate values for the expression on the right side of (8).

$d\left(u_5\right)$	2	3	4
$ER\left(T_9\right)-ER\left(T_9^{\prime }\right)$	0.0525	0.0717	0.0819

From (8) and Table 7, we see that $ER\left(T_9\right)-ER\left(T_9^{\prime }\right)>0$, a contradiction.

• Case 2. $\left(d\left(u_1\right),d\left(u_2\right)\right)=(4,4)$. The tree $T_8$ is illustrated as $T_{10}$ in Figure 9, where $N_{T_{10}}\left(u_1\right)\backslash \left\{v_1\right\}=\{u_3,u_4,u_5\}$ and $N_{T_{10}}\left(u_2\right)\backslash \left\{v_1\right\}=\{u_6,u_7,u_8\}$. As $n\ge 11$, $(d\left(u_3\right),d\left(u_4\right),$ $d\left(u_5\right),d\left(u_6\right),d\left(u_7\right),d\left(u_8\right))\ne (1,1,1,1,1,1)$. Thus, $d\left(u_j\right)>1$ for at least one $j\in \{3,4,5,6,7,8\}$. Without loss of generality, we can assume that $d\left(u_8\right)>1$. Now the following three subcases may arise.
  

  Figure 9. The molecular tree $T_{10}$.

• Case 1.1. $d\left(u_8\right)=2$. We represent the tree $T_{10}$ as $T_{11}$ in Figure 10, with $N_{T_{11}}\left(u_8\right)\backslash$$\left\{u_2\right\}=\left\{u_9\right\}$. Now we form a new tree $T_{11}^{\prime }$ (see Figure 10) from $T_{11}$ by deleting the edge $u_9{u}_8$ and adding the edge $u_9{v}_2$. Thus, applying (1), we obtain:

  $$\begin{array}{cc}\hfill ER\left(T_{11}\right)-ER\left(T_{11}^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_{11}\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_{11}^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =e^{{\displaystyle \frac{1}{\sqrt{3}}}}+e^{{\displaystyle \frac{1}{\sqrt{8}}}}-e^{{\displaystyle \frac{1}{\sqrt{6}}}}-e^{{\displaystyle \frac{1}{\sqrt{4}}}}>0,\hfill \end{array}$$

  which is a contradiction.
  

  Figure 10. Two molecular trees $T_{11}$ and $T_{11}^{\prime }$.

• Case 1.2. $d\left(u_8\right)=3$. We display the tree $T_{10}$ as $T_{12}$ in Figure 11, where $N_{T_{12}}\left(u_8\right)\backslash$$\left\{u_2\right\}=\{u_9,u_{10}\}$. Now we transform $T_{12}$ to $T_{12}^{\prime }:=T_{12}-u_8{u}_9-u_8{u}_{10}+u_9{v}_2+u_{10}{v}_2$, as displayed in Figure 11.
  

  Figure 11. Two molecular trees $T_{12}$ and $T_{12}^{\prime }$.

Thus, applying (1), we get:

$$\begin{array}{cc}\hfill ER\left(T_{12}\right)-ER\left(T_{12}^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_{12}\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_{12}^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =e^{{\displaystyle \frac{1}{\sqrt{3}}}}+e^{{\displaystyle \frac{1}{\sqrt{12}}}}-e^{{\displaystyle \frac{1}{\sqrt{9}}}}-e^{{\displaystyle \frac{1}{\sqrt{4}}}}>0,\hfill \end{array}$$

which is a contradiction.

• Case 1.3. $d\left(u_8\right)=4$. We represent the tree $T_{10}$ as $T_{13}$ in Figure 12, where $N_{T_{13}}\left(u_8\right)\backslash$$\left\{u_2\right\}=\{u_9,u_{10},u_{11}\}$. Now we transform $T_{13}$ to $T_{13}^{\prime }:=T_{13}-u_8{u}_9-u_8{u}_{10}-u_8{u}_{11}+u_9{v}_2+u_{10}{v}_2+u_{11}{v}_2$, as displayed in Figure 12.
  

  Figure 12. Two molecular trees $T_{13}$ and $T_{13}^{\prime }$.

Thus, applying (1), we get:

$$\begin{array}{cc}\hfill ER\left(T_{13}\right)-ER\left(T_{13}^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_{13}\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_{13}^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =e^{{\displaystyle \frac{1}{\sqrt{3}}}}+e^{{\displaystyle \frac{1}{\sqrt{16}}}}-e^{{\displaystyle \frac{1}{\sqrt{12}}}}-e^{{\displaystyle \frac{1}{\sqrt{4}}}}>0,\hfill \end{array}$$

which is a contradiction. Thus, the proof is finished. □

Lemma 5. 

Let $n\ge 7$. If $T\in {\mathcal{C}}_n^{\mathit{MIN}}$, then $m_{12}=0$.

Proof. 

Assume to the contrary that $T_{14}\in {\mathcal{C}}_n^{\mathrm{MIN}}$ such that $v_1{v}_2\in E\left(T_{14}\right)$ with $d\left(v_1\right)=2$ and $d\left(v_2\right)=1$. Suppose that $N_{T_{14}}\left(v_1\right)\backslash \left\{v_2\right\}=\left\{u_1\right\}$. Since $n\ge 7$, so $d\left(u_1\right)\ne 1$. According to Lemmas 1 and 2, $d\left(u_1\right)\notin \{2,3\}$, and hence, $d\left(u_1\right)=4$ as $T_{14}$ is a molecular tree. We display the tree $T_{14}$ as $T_{15}$ in Figure 13, with $N_{T_{15}}\left(u_1\right)\backslash \left\{v_1\right\}=\{u_2,u_3,u_4\}$. Without loss of generality, we can assume that $d\left(u_4\right)\ge max\{d\left(u_2\right),d\left(u_3\right)\}$. Now we transform the tree $T_{15}$ into a tree $T_{15}^{\prime }$ (see, Figure 13) by deleting the edges $u_1{u}_2$, $u_1{u}_3$ and adding the edges $u_2{v}_1$, $u_3{v}_1$. By applying (1), we get from Figure 13 that:

$$\begin{array}{cc}\hfill ER\left(T_{15}\right)-ER\left(T_{15}^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_{15}\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_{15}^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =\left(e^{{\displaystyle \frac{1}{\sqrt{4d\left(u_4\right)}}}}-e^{{\displaystyle \frac{1}{\sqrt{2d\left(u_4\right)}}}}\right)+\left(e^{{\displaystyle \frac{1}{\sqrt{2}}}}-e^{{\displaystyle \frac{1}{\sqrt{4}}}}\right).\hfill \end{array}$$

(9)

Figure 13. Two molecular trees $T_{15}$ and $T_{15}^{\prime }$.

If $d\left(u_4\right)=1$, then $d\left(u_2\right)=d\left(u_3\right)=1$, and hence, $n=6$, a contradiction as $n\ge 7$. Otherwise, $d\left(u_4\right)\ge 2$. The values of $ER\left(T_{15}\right)-ER\left(T_{15}^{\prime }\right)$ for three cases of $d\left(u_4\right)$ are shown in Table 8.

Table 8. Approximate values for the expression on the right side of (9).

$d\left(u_4\right)$	2	3	4
$ER\left(T_{15}\right)-ER\left(T_{15}^{\prime }\right)$	0.1548	0.2099	0.2393

From (9) and Table 8, we find that $ER\left(T_{15}\right)-ER\left(T_{15}^{\prime }\right)>0$, a contradiction. Thus, the proof is finished. □

Lemma 6. 

Let $n\ge 7$. If $T\in {\mathcal{C}}_n^{\mathit{MIN}}$, then $n_2\le 1$.

Proof. 

Assume to the contrary that $T_{16}\in {\mathcal{C}}_n^{\mathrm{MIN}}$ with $n_2>1$. There exist vertices $v_1$, $v_2$ in $T_{16}$, where the degree of $v_1$ is 2 and the degree of $v_2$ is 2. By Lemma 1, we have $v_1{v}_2\notin E\left(T_{16}\right)$. Suppose that $N_{T_{16}}\left(v_1\right)=\{u_1,u_2\}$ and $N_{T_{16}}\left(v_2\right)=\{u_3,u_4\}$. Based on Lemmas 1, 2, and 5, one can easily see that $d\left(u_j\right)\notin \{1,2,3\}$ for $1\le j\le 4$. Hence, $d\left(u_j\right)=4$ for $1\le j\le 4$. We express the tree $T_{16}$ as $T_{17}$ in Figure 14, with $N_{T_{17}}\left(u_1\right)\backslash$$\left\{v_1\right\}=\{u_8,u_9,u_{10}\}$, $N_{T_{17}}\left(u_2\right)\backslash \left\{v_1\right\}=\{u_5,u_6,u_7\}$, $N_{T_{17}}\left(u_3\right)\backslash \left\{v_2\right\}=\{u_{11},u_{12},u_{13}\}$, and $N_{T_{17}}\left(u_4\right)\backslash \left\{v_2\right\}=\{u_{14},u_{15},u_{16}\}$. Now we transform $T_{17}$ to $T_{17}^{\prime }$, where $T_{17}^{\prime }:=T_{17}-u_4{v}_2+u_4{v}_1$, as illustrated in Figure 14. Thus, applying (1), we get:

$$\begin{array}{cc}\hfill ER\left(T_{17}\right)-ER\left(T_{17}^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_{17}\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_{17}^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =4e^{{\displaystyle \frac{1}{\sqrt{8}}}}-3e^{{\displaystyle \frac{1}{\sqrt{12}}}}-e^{{\displaystyle \frac{1}{\sqrt{4}}}}>0,\hfill \end{array}$$

which contradicts our assumption. Thus, the proof is finished. □

Figure 14. Two molecular trees $T_{17}$ and $T_{17}^{\prime }$.

Lemma 7. 

Let $n\ge 11$. If $T\in {\mathcal{C}}_n^{\mathit{MIN}}$, then $n_3\le 1$.

Proof. 

Assume to the contrary that $T_{18}\in {\mathcal{C}}_n^{\mathrm{MIN}}$ with $n_3>1$. Consider $v_1,v_2\in V\left(T_{18}\right)$ with $d\left(v_1\right)=3$ and $d\left(v_2\right)=3$. By Lemma 3, we have $v_1{v}_2\notin E\left(T_{18}\right)$. Suppose that $N_{T_{18}}\left(v_1\right)=\{u_1,u_2,u_3\}$ and $N_{T_{18}}\left(v_2\right)=\{u_4,u_5,u_6\}$. By Lemmas 2–4, we obtain $d\left(u_j\right)\notin \{1,2,3\}$ for $1\le j\le 6$. So, $d\left(u_j\right)=4$ for $1\le j\le 6$. We represent the tree $T_{18}$ as $T_{19}$ in Figure 15, with $N_{T_{19}}\left(u_1\right)\backslash \left\{v_1\right\}=\{u_{10},u_{11},u_{12}\}$, $N_{T_{19}}\left(u_2\right)\backslash \left\{v_1\right\}=\{u_7,u_8,u_9\}$, $N_{T_{19}}\left(u_3\right)\backslash \left\{v_1\right\}=\{u_{13},u_{14},u_{15}\}$, $N_{T_{19}}\left(u_4\right)\backslash \left\{v_2\right\}=\{u_{19},u_{20},u_{21}\}$, $N_{T_{19}}\left(u_5\right)\backslash$$\left\{v_2\right\}=\{u_{16},u_{17},u_{18}\}$, and $N_{T_{19}}\left(u_6\right)\backslash \left\{v_2\right\}=\{u_{22},u_{23},u_{24}\}$. Now we transform $T_{19}$ to $T_{19}^{\prime }$, where $T_{19}^{\prime }:=T_{19}-u_4{v}_2+u_4{v}_1$, as shown in Figure 15. Thus, from (1), we obtain:

$$\begin{array}{cc}\hfill ER\left(T_{19}\right)-ER\left(T_{19}^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_{19}\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_{19}^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =6e^{{\displaystyle \frac{1}{\sqrt{12}}}}-4e^{{\displaystyle \frac{1}{\sqrt{16}}}}-2e^{{\displaystyle \frac{1}{\sqrt{8}}}}>0,\hfill \end{array}$$

which contradicts our assumption. Thus, the proof is finished. □

Figure 15. Two molecular trees $T_{19}$ and $T_{19}^{\prime }$.

Lemma 8. 

Let $n\ge 11$. If $T\in {\mathcal{C}}_n^{\mathit{MIN}}$, then $n_2=0$ or $n_3=0$ or $n_2=n_3=0$.

Proof. 

Assume to the contrary that $T_{20}\in {\mathcal{C}}_n^{\mathrm{MIN}}$ with $n_2\ge 1$ and $n_3\ge 1$. There exist vertices $v_1,v_2$ in $T_{20}$, where the degree of $v_1$ is 3 and the degree of $v_2$ is 2. By Lemma 2, we have $v_1{v}_2\notin E\left(T_{20}\right)$. Suppose that $N_{T_{20}}\left(v_1\right)=\{u_1,u_2,u_3\}$ and $N_{T_{20}}\left(v_2\right)=\{u_4,u_5\}$. According to Lemmas 1–5, it is evident that $d\left(u_j\right)\notin \{1,2,3\}$ for $1\le j\le 5$. Thus, $d\left(u_j\right)=4$ for $1\le j\le 5$. We display the tree $T_{20}$ as $T_{21}$ in Figure 16, with $N_{T_{21}}\left(u_1\right)\backslash$$\left\{v_1\right\}=\{u_9,u_{10},u_{11}\}$, $N_{T_{21}}\left(u_2\right)\backslash \left\{v_1\right\}=\{u_6,u_7,u_8\}$, $N_{T_{21}}\left(u_3\right)\backslash \left\{v_1\right\}=\{u_{12},u_{13},u_{14}\}$, $N_{T_{21}}\left(u_4\right)\backslash \left\{v_2\right\}=\{u_{18},u_{19},u_{20}\}$, and $N_{T_{21}}\left(u_5\right)\backslash \left\{v_2\right\}=\{u_{15},u_{16},u_{17}\}$. Now we transform $T_{21}$ to $T_{21}^{\prime }$, where $T_{21}^{\prime }:=T_{21}-u_4{v}_2+u_4{v}_1$, as displayed in Figure 16.

Figure 16. Two molecular trees $T_{21}$ and $T_{21}^{\prime }$.

By applying (1), we get:

$$\begin{array}{cc}\hfill ER\left(T_{21}\right)-ER\left(T_{21}^{\prime }\right)& =\sum _{v_i{v}_j\in E\left(T_{21}\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}-\sum _{v_i{v}_j\in E\left(T_{21}^{\prime }\right)}{e}^{{\displaystyle \frac{1}{\sqrt{d\left(v_i\right)d\left(v_j\right)}}}}\hfill \\ \hfill & =3e^{{\displaystyle \frac{1}{\sqrt{12}}}}-4e^{{\displaystyle \frac{1}{\sqrt{16}}}}+2e^{{\displaystyle \frac{1}{\sqrt{8}}}}-e^{{\displaystyle \frac{1}{\sqrt{4}}}}>0,\hfill \end{array}$$

which contradicts our assumption. This completes the proof. □

We define a class of molecular trees ${\mathsf{\Lambda }}_1$ as follows:

$${\mathsf{\Lambda }}_1=\{T\in {\mathcal{C}}_n:n_4+n_1=n-1,n_2=1,m_{24}=2\}.$$

Figure 17 presents examples of all molecular trees in ${\mathsf{\Lambda }}_1$ for $n=18$. For $T\in {\mathsf{\Lambda }}_1$, we obtain:

$$ER\left(T\right)={\displaystyle \frac{n-9}{3}}{e}^{{\displaystyle \frac{1}{4}}}+{\displaystyle \frac{2n}{3}}{e}^{{\displaystyle \frac{1}{2}}}+2e^{{\displaystyle \frac{1}{2\sqrt{2}}}}.$$

Figure 17. Molecular trees $Q_1,Q_2,Q_3,Q_4,Q_5$, and $Q_6$ in ${\mathsf{\Lambda }}_1$ for $n=18$.

We define a class of molecular trees ${\mathsf{\Lambda }}_2$ as follows:

$${\mathsf{\Lambda }}_2=\{T\in {\mathcal{C}}_n:n_4+n_1=n-1,n_3=1,m_{34}=3\}.$$

Figure 18 presents examples of all molecular trees in ${\mathsf{\Lambda }}_2$ for $n=19$. For $T\in {\mathsf{\Lambda }}_2$, we obtain:

$$ER\left(T\right)={\displaystyle \frac{n-13}{3}}{e}^{{\displaystyle \frac{1}{4}}}+{\displaystyle \frac{2n+1}{3}}{e}^{{\displaystyle \frac{1}{2}}}+3e^{{\displaystyle \frac{1}{2\sqrt{3}}}}.$$

Figure 18. Molecular trees $Q_7,Q_8$, and $Q_9$ in ${\mathsf{\Lambda }}_2$ for $n=19$.

We define a class of molecular trees ${\mathsf{\Lambda }}_3$ as follows:

$${\mathsf{\Lambda }}_3=\{T\in {\mathcal{C}}_n:n_4+n_1=n\}.$$

Figure 19 presents examples of all molecular trees in ${\mathsf{\Lambda }}_3$ for $n=17$. For $T\in {\mathsf{\Lambda }}_3$, we obtain:

$$ER\left(T\right)={\displaystyle \frac{n-5}{3}}{e}^{{\displaystyle \frac{1}{4}}}+{\displaystyle \frac{2n+2}{3}}{e}^{{\displaystyle \frac{1}{2}}}.$$

Figure 19. Molecular trees $Q_{10},Q_{11}$, and $Q_{12}$ in ${\mathsf{\Lambda }}_3$ for $n=17$.

We now present the minimal value of the $ER$ for molecular trees and identify the trees that attain this extremal value.

Theorem 1. 

For a positive integer n, the minimum value of $ER$ over ${\mathcal{C}}_n$ is achieved as follows:

(i) 

When $n\equiv 0\left(mod3\right)(n\ge 9)$, the minimum value is achieved in ${\mathsf{\Lambda }}_1$ with the value:

$${\displaystyle \frac{n-9}{3}}{e}^{{\displaystyle \frac{1}{4}}}+{\displaystyle \frac{2n}{3}}{e}^{{\displaystyle \frac{1}{2}}}+2e^{{\displaystyle \frac{1}{2\sqrt{2}}}},$$

(ii) 

When $n\equiv 1\left(mod3\right)(n\ge 13)$, the minimum value is achieved in ${\mathsf{\Lambda }}_2$ with the value:

$${\displaystyle \frac{n-13}{3}}{e}^{{\displaystyle \frac{1}{4}}}+{\displaystyle \frac{2n+1}{3}}{e}^{{\displaystyle \frac{1}{2}}}+3e^{{\displaystyle \frac{1}{2\sqrt{3}}}},$$

(iii) 

When $n\equiv 2\left(mod3\right)(n\ge 5)$, the minimum value is achieved in ${\mathsf{\Lambda }}_3$ with the value:

$${\displaystyle \frac{n-5}{3}}{e}^{{\displaystyle \frac{1}{4}}}+{\displaystyle \frac{2n+2}{3}}{e}^{{\displaystyle \frac{1}{2}}}.$$

Proof. 

By Sage [45], we obtain the minimal molecular trees $Q_{13}(\in {\mathsf{\Lambda }}_3)$ with $ER\left(Q_{13}\right)=4e^{1/2}$ for $n=5$, $Q_{14}(\in {\mathsf{\Lambda }}_3)$ with $ER\left(Q_{14}\right)=e^{1/4}+6e^{1/2}$ for $n=8$, and $Q_{15}(\in {\mathsf{\Lambda }}_1)$ with $ER(Q_{15}=2e^{1/\sqrt{8}}+6e^{1/2}$ for $n=9$ (see Figure 20).

Figure 20. The molecular trees $Q_{13}$, $Q_{14}$, and $Q_{15}$ for $n=5,8,9$, respectively.

Now we consider $n\ge 11$. For any molecular tree with n vertices, the following conditions must be satisfied:

$$\begin{array}{ccc}\hfill n_1+n_2+n_3+n_4& =& n,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill n_1+2n_2+3n_3+4n_4& =& 2n-2,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill m_{12}+m_{13}+m_{14}& =& n_1,\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill m_{12}+2m_{22}+m_{23}+m_{24}& =& 2n_2,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill m_{31}+m_{32}+2m_{33}+m_{34}& =& 3n_3,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill m_{14}+m_{24}+m_{34}+2m_{44}& =& 4n_4.\hfill \end{array}$$

(15)

From (10), (11) we get:

$$\begin{array}{ccc}\hfill n_2+2n_3+3n_4& =& n-2.\hfill \end{array}$$

(16)

Let $T\in {\mathcal{C}}_n^{\mathrm{MIN}}$. Then, applying Lemmas 1–5, we get from (12)–(15):

$$\begin{array}{ccc}\hfill m_{14}& =& n_1,\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill m_{24}& =& 2n_2,\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill m_{34}& =& 3n_3,\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill m_{14}+m_{24}+m_{34}+2m_{44}& =& 4n_4.\hfill \end{array}$$

(20)

We now continue the proof for the following three cases.

• Case 1. $n\equiv 0\left(mod3\right)$. Since $n\ge 11$, in this case $n=3P$ for some positive integer $P\ge 4$. From (16), we obtain $n_2+2n_3-1=3\left(P-n_4-1\right)$, that is, $n_2+2n_3-1\equiv 0\left(mod3\right)$. Combining this result with Lemmas 6–8, we must have $n_2=1$ and $n_3=0$ for $n\ge 12$. Using these results with (10) and (18), we obtain $n_1+n_4=n-1$, $m_{24}=2$, and hence, $T\in {\mathsf{\Lambda }}_1$. From (16), we have $n_4={\displaystyle \frac{n-3}{3}}$, and then from (10), $n_1={\displaystyle \frac{2n}{3}}$. Using (17)–(20), we obtain $m_{14}={\displaystyle \frac{2n}{3}}$ and $m_{14}+2m_{44}={\displaystyle \frac{4(n-3)}{3}}-2,$ that is, $m_{44}={\displaystyle \frac{n-9}{3}}.$ Thus, we obtain:

  $$ER\left(T\right)={\displaystyle \frac{n-9}{3}}{e}^{{\displaystyle \frac{1}{4}}}+{\displaystyle \frac{2n}{3}}{e}^{{\displaystyle \frac{1}{2}}}+2e^{{\displaystyle \frac{1}{2\sqrt{2}}}}.$$
• Case 2. $n\equiv 1\left(mod3\right)$. In this case $n=3P+1$ for some positive integer $P\ge 4$ as $n\ge 11$. According to (16), we get $n_2+2n_3-2\equiv 0\left(mod3\right)$. Considering this and Lemmas 6–8, one can easily see that $n_3=1$ and $n_2=0$ for $n\ge 13$. From (10) and (19), we obtain $n_1+n_4=n-1$ and $m_{34}=3$, and hence, $T\in {\mathsf{\Lambda }}_2$. Similar to $\mathbf{Case}\mathbf{1}$, we obtain $n_4={\displaystyle \frac{n-4}{3}}$ and $n_1={\displaystyle \frac{2n+1}{3}}$. Using (17)–(20), we obtain $m_{14}={\displaystyle \frac{2n+1}{3}}$ and $m_{44}={\displaystyle \frac{n-13}{3}}$. Thus, we obtain:

  $$ER\left(T\right)={\displaystyle \frac{n-13}{3}}{e}^{{\displaystyle \frac{1}{4}}}+{\displaystyle \frac{2n+1}{3}}{e}^{{\displaystyle \frac{1}{2}}}+3e^{{\displaystyle \frac{1}{2\sqrt{3}}}}.$$
• Case 3. $n\equiv 2\left(mod3\right)$. In this case $n=3P+2$ for some positive integer $P\ge 3$. By (16), we obtain $n_2+2n_3\equiv 0\left(mod3\right)$. Applying Lemmas 6–8, it follows that $n_2=n_3=0$, that is, $n_1+n_4=n$. Hence $T\in {\mathsf{\Lambda }}_3$. Similar to Case 1, we obtain $n_4={\displaystyle \frac{n-2}{3}}$, $n_1={\displaystyle \frac{2(n+1)}{3}}$, $m_{14}={\displaystyle \frac{2n+2}{3}}$ and $m_{44}={\displaystyle \frac{n-5}{3}}$. Thus, we obtain:

  $$ER\left(T\right)={\displaystyle \frac{n-5}{3}}{e}^{{\displaystyle \frac{1}{4}}}+{\displaystyle \frac{2n+2}{3}}{e}^{{\displaystyle \frac{1}{2}}}.$$

This completes the proof of the theorem. □

Remark 1. 

Theorem 1 does not generate Minimal molecular trees of $ER$ for $n=4,6,7,10$. For these values, Figure 21 shows the minimal molecular trees.

Figure 21. Minimal molecular trees $Q_{16}$, $Q_{17}$, $Q_{18}$, and $Q_{19}$ of $ER$ for $n=4,6,7,10$, respectively.

3. Chemical Relevance

In this section, we examine the chemical relevance of the exponential Randić ($ER$) index, a key topological index. Following the methodology suggested by Randić and Trinajstić [46], we assess the chemical significance of the $ER$ index by correlating it with experimental properties of octane isomers. We observed that the $ER$ index demonstrates strong correlations with key molecular properties, such as enthalpy of vaporization (HVAP) and standard enthalpy of vaporization (DHVAP) for octanes. The correlation coefficient between HVAP and $ER$ is 0.935, and that between DHVAP and $ER$ is 0.945. Based on these findings, we conclude that the $ER$ index effectively explains HVAP and DHVAP of octane isomers. Thus, we can state that the $ER$ index is chemically significant.

4. Concluding Remarks

The exponential Randić $\left(ER\right)$ index is an important topological index with growing interest. In this study, we identified the lowest values of the $ER$ index for molecular trees and determined the structures that achieved these values through various graph transformations. Additionally, we explored the chemical relevance of the $ER$ index, showing its strong correlation with properties like standard enthalpy of vaporization (DHVAP) and enthalpy of vaporization (HVAP) for octane isomers. These findings extend the applicability of the $ER$ index beyond theoretical graph analysis and suggest its potential for practical use in chemistry and pharmacology. Future research should explore more complex molecular structures to build on these findings and focus on identifying extremal graphs within various families of graphs for the $ER$ index.