Detour Eccentric Sum Index for QSPR Modeling in Molecular Structures

Abstract

In this paper, we study the detour eccentric sum index (DESI) to obtain the Quantitative Structure–Property Relationship (QSPR) for different molecular structures. We establish theoretical bounds for this index and compute its values across fundamental graph families. Through correlation analyses between the physicochemical properties of molecular structures representing anti-malarial and breast cancer drugs, we show the high predictive value of two topological parameters, detour diameter (DD) and detour radius (DR). Specifically, DR shows strong positive correlations with boiling point, enthalpy, and flash point (up to 0.94), while DD is highly correlated with properties such as molar volume, molar refraction, and polarizability (up to 0.97). The DESI was then selected for detailed curvilinear regression modeling and comparison against the established eccentric distance sum index. For anti-malarial drugs, the second-order model yields the best fit. The DESI provides optimal prediction for boiling point, enthalpy, and flash point. In breast cancer drugs, the second-order model is again favored for properties except for melting point, best described by a third-order model. The results highlight how well the index captures subtle structural characteristics.

1. Introduction

This study focuses exclusively on finite graphs without self-loops or parallel edges, undirected in nature and ensuring connectivity among all vertices. The study of distances within graphs serves a multitude of purposes. While the common objective of moving from point A to point B involves identifying the shortest path, real-world scenarios often present obstacles that necessitate alternative, longer routes or detours. In 1993, Chartrand and Zhang [1] introduced the concept of a detour distance in graphs, marking the beginning of a research trajectory that has since been pursued by numerous scholars. Distance in graph theory finds practical application in predicting the properties of chemical compounds [2]. The chemical and physical attributes of compounds depend on the arrangement of atoms within the molecule. Molecular descriptors quantify molecular structures, often modeled using graphs. Topological indices assign numerical values to these structures, which are crucial in theoretical chemistry and biology for characterizing compound properties. These indices aid in studying chemical materials and drug structures, guiding their manufacturing processes. These indices, derived from various properties like node connections, path lengths, and spectral characteristics, offer unique perspectives. The term ’topological index’ was introduced by Hosoya [3] to characterize the topological nature of a graph. In a study, ref. [4], topological indices correlated strongly with chemical and physical characteristics of medications used to treat blood cancer. Distance-based topological indices are used for various studies in physics, chemistry, and biology [5]. The classic Wiener index captures overall connectivity, while its counterpart, the detour index, explores the longest possible paths within the network. In [6] the authors described the Wiener index as a measure of the compactness of a molecule. The composite of the Wiener index and detour index serves as an effective molecular descriptor for characterizing the properties of both acyclic and cyclic compound classes in [7]. In 2017, Kavitha et al. [8] extended the graph invariant Gutman index based on the distance to detour analog by considering detour distance instead of distance in the Gutman index. Kumar [9] and Elakkiya [10] explored the concept of detour distance. The first and second Zagreb indices, which are degree-based topological indices, were examined on the total graphs of some graphs in [11]. Also, the hyper-Zagreb index of some graph operations is determined in [12]. The chemical significance of the variants of the Zagreb index were discussed in [13]. A topological index described in [14], the weighted Padmakar–Ivan (PI) index of a connected graph, is found for the tensor product and strong product of graphs. The author of [15] finds the PI index of a few perfect graphs. Arockiaraj et al. [16] showed effective QSPR modeling for anti-tuberculosis drugs, where selected distance indices markedly improved the prediction of drug activity. Boraiah [17] analyzed distance-2 topological models of alkanes, reinforcing the utility of higher-order distance indices in capturing molecular structural nuances relevant for property prediction. Rasheed et al. [18] explored degree-based indices in drug QSPR models and found these parameters to be robust predictors of molecular characteristics influencing drug efficacy. Shirakol et al. [19] provided comprehensive QSPR analyses incorporating distance-based indices that correlated well with key physicochemical descriptors. Overall, these studies emphasize the growing importance of distance and related detour indices in accurate and computationally efficient drug property prediction, supporting their integration into cheminformatics workflows.

The eccentric distance sum (EDS), presented in [20] as a distance-based topological index, has proven to be a valuable tool for understanding structure–activity and property relationships, surpassing Wiener’s index in effectiveness. In 2014, Monika et al. [21] considered a variation of EDS based on the detour matrix for QSAR/QSPR. In this article we study a topological index based on detour distance called the detour eccentric sum (DESI) of graphs. This is an extended notion of the eccentric distance sum (EDS).

We refer to Harary and Buckley [22] for definitions of graph theory terms and notation that are not provided here. A graph, denoted as G, is described by its vertices, $V\left(G\right)$, and its edges, $E\left(G\right).$ The vertex count, $n\left(V\right(G\left)\right)$, defines the graph’s size or order. Within a graph G that is connected, the detour distance [23], $D(x,y),$ between two vertices x and y is the longest path length connecting them, whereas the distance $d(x,y)$ is the shortest path length connecting them. The detour eccentricity, $e_D\left(x\right)$, of a vertex is the maximum detour distance from that vertex x to any other vertex in G.

• Key contributions:
  • A new topological index, called the “detour eccentric sum index (DESI),” is introduced, and the bounds for the index are obtained.
  • The DESI is obtained for certain classes of graphs.
  • The Quantitative Structure–Property Relationship (QSPR) for different molecular structures is studied through the DESI.
  • The physical characteristics of anti-malarial and breast cancer drugs are obtained from the DESI.
  • We obtain the best predictive fits from the curvilinear regression model of malaria and breast cancer drugs using the DESI.
  • Our experimental results for malaria drugs show that the second-order model is the best for the DESI, providing optimal prediction for boiling point, enthalpy, and flash point.
  • For breast cancer drugs, the second-order model best fits all properties except melting point, which is best described by a third-order model.

2. Detour Eccentric Sum of Graphs

For a graph G that is connected, the detour eccentric sum denoted by ${\xi }^D\left(G\right)$ is defined as follows:

$${\xi }^D\left(G\right)=\sum _{x\in V\left(G\right)}{e}_D\left(x\right)S_D\left(x\right),$$

(1)

where $e_D\left(x\right)$ denotes the detour eccentricity of x and $S_D\left(x\right)$ denotes the sum of detour distances from x to all other vertices. Equation (1) can also be expressed as

$${\xi }^D\left(G\right)=\sum _{\{x,y\}\subseteq \mathbf{V}}[e_D\left(x\right)+e_D\left(y\right)]D(x,y),$$

(2)

where $D(x,y)$ denotes the detour distance between x and y.

The following are some basic results on the detour distances [1]. For example, a graph G is a tree iff $D(x,y)=d(x,y)$ for all pairs of vertices ‘x’ and ‘y’ in G. Therefore, ${\xi }^d\left(G\right)={\xi }^D\left(G\right)$ iff G is a tree. Here ${\xi }^d\left(G\right)$ represents the EDS of $G.$ It is defined as ${\xi }^d\left(G\right)=\sum e_d\left(v\right)·S_d\left(v\right)$, where $e_d\left(v\right)$ denotes the eccentricity of the vertex $v\in V\left(G\right)$ and $S_d\left(v\right)$ denotes the sum of the distances from v to the remaining vertices in $G.$ For a cycle $C_n$, $D(x,y)+d(x,y)=n.$ Similarly, if the detour distance between any two distinct vertices in a graph G is $n-1$, where n is the number of vertices in G, then G is Hamiltonian. For any graph G with n vertices, it is easy to see that $S_D\left(v\right)\le {(n-1)}^2$, $v\in V\left(G\right)$.

Detour Eccentric Sum for Some Graph Classes

The following theorem gives the bounds on the detour eccentric sum index of a graph.

Theorem 1. 

For any graph G of order n, $n(n-1)\le {\xi }^D\left(G\right)\le n{(n-1)}^3.$

Proof. 

To see the validity of the upper bound prescribed on ${\xi }^D\left(G\right)$ in the statement of the theorem, consider any graph G with $V\left(G\right)=\{v_i:1\le i\le n\}.$ Since the maximum of $D(v_i,v_j)$ is $n-1$ for every pair of vertices in $V\left(G\right)$, $e_D\left(v_i\right)\le n-1.$ Hence,

$$\begin{array}{ccc}\hfill {\xi }^D\left(G\right)& =& \sum _{v_i\in V\left(G\right)}{e}_D\left(v_i\right)S_D\left(v_i\right)\hfill \\ & \le & \sum _{v_i\in V\left(G\right)}(n-1)·{(n-1)}^2\hfill \\ & =& n{(n-1)}^3.\hfill \end{array}$$

The validity of the lower bound follows from the fact that $D(v_i,v_j)$ is at least one for every pair $(v_i,v_j)\in V\left(G\right)$. □

It is easy to see that ${\xi }^D\left(G\right)=n(n-1)$ if $G\cong P_2$. A graph G with n vertices is said to be Hamiltonian-connected if $D(u,v)=n-1$ for every pair of vertices $(u,v)$ in G [24]. Hamiltonian-connected graphs attain the upper bound prescribed in the above theorem.

Theorem 2. 

A graph G with $n\ge 3$ is Hamiltonian-connected iff ${\xi }^D\left(G\right)=n{(n-1)}^3.$

Proof. 

Consider a Hamiltonian-connected graph G of order n. Hence $D(u,v)=(n-1)$ for every $(u,v)\in V\left(G\right)$ and $e_D\left(u\right)=(n-1)$ for all $u\in V\left(G\right).$ Also, there are $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)$ pairs of vertices in G. Therefore

$$\begin{array}{ccc}\hfill {\xi }^D\left(G\right)& =& \sum _{\{u,v\}\subset V\left(G\right)}[e_D\left(u\right)+e_D\left(v\right)]D(u,v)\hfill \\ & =& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)[e_D\left(u\right)+e_D\left(v\right)]D(u,v)\hfill \\ & =& \left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)\left[(n-1)+(n-1)\right](n-1)\hfill \\ & =& n{(n-1)}^3.\hfill \end{array}$$

Conversely, let G be a graph with ${\xi }^D\left(G\right)=n{(n-1)}^3$. Suppose that G is not Hamiltonian-connected; then there exists a pair $(x,y)\in V\left(G\right)$ such that $D(x,y)\le (n-2)$ and therefore ${\xi }^D\left(G\right)<n{(n-1)}^3.$ This is a contradiction and therefore G is Hamiltonian-connected. □

Corollary 1. 

For $n\ge 3,$ we have ${\xi }^D\left(K_n\right)=n{(n-1)}^3$, where $K_n$ is the complete graph with n vertices.

Corollary 2. 

For the wheel graph $W_n$ with n vertices $n>3,$ ${\xi }^D\left(W_n\right)=n{(n-1)}^3$.

Theorem 3. 

$$\begin{array}{ccc}\hfill {\xi }^D\left(C_n\right)& =& \left\{\begin{array}{cc}{\displaystyle \frac{1}{4}}{n}^2(n-1)(3n-4)\hfill & \mathit{if}n\mathit{is}\mathit{even},\hfill \\ \\ {\displaystyle \frac{1}{4}}n(3n-1){(n-1)}^2\hfill & \mathit{if}n\mathit{is}\mathit{odd}.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

Consider the vertex set $V=\{u_i:1\le i\le n\}$ of $C_n.$ For any pair of vertices $u_i,u_j\in V,$ the detour distance between them is given by

$$\begin{array}{ccc}\hfill D(u_i,u_j)& =& \left\{\begin{array}{cc}n-|j-i|\hfill & ,\mathrm{if}|j-i|<\lceil {\displaystyle \frac{n}{2}}\rceil \hfill \\ |j-i|\hfill & ,\mathrm{if}|j-i|\ge \lceil {\displaystyle \frac{n}{2}}\rceil .\hfill \end{array}\right.\hfill \end{array}$$

Also for any $u_i\in V,$ $e_D\left(u_i\right)=(n-1).$

• Case 1: When n is even, there are n pairs of vertices in $C_n$ with detour distance $(n-1)$, $(n-2)$, $(n-3),\dots ,({\displaystyle \frac{n}{2}}+1)$ and ${\displaystyle \frac{n}{2}}$ pairs of vertices with detour distance ${\displaystyle \frac{n}{2}}$. Hence,

  $$\begin{array}{ccc}\hfill {\xi }^D\left(G\right)& =& \sum _{\{u_i,u_j\}\subseteq V}[e_D\left(u_i\right)+e_D\left(u_j\right)]D(u_i,u_j)\hfill \\ & =& \sum _{\{u_i,u_j\}\subseteq V}[(n-1)+(n-1)]D(u_i,u_j)\hfill \\ & =& 2(n-1)\left[n(n-1)+n(n-2)+\dots +n\left({\displaystyle \frac{n}{2}}+1\right)+\left({\displaystyle \frac{n}{2}}\right)\left({\displaystyle \frac{n}{2}}\right)\right]\hfill \\ & =& {\displaystyle \frac{1}{4}}{n}^2(n-1)(3n-4).\hfill \end{array}$$
• Case 2: When n is odd, there are n pairs of vertices in $C_n$ with detour distance $(n-1)$, $(n-2)$, $(n-3),\dots ,\left({\displaystyle \frac{n+1}{2}}\right)$.

  $$\begin{array}{ccc}\hfill {\xi }^D\left(G\right)& =& \sum _{\{u_i,u_j\}\subseteq V}[e_D\left(u_i\right)+e_D\left(u_j\right)]D(u_i,u_j)\hfill \\ & =& \sum _{\{u_i,u_j\}\subseteq V}[(n-1)+(n-1)]D(u_i,u_j)\hfill \\ & =& 2(n-1)\left[n(n-1)+n(n-2)+\dots +{\displaystyle \frac{1}{2}}n(n+1)\right]\hfill \\ & =& {\displaystyle \frac{1}{4}}n(3n-1){(n-1)}^2.\hfill \end{array}$$
  □

Let $C_n^{*}$ be a graph obtained by subdividing the edges of $C_n$ and $L\left(C_n\right)$ be the line graph of $C_n$. Then it is easy to see that ${\xi }^D\left(C_n^{*}\right)=2n^2(2n-1)(3n-2)$ and ${\xi }^D\left(L\left(C_n\right)\right)={\xi }^D\left(C_n\right)$.

In the next theorem, we will obtain the detour eccentric sum for the complete bipartite graph $K_{m,n}.$

Theorem 4. 

$$\begin{array}{ccc}\hfill {\xi }^D\left(K_{m,n}\right)& =& \left\{\begin{array}{cc}2m\left(8m^3-14m^2+9m-2\right)\hfill & ,m=n\hfill \\ \\ m\left(4m^3+8m^{2}n-10m^2+4m{n}^2+8m-10mn+n-2\right)\hfill & ,m\ne n.\hfill \end{array}\right.\hfill \end{array}$$

Proof. 

Suppose $m=n$. Let $V_1=\{u_i:1\le i\le m\}$ and $V_2=\{v_i:1\le i\le m\}$ be the equipotent sets of the partition of $K_{m,m}$. The detour distance matrix $M_D\left(G\right)$ of G is

$$M_D\left(G\right)=\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em} u_i \hspace{1em} u_j \hspace{1em} v_i\hspace{1em} v_j\hfill \\ \begin{array}{cc}\begin{array}{c}{u}_i\\ u_j\\ v_i\\ v_j\end{array}& \left[\begin{array}{cccc}0& 2m-2& 2m-1& 2m-1\\ 2m-2& 0& 2m-1& 2m-1\\ 2m-1& 2m-1& 0& 2m-2\\ 2m-1& 2m-1& 2m-2& 0\end{array}\right]\end{array}\hfill \end{array}$$

and $e_D\left(u_i\right)=e_D\left(v_j\right)=2m-1$ for all $u_i\in V_1$ and $v_i\in V_2.$ Since $|V_1|=|V_2|=m,$ $\left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)$ pairs of vertices can be obtained from each of $V_1$ and $V_2$, and ${\displaystyle m^2}$ pairs of vertices can be obtained by selecting one vertex each from $V_1$ and $V_2$. Hence,

$$\begin{array}{ccc}\hfill {\xi }^D\left(G\right)& =& \sum _{\{u_i,u_j\}\subseteq V_1}[e_D\left(u_i\right)+e_D\left(u_j\right)]D(u_i,u_j)+\sum _{\{v_i,v_j\}\subseteq V_2}[e_D\left(v_i\right)+e_D\left(v_j\right)]D(v_i,v_j)+\hfill \\ & & \sum _{u_i\in V_1,v_j\in V_2}[e_D\left(u_i\right)+e_D\left(v_j\right)]D(u_i,v_j)\hfill \\ & =& \left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)[e_D\left(u_i\right)+e_D\left(u_j\right)]D(u_i,u_j)]+\left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)[e_D\left(v_i\right)+e_D\left(v_j\right)]D(v_i,v_j)+\hfill \\ & & m^2[e_D\left(u_i\right)+e_D\left(v_j\right)]D(u_i,v_j)\hfill \\ & =& \left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)\left[(2m-1)+(2m-1)\right](2m-2)+\left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)\left[(2m-1)+(2m-1)\right](2m-2)+\hfill \\ & & m^2[(2m-1)+(2m-1)](2m-1)\hfill \\ & =& 4\left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)(2m-1)(2m-2)+2m^2{(2m-1)}^2\hfill \\ & =& 2m(8m^3-14m^2+9m-2).\hfill \end{array}$$

Suppose $m\ne n$. Without loss of generality let $m<n.$ Let $V_1=\{u_i:1\le i\le m\}$ and $V_2=\{v_j:1\le j\le n\}$ be the two sets of the partition of $K_{m,n}.$ Since the circumference (length of the longest cycle) of $K_{m,n}$ is equal to $2min\{m,n\},$ for any adjacent pair of vertices $u_i,v_j$ in $K_{m,n}$ the detour distance between them is $D(u_i,v_j)=2min\{m,n\}-1=2m-1,$ the detour distance between any pair of vertices in $V_1$ is $D(u_i,u_j)=2m-2$, and the detour distance between any pair of vertices in $V_2$ is $D(v_i,v_j)=2m.$ Also $e_D\left(u_i\right)=2m-1$ for all $u_i\in V_1$ and $e_D\left(v_j\right)=2m$ for all $v_j\in V_2.$ Since $|V_1|=m$ and $|V_2|=n,$ $\left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)$ pairs of vertices can be obtained from $V_1,$ $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)$ pairs of vertices can be obtained from $V_2$, and ${\displaystyle mn}$ pairs of vertices can be obtained by selecting one vertex each from $V_1$ and $V_2$. Hence,

$$\begin{array}{ccc}\hfill {\xi }^D\left(G\right)& =& \sum _{\{u_i,u_j\}\subseteq V_1}[e_D\left(u_i\right)+e_D\left(u_j\right)]D(u_i,u_j)+\sum _{\{v_i,v_j\}\subseteq V_2}[e_D\left(v_i\right)+e_D\left(v_j\right)]D(v_i,v_j)+\hfill \\ & & \sum _{u_i\in V_1,v_j\in V_2}[e_D\left(u_i\right)+e_D\left(v_j\right)]D(u_i,v_j)\hfill \\ & =& \left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)[e_D\left(u_i\right)+e_D\left(u_j\right)]D(u_i,u_j)+\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)[e_D\left(v_i\right)+e_D\left(v_j\right)]D(v_i,v_j)+\hfill \\ & & mn[e_D\left(u_i\right)+e_D\left(v_j\right)]D(u_i,v_j)\hfill \\ & =& 2\left({\displaystyle \genfrac{}{}{0pt}{}{m}{2}}\right)(2m-1)(2m-2)+8\left({\displaystyle \genfrac{}{}{0pt}{}{n}{2}}\right)m^2+mn(4m-1)(2m-1)\hfill \\ & =& m\left(4m^3+8m^{2}n-10m^2+4m{n}^2+8m-10mn+n-2\right).\hfill \end{array}$$

□

The following corollary is a particular case of Theorem 4 and we get the eccentric distance sum of stars for $n\ge 3$, as established in [25].

Corollary 3. 

${\xi }^D\left(K_{1,n-1}\right)=4n^2-9n+5,$ where $n\ge 3.$

A graph is defined as n-partite if its vertex set can be partitioned into n mutually disjoint subsets such that no two vertices within any given subset are adjacent [26]. Here we consider a complete n-partite graph $K_{1,2,\dots ,n}$ with vertex set V = $V_1\cup V_2\cup V_3,\dots ,\cup V_n$ and $n\left(V_i\right)=i,1\le i\le n.$ Figure 1 shows the complete 4-partite graph $K_{1,2,3,4}$.

Theorem 5. 

For $n>2,$ ${\xi }^D\left(K_{1,2,\dots ,n}\right)={\displaystyle \frac{1}{16}}(n^2+n){(n^2+n-2)}^3.$

Proof. 

Let the partite sets be $X_1=\left\{a\right\},$ $X_2=\{b_1,b_2\},$ $X_3=\{c_1,c_2,c_3\},$..., $X_n=\{n_1,n_2,n_3,\dots ,n_n\}$. For any pair of vertices in $X_i$ for $1\le i\le n,$ $D(u,v)=(S_n-1)$, where $S_n$ is the sum of the first n natural numbers. It is easy to see that $e\left(v\right)=(S_n-1$) and $S_D\left(v\right)={(S_n-1)}^2$.

$$\begin{array}{ccc}\hfill {\xi }^D\left(K_{1,2,...n}\right)& =& \sum _{v\in X_i}e\left(v\right)S_D\left(v\right)\hfill \\ & =& S_n{(S_n-1)}^3\hfill \\ & =& {\displaystyle \frac{1}{16}}(n^2+n){(n^2+n-2)}^3.\hfill \end{array}$$

□

The generalized Sierpiński graph of $G=(V,E)$, where $V=\{1,2,\dots ,k\}$ [27] of dimension n denoted by $S(n,G)$, is a graph with a vertex set ${1,2,\dots ,k}^n$ and an edge set defined by $u,v,$ where $u=(u_1,\dots ,u_n)$ and $v=(v_1,\dots ,v_n)$ is an edge of $S(n,G)$ iff there exists $i\in 1,2,\dots ,n$ such that

1

$u_j=v_j$ when $j<i.$

2

$u_i\ne v_i$ and $(u_i,v_i)\in E\left(G\right).$

3

$u_j=v_i$ and $v_j=u_i$ if $j>i$.

We denote $S(t,K_n)$ as the generalized Sierpiński of $K_n$ of dimension t. Figure 2 shows the two-level Sierpinski of $K_5$, that is, $S(2,K_5)$.

Theorem 6. 

For $n>3$, ${\xi }^D\left(S(t,K_n)\right)=n^t{(n^t-1)}^3$.

Proof. 

Consider the generalized Sierpiński $S(t,K_n)$, $n>3$. Let $u,v\in V\left(S\right(t,G\left)\right)$. Then there exists a Hamilton path connecting each pair of vertices in $S(t,K_n)$. There are $n^t$ vertices in $S(t,K_n)$ and $D(u,v)$ is $(n^t-1)$. $S_D\left(u\right)={(n^t-1)}^2$ and $e\left(u\right)=(n^t-1).$ Hence ${\xi }^D\left(S(t,G)\right)=\sum e\left(u\right)S_D\left(u\right)=n^t{(n^t-1)}^3.$ □

The triangular snake graph $T_n$ [28] is defined as a graph obtained from a path $u_1,u_2,\dots ,u_{n+1}$ by joining $u_i$ and $u_{i+1}$ to a new vertex $v_i$ for $1\le i\le n$. Figure 3 shows the triangular snake $T_n$.

A computation similar to the preceding ones proves the following theorems.

Theorem 7. 

If G is a triangular snake graph of order $2n+1$, then

$$\begin{array}{ccc}\hfill {\xi }^D\left(G\right)& =& \left\{\begin{array}{cc}{\displaystyle \frac{1}{6}}(25n^4+86n^3+44n^2-8n)\hfill & ,n\mathit{is}\mathit{even}\hfill \\ \\ {\displaystyle \frac{1}{6}}(25n^4+86n^3+44n^2-14n+3)\hfill & ,n\mathit{is}\mathit{odd}.\hfill \end{array}\right.\hfill \end{array}$$

A graph G is said to be a starred complete graph [29] if it is obtained from $K_n$ by adding m pendant edges with a vertex in $K_n$. It is denoted by $K_n{S}_m.$ Figure 4 shows the starred complete graph $K_5{S}_3.$

Theorem 8. 

For $n>3$ and $m>1$, ${\xi }^D\left(K_n{S}_m\right)=n^4+2m{n}^3-2m{n}^2+2m^{2}n-2n^3+2n-m-1.$

Building upon the exploration of the detour eccentric sum index across the various networks presented, the next section delves deeper into specific detour parameters, such as detour diameter and detour radius. We investigated the correlation of these parameters with the physicochemical properties of drugs. This progression allows us to demonstrate the practical utility of detour-based graph indices, particularly in predicting properties of drugs.

3. Correlation of Detour Graph Parameters with Drug Properties

Distance-based topological indices serve as valuable tools for deciphering the intricate connectivity and interatomic distances within molecular structures. Specifically, the graph parameter detour distance offers insights into molecular stability and facilitates the differentiation of isomers. Furthermore, detour eccentricity plays a predictive role in assessing the reactivity of molecules. Notably, molecules exhibiting larger detour eccentricity values may possess potentially intricate reaction pathways, a significant consideration for optimizing chemical reaction design. In this study, we focus on two key parameters, detour diameter (DD) and detour radius (DR), the maximum and minimum values of the detour eccentricities among the vertices of the graph. We investigate the physical and chemical characteristics of drugs used to treat breast cancer and malaria, conducting a correlation analysis to explore the relationship between these detour-based parameters and drug efficacy. The outcome of this analysis give valuable insights into the structural determinants of drug activity.

3.1. Correlation with Anti-Malaria Drugs

Female Anopheles mosquitoes are the source of malaria, a potentially fatal illness that affects people. P. falciparum and P. vivax are two of the deadliest parasites that cause malaria. Significant health risks are associated with these parasites, particularly in Saharan parts of Africa. In particular, malaria remains a global health challenge due to the emergence of drug-resistant Plasmodium strains, underscoring the urgent need for efficient drug discovery methods [30].

This study investigates the structural characteristics of nine anti-malarial drugs: chloroquine, doxycycline, mefloquine, pyrimethamine, piperaquine, amodiaquine, primaquine, atovaquone, and lumefantrine. Refer to Figure 5, Figure 6, Figure 7 and Figure 8 for the molecular graphs of these drugs.

We examined their physical characteristics, specifically boiling point (BOP), molar volume (MOV), molar refraction (MRF), enthalpy (EP), flash point (FLP), and polarizability (PO). The experimental values for these properties, obtained from PubChem, are presented in

Table 1. Physical properties of anti-malaria drugs (source: PubChem).

Drugs (Chemical Formula)	BOP (°C at 760 mmHg)	EP (kJ/mol)	FLP (°C)	MRF (m3/mol)	MOV (cm3)	PO 10−24 (cm3)
Chloroquine (C18H26ClN3)	460.6	72.1	232.23	97.4	287.9	38.6
Amodiaquine (C20H22ClN3O)	478	77	242.9	105.5	282.8	41.8
Mefloquine (C17H16F6N2O)	415.7	70.5	205.2	83	273.4	32.9
Piperaquine (C40H52N6)	721.1	105.3	389.9	153.7	414.2	60.9
Primaquine (C15H21N3O)	451.1	71	226.6	80.5	230.3	31.9
Lumefantrine (C30H32Cl3NO)	642.5	99.6	342.3	151	422.3	59.9
Atovaquone (C22H19ClO3)	535	85.4	277.3	99.5	271.8	39.5
Pyrimethamine (C12H13N5)	368.4	61.5	176.6	67.1	180.2	26.6
Doxycycline (C22H24N2O8)	762.6	116.5	415	109	271.1	43.2

. We analyzed the correlation between the detour diameter (DD) and detour radius (DR) of the molecular graphs of these drugs and their respective physical properties. The results of this correlation analysis are presented in

Table 2. The correlation between DD and DR of the molecular structures of anti-malaria drugs and six physical properties.

Properties	DD	DR
BOP	0.79522	0.930662
EP	0.764841	0.94114
FLP	0.79518	0.930658
MRF	0.97228	0.81235
MOV	0.943883	0.737934
PO	0.972228	0.812386

. Figure 9 shows the DD and DR values for the molecular structures of anti-malaria drugs.

Our analysis revealed distinct correlations between detour-based parameters and the physical characteristics of the anti-malarial drugs. Specifically, the detour radius exhibited a high positive correlation of 0.93, 0.94, and 0.93, respectively, with boiling point (BOP), enthalpy (EP), and flash point (FLP). Conversely, the detour diameter demonstrated a high positive correlation of 0.94, 0.97, and 0.97, respectively, with molar volume (MOV), molar refraction (MRF), and polarizability (PO).

3.2. Correlation with Breast Cancer Drugs

Cancer ranks as one of the leading causes of mortality globally, impacting individuals across all demographics. Cancer develops when cells within a particular tissue begin to proliferate uncontrollably for hormonal or other reasons, which can result in the creation of abnormal growths or masses. Sustained investments in research and widespread awareness campaigns for breast cancer have advanced the field’s understanding of diagnosis and therapy.

Subsequently, we extended our analysis to fourteen breast cancer medications: abemaciclib, megestrol acetate, anastrozole, capecitabine, abraxane, everolimus, exemestane, fulvestrant, ixabepilone, cyclophosphamide, tamoxifen, letrozole, methotrexate, and thiotepa (refer to Figure 10, Figure 11, Figure 12, Figure 13 and Figure 14).

The experimental values for seven key properties—boiling point (BOP), melting point (MLP), enthalpy (EP), flash point (FLP), molar refractivity (MRF), molar volume (MOV), and polarizability (PO) —were compiled and are presented in

Table 3. Physicochemical attributes of breast cancer pharmaceuticals (source: PubChem).

Drugs (Chemical Formula)	BOP (°C at 760 mmHg)	MLP (°C)	EP (kJ/mol)	FLP (°C)	MRF (m3/mol)	MOV (cm3)	PO 10−24 (cm3)
Abemaciclib (C27H32N8O)	689.3	-	101	370.7	140.4	382.3	55.7
Abraxane (C47H51NO14)	957.1	-	146	532.6	219.3	610.6	86.9
Anastrozole (C17H19N5)	469.7	81.5	73.2	237.9	90	270.3	35.7
Capecitabine (C15H22FN3O6)	517	115.5	112	315.5	82.3	256.5	32.2
Cyclophosphamide (C7H15Cl2N2O2P)	336.1	51	57.9	157.8	38.4	156.2	23
Everolimus (C53H83NO14)	998.7	998.7	165.1	557.8	257.7	811.2	102.2
Exemestane (C20H24O2)	453.7	155.3	101.3	367.4	197.5	495.1	89.6
Fulvestrant (C32H47F5O3S)	674.8	104	104.1	361.9	154	505.1	61.1
Ixabepilone (C27H42N2O5S)	593.5	-	110	385.6	121	404.3	50.5
Letrozole (C17H11N5)	563.5	181	84.7	294.1	124.9	351.4	41.4
Megestrol Acetate (C24H32O4)	507.1	214	77.7	77.7	107.2	318.6	28.4
Methotrexate (C20H22N8O5)	537.0	192	144	337.4	140.7	393.0	61.6
Tamoxifen (C26H29NO)	482.3	96	74.7	140	118.9	118.9	47.1
Theotepa (C6H12N3P)	270.2	51.5	50.8	117.2	49.1	125.8	19.5

.

We then calculated the detour diameter (DD) and detour radius (DR) for each of these drugs and investigated their correlation with the aforementioned properties. The results of this correlation analysis are summarized in

Table 4. The correlation between DD and DR of the molecular structures of breast cancer drugs and their properties.

Properties	DD	DR
BOP	0.939224	0.861037
MLP	0.795271	0.879994
EP	0.943927	0.886779
FLP	0.83829	0.713036
MRF	0.96088	0.900722
MOV	0.961042	0.926225
PO	0.961042	0.900802

. Also, a comparison bar chat is shown in Figure 15.

The detour diameter exhibits strong positive correlations of 0.94, 0.94, 0.96, 0.96, 0.96, and 0.83 with boiling point, enthalpy, molar refraction, molar volume, polarizability, and flash point, respectively. In contrast, the detour radius demonstrates a comparatively high correlation of 0.87 with the melting point.

The strong correlations observed between detour parameters and the physicochemical properties of both breast cancer and malaria drugs provide a solid foundation for further quantitative analysis. As reliable graph invariants that improve molecular property estimation by reflecting long-range connectivity and network durability in molecular graphs, this demonstrates the usefulness of detour-based indices in cheminformatics studies. Building on these findings, the next section presents a detailed Quantitative Structure–Property Relationship (QSPR) study comparing the predictive performance of the two indices: the novel detour-based index DESI and the traditional distance-based index EDS. This analysis aims to validate and demonstrate the practical applicability of these indices in modeling drug properties, thereby supporting their use as effective computational tools in drug design and development.

4. Comparative QSPR Modeling of Drug Properties Using DESI and EDS

We conducted curvilinear regression analysis—including linear, quadratic, and cubic models—to rigorously evaluate the relationships between the proposed topological indices and the physicochemical properties of anti-malarial and breast cancer drugs. We predict the best fit topological descriptor among the DESI and EDS for pharmaceutical properties using a curvilinear regression analysis of order $n,n=1,2,3$.

The equation of curvilinear regression of the properties of the drugs under consideration is

$$\begin{array}{c}\hfill P\left(D\right)=\left[\sum _{i=1}^n{\lambda }_i{\left(T\right)}^i\right]+\omega \end{array}$$

(3)

where P(D) represents a specific property associated with the drug molecule, ${\lambda }_i$s are the regression coefficients of the topological descriptors, and $\omega$ is the regression constant.

4.1. Curvilinear Regression Analysis of the Properties of Anti-Malaria Drugs Using DESI and EDS

Using our indices, DESI and EDS, and six physicochemical characteristics of the anti-malaria medications, we created regression models. The DESI and EDS for the drugs were calculated and are presented in

Table 5. The DESI and EDS values of anti-malaria drugs.

Drugs	DESI	EDS
Chloroquine (C18H26ClN3)	62,730	25,550
Amodiaquine (C20H22ClN3O)	102,153	33,161
Mefloquine (C17H16F6N2O)	53,396	11,184
Piperaquine (C40H52N6)	497,756	194,916
Primaquine (C15H21N3O)	41,168	11,964
Lumefantrine (C30H32Cl3NO)	328,156	91,420
Atovaquone (C22H19ClO3)	122,649	37,819
Pyrimethamine (C12H13N5)	20,756	7136
Doxycycline (C22H24N2O8)	288,272	41,176

.

Curvilinear regression analysis of order $n,n=1,2,3$ is performed to predict the best fit for the physicochemical properties. Based on the $R^2$ and RMSE values, we observed that the second-order regression model is the best fit for the mentioned properties using our topological descriptors DESI and EDS. These observations are presented in

Table 6. Predictive fits from second-order regression model of anti-malaria drugs.

Physicochemical Properties	DESI		EDS
	$R^{\mathbf{2}}$	RMSE	$R^{\mathbf{2}}$	RMSE
BOP	0.9191	45.82	0.67	92.1782
EP	0.924	5.95	0.6451	12.87
FLP	0.9191	27.72	0.6725	55.76
MRF	0.8623	12.77	0.9786	5.0417
MOV	0.7095	48.95	0.8714	32.56
PO	0.862	5.0713	0.9789	1.9837

. The second-order best-fit regression equations for property prediction using the topological descriptors DESI and EDS are represented by the following Equations (4)–(9). The best predictor for boiling point (BOP), enthalpy (EP) and flash point (FLP) is DESI (${\xi }^D\left(G\right)$) (refer to Equations (4)–(6)). But the best predictor for molar refraction (MRF), polarizability (PO), and molar volume (MOV) is EDS (${\xi }^d\left(G\right)$) (refer to Equations (7)–(9)). The findings [31] indicate that molar volume cannot be accurately predicted using these degree-based topological indices. However, our EDS index proves to be a reliable predictor for this property.

$$\begin{array}{c}\hfill BOP=343.661+0.0018\left({\xi }^D\left(G\right)\right)-2.1537\times {10}^{-9}{\left({\xi }^D\left(G\right)\right)}^2\end{array}$$

(4)

$$\begin{array}{c}\hfill EP=56.7428+2.7641\times {10}^{-4}\left({\xi }^D\left(G\right)\right)-3.618\times {10}^{-10}{\left({\xi }^D\left(G\right)\right)}^2\end{array}$$

(5)

$$\begin{array}{c}\hfill FLP=161.6218+0.0011\left({\xi }^D\left(G\right)\right)-1.3021\times {10}^{-9}{\left({\xi }^D\left(G\right)\right)}^2\end{array}$$

(6)

$$\begin{array}{c}\hfill MRF=63.0628+0.0013\times \left({\xi }^d\left(G\right)\right)-4.515\times {10}^{-9}{\left({\xi }^d\left(G\right)\right)}^2\end{array}$$

(7)

$$\begin{array}{c}\hfill MOV=189.2628+0.0033\left({\xi }^d\left(G\right)\right)-1.077\times {10}^{-8}{\left({\xi }^d\left(G\right)\right)}^2\end{array}$$

(8)

$$\begin{array}{c}\hfill PO=24.9784+5.35\times {10}^{-4}\left({\xi }^d\left(G\right)\right)-1.7959\times {10}^{-9}{\left({\xi }^d\left(G\right)\right)}^2\end{array}$$

(9)

4.2. Curvilinear Regression Analysis of the Properties of Breast Cancer Drugs Using DESI and EDS

Table 7. The DESI and EDS values of breast cancer drugs.

Drugs	DESI	EDS
Abemaciclib (C27H32N8O)	359,794	163,238
Abraxane (C47H51NO14)	1,930,037	4,545,804
Anastrozole (C17H19N5)	35,632	15,488
Capecitabine (C15H22FN3O6)	64,841	40,511
Cyclophosphamide (C7H15Cl2N2O2P)	6758	3478
Everolimus (C53H83NO14)	4,522,920	785,429
Exemestane (C20H24O2)	110,916	13,740
Fulvestrant (C32H47F5O3S)	779,714	274,628
Ixabepilone (C27H42N2O5S)	329,896	84,211
Letrozole (C17H11N5)	35,481	19,481
Megestrol Acetate (C24H32O4)	198,979	33,209
Methotrexate (C20H22N8O5)	209,154	116,770
Tamoxifen (C26H29NO)	81,847	48,709
Theotepa (C6H12N3P)	2062	992

shows the computed DESI and EDS values of the molecular graphs of breast cancer drugs.

We discovered that, with the exception of MLP, every other physical property had the second-order model as the best match when we performed the curvilinear regression analysis of order $n,n=1,2,3$. However, order three is the model that fits MLP the best.

Table 8. Predictive fits from curvilinear regression model of Breast cancer drugs.

Physicochemical Properties	Order	DESI		EDS
		${\mathit{R}}^{\mathbf{2}}$	RMSE	${\mathit{R}}^{\mathbf{2}}$	RMSE
BOP	2	0.84	94.19	0.82	99.59
MLP	3	0.97	55.06	0.94	73.71
EP	2	0.90	11.92	0.89	12.47
FLP	2	0.75	87.88	0.8	87.20
MRF	2	0.91	19.31	0.91	19.43
MOV	2	0.88	72.82	0.86	78.55
PO	2	0.91	7.66	0.91	7.70

presents these observations. The best-fit regression equations for predicting the characteristics of breast cancer medications using the topological descriptors DESI and EDS are represented by the following Equations (10)–(16). It should be noted that DESI is the best predictor index for all other attributes, with the exception of FLP. In contrast, EDS, which is shown in Equation (13), is the best predictor of FLP.

$$\begin{array}{c}\hfill BOP=442.1849+4.1060\times {10}^{-4}\left({\xi }^D\left(G\right)\right)-6.3935\times {10}^{-11}{\left({\xi }^D\left(G\right)\right)}^2\end{array}$$

(10)

$$\begin{array}{c}\hfill MLP=0.0015\left({\xi }^D\left(G\right)\right)-2.1087\times {10}^{-9}{\left({\xi }^D\left(G\right)\right)}^2+4.0229\times {10}^{-16}{\left({\xi }^D\left(G\right)\right)}^3\end{array}$$

(11)

$$\begin{array}{c}\hfill EP=69.2583+5.8417\times {10}^{-5}\left({\xi }^D\left(G\right)\right)-8.2765\times {10}^{-12}{\left({\xi }^D\left(G\right)\right)}^2\end{array}$$

(12)

$$\begin{array}{c}\hfill FLP=188.1652+6.0735\times {10}^{-4}\left({\xi }^d\left(G\right)\right)-1.1698\times {10}^{-10}{\left({\xi }^d\left(G\right)\right)}^2\end{array}$$

(13)

$$\begin{array}{c}\hfill MRF=82.98+1.0761\times {10}^{-4}\left({\xi }^D\left(G\right)\right)-1.5368\times {10}^{-11}{\left({\xi }^D\left(G\right)\right)}^2\end{array}$$

(14)

$$\begin{array}{c}\hfill MOV=221.32+3.3049\times {10}^{-4}\left({\xi }^D\left(G\right)\right)-4.4833\times {10}^{-11}{\left({\xi }^D\left(G\right)\right)}^2\end{array}$$

(15)

$$\begin{array}{c}\hfill PO=32.8956+4.2655\times {10}^{-5}\left({\xi }^D\left(G\right)\right)-6.0894\times {10}^{-12}{\left({\xi }^D\left(G\right)\right)}^2\end{array}$$

(16)

5. Conclusions

In conclusion, this study thoroughly investigated the detour eccentric sum (DESI), establishing its bounds and exploring its behavior across diverse graph classes that serve as models for various interconnection network topologies, including ring and client–server configurations. Furthermore, we demonstrated the efficacy of graph parameters based on detour, particularly detour diameter and detour radius, in predicting the physicochemical properties of both anti-malarial and breast cancer drugs. The strong correlations observed highlight the potential of these parameters as valuable tools in drug discovery and development, offering a computationally efficient alternative to traditional, time-consuming experimental methods. This bridges graph theory with chemical property prediction, advancing modeling techniques for drug discovery and material science applications. This study highlighted the robust predictive capabilities of the DESI and EDS index in modeling critical physicochemical properties relevant to drug efficacy. The results highlight how indices derived from detour can be used as computationally efficient and cost-effective substitutes for experimental techniques, facilitating rapid drug development, particularly in the fight against malaria, where it is crucial to identify promising candidates. The computational time is linear for all the experiments. This study can be extended to molecular graphs of other cancer drugs. In addition, simulation results can be correlated with clinical studies.