Computing Some Eccentric Connectivity Indices Based on Vertices and Edges of Backbone DNA Graphs

Abstract

Graph theory plays a central role in mathematics, biology, chemistry, computer science, and related disciplines. It has many applications in everyday life, particularly in chemistry, biology, and network theory. Chemical graph theory is a subfield devoted to the mathematical representation and analysis of molecular structures. It is also used in the calculation of topological indices and the prediction of many chemical properties. A topological index is a numerical parameter that characterizes the molecular structure based on its corresponding molecular graph. Consider a simple molecular graph $G=(V(G),E(G))$ with no loops, multiple edges, or directed edges. Numerous topological indices have been defined and studied for molecular graphs. The vertex and edge eccentric connectivity indices, along with their modified versions, play a significant role in QSPR/QSAR studies within the framework of chemical graph theory. Recently, various studies have been conducted on the backbone DNA graphs. The repeating cycles in the backbone DNA graphs indicate that the graph possesses a periodic and regular symmetry. This symmetry is taken into account in deriving closed formulas for topological index values such as the eccentric connectivity indices. In this paper, some eccentric connectivity indices based on vertices and edges of backbone DNA graphs $DN{A}_n$ have been computed. Furthermore, the two-dimensional plots of $DN{A}_n$ were generated using Cartesian coordinates.

1. Introduction

Graph theory is the most powerful tool for chemical networks, biological networks, and mathematical networks. Specifically, it is utilized in the assessment of molecular descriptors for quantitative structure–property relationship (QSPR) and quantitative structure–activity relationship (QSAR) analyses in chemical research. During this examination, graph–theoretic parameters called topological indices are calculated [1]. A topological index represents a numerical descriptor of molecular structure designed to establish quantitative correlations between a compound’s chemical constitution and its physical, chemical, or biological properties [2,3]. Additionally, topological indices are molecular descriptors that characterize the structures of chemical compounds and help us predict specific physicochemical properties such as boiling point, vaporization enthalpy, stability, etc. [4].

Recent advancements in the study of topological indices have garnered substantial attention. Numerous topological descriptors have been developed, many of which have demonstrated significant utility in the field of theoretical chemistry, particularly within the context of quantitative structure–activity relationship (QSAR) and quantitative structure–property relationship (QSPR) modeling [5,6,7,8,9,10,11,12,13,14,15]. A chemical graph is defined as a graph where each vertex represents an atom within the molecular structure, and the edges between these vertices correspond to the covalent bonds that link the atoms together [2,16].

Let $G=(V(G),E(G))$ be a simple undirected graph with order n (number of vertices) and size m (number of edges). In the following, we present several definitions that are fundamental to the study of graph–theoretical properties. Let $v\in V(G)$. The open neighborhood of v is $N_G(v)=\{u\in V(G)| uv\in E(G)\}$ and the closed neighborhood of v is $N_G[v]=N_G(v)\cup \left\{v\right\}$. The degree of a vertex v in a graph G, denoted by $de{g}_G(v)$, is defined as the number of edges incident to v, or equivalently, the size of its open neighborhood [17]. The distance between two vertices u and v in a graph G, denoted as $d_G\left(u,v\right)$, is defined as the length of the shortest path connecting them. The maximum distance between the vertex v and any other vertex u in G is called the eccentricity of v and it is denoted by ${\epsilon }_G(v)$. Clearly, we have ${\epsilon }_G(v)={\max }_{u\in V(G)}{d}_G(u,v)$. Let $f=uv\in E(G)$. Similarly, the degrees and eccentricity values of the edge f can be calculated. The degree of an edge f is computed as ${\mathrm{deg}}_G(f)={\mathrm{deg}}_G(u)+{\mathrm{deg}}_G(v)-2$ and it is denoted by ${\mathrm{deg}}_G(f)$. Let $f_1$ and $f_2$ be two edges in the graph G, and let $f_1=u_1{v}_1$ and $f_2=u_2{v}_2$. The maximum distance between the edge $f_1$ and any other edge $f_2$ is defined as $d_G\left(f_1,f_2\right)=\min \left\{d_G\left(u_1,u_2\right),d_G\left(u_1,v_2\right),d_G\left(v_1,u_2\right),d_G\left(v_1,v_2\right)\right\}$ and it is also denoted by $d_G\left(f_1,f_2\right)$. The largest distance between the edge $f_1$ and any other edge $f_2$ of the graph G is called the eccentricity value of the edge $f_1$ and it is also denoted by ${\epsilon }_G(f_1)$. That is, ${\epsilon }_G(f_1)=\max \left\{d_G(f_1,f_2)| f_2\in E(G)\right\}$ [18].

The Wiener index, which was first introduced by H. Wiener [19], represents the earliest topological index used in chemical graph theory. It is mathematically defined as one half of the total sum of distances between all vertex pairs in a graph G. So, it is defined as:

$$W(G)={\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^n{d}_G(v_i,v_j)}}.$$

(1)

Since the introduction of the Wiener index, many additional topological indices have been formulated, some degree-based and others distance-based, reflecting different structural characteristics of molecular graphs. The eccentric connectivity indices based on vertex and edge have been defined and investigated. The eccentric connectivity index ${\xi }^c(G)$ was first defined by Sharma et al. [20]. This index has proven to be a valuable tool in constructing a wide range of mathematical models for the quantitative prediction of diverse biological activities and it has been further studied by many authors [21,22,23,24,25].

The eccentric connectivity index ${\xi }^c(G)$ for any graph G is defined as:

$${\xi }^c(G)={\displaystyle \sum _{u\in V(G)}{\epsilon }_G(u)}{\mathrm{deg}}_G(u)$$

(2)

Then, in [26], the modified eccentric connectivity index ${\xi }_c(G)$ has been defined as follows:

$${\xi }_c(G)={\displaystyle \sum _{u\in V(G)}{\epsilon }_G(u)} {\delta }_G(u),$$

(3)

where ${\delta }_G(u)={\displaystyle \sum _{v\in N_G(u)}{\mathrm{deg}}_G(v)}$, that is, the sum of degrees of vertices which is the vertex u neighbor’s; furthermore, it was studied in [27].

Subsequently, a novel topological descriptor known as the edge eccentric connectivity index was proposed by Xu et al. [28] and has since been extensively examined by various authors [29,30,31,32,33,34]. For a given graph G, the edge eccentric connectivity index, denoted by ${\xi }_e^c(G)$, is defined as follows:

$${\xi }_e^c(G)={\displaystyle \sum _{f\in E(G)}{\epsilon }_G(f)}{\mathrm{deg}}_G(f),$$

(4)

where ${\epsilon }_G(f)$ is the eccentricity value and ${\mathrm{deg}}_G(f)$ is the degree of an edge f in graph G.

Then, the modified edge eccentric connectivity index ${\wedge }_e^c(G)$ has been defined in [35,36] and it has been defined as follows:

$${\wedge }_e^c(G)={\displaystyle \sum _{f\in E(G)}{\epsilon }_G(f)} S_G(f),$$

(5)

where $S_G(f)$ is the sum of degrees of edges which is the neighbor’s edges of edge f. It has been further studied by Alishahi and Shalmaee in [30].

The eccentric connectivity index and its edge-based counterpart are distance-dependent topological descriptors that have gained considerable attention in QSAR/QSPR modeling due to their strong predictive capability for the biological activities of various classes of chemical compounds.

Wang et al. [37] examined the structural configuration of the DNA backbone and its conformational stability within complex biological systems. The DNA molecule is composed of a sequential arrangement of nucleotides which constitute its essential molecular components. Each nucleotide is covalently bonded to the next through phosphodiester linkages that alternate between phosphate and sugar (deoxyribose) residues, forming a repetitive sugar–phosphate backbone. Within the framework of graph theory, this molecular structure can be modeled as a graph where nucleotides are represented by vertices and phosphodiester bonds correspond to the edges connecting them. Such graphical representations effectively demonstrate the organization and connectivity patterns among nucleotides in DNA networks. The backbone DNA graph is therefore an important aspect of DNA biophysics and nanotechnology [38]. Certain chemical structures can be modeled using specific classes of graphs, revealing distinct relationships between molecular architecture and graph–theoretical representations. The backbone DNA graphs are five-membered rings, as shown in Figure 1, and they are denoted by $DN{A}_n$ in [39]. Clearly, we have $\left|V(DN{A}_n)\right|=7n-2$ and $\left|E(DN{A}_n)\right|=8n-3$.

The backbone DNA graphs serve as pivotal models in chemical graph theory, facilitating the exploration of DNA’s structural and functional attributes through topological indices. These indices, such as the Zagreb connection indices, offer insights into the structural characteristics of the DNA backbone network, aiding in the understanding of molecular stability and interactions [40]. Moreover, the application of these topological descriptors extends to the development of QSAR and QSPR models, which are instrumental in predicting the biological activities and physicochemical properties of various chemical compounds. In a recent contribution, in [39], several degree-based topological parameters to characterize the structural attributes of backbone DNA networks have been computed. Another study also delved into the Van, R, and S topological indices for networks with backbone DNA, employing direct techniques to derive these descriptors and furthering the comprehension of DNA’s molecular topology [41]. Additionally, the backbone DNA graphs have been further studied by many authors [42,43,44]. Collectively, these research endeavors underscore the significance of backbone DNA graphs in advancing the field of chemical graph theory and their practical applications in molecular modeling and drug discovery.

Following previous investigations, this work focuses on the calculation of several eccentricity-based topological indices, such as the vertex eccentric connectivity index, the edge eccentric connectivity index, the modified vertex, and edge eccentric connectivity indices, applied to backbone DNA graph models. Furthermore, this study calculates some eccentric connectivity indices for backbone DNA graphs, using generalized closed formulas that are valid for any graph length n, thereby translating the limited numerical data in the literature into a theoretical general model. These precise values standardize the molecular distance and connectivity relationship in the DNA structure, providing a highly accurate theoretical basis for QSPR/QSAR studies, in which physicochemical properties such as biological stability are predicted.

2. Main Results

In this section, main four theorems regarding some eccentric connectivity indices of backbone DNA graphs are given. The backbone DNA graphs are connected in all the proofs.

Theorem 1.

The eccentric connectivity index of backbone  $DN{A}_n$ graph of length n is

$${\xi }^c(DN{A}_n)=\left\{\begin{array}{c}48n^2-25n+3 , if n is odd;\\ 48n^2-25n+ 8 , if n is even.\end{array}\right.$$

Proof. 

Considering the definition of backbone $DN{A}_n$ graphs, we have seven types of vertices in $DN{A}_n$ according to location of the vertices. In Figure 2 and Figure 3, labels of the vertices of backbone $DN{A}_n$ graphs are given. When labeling the vertices, on cycle graphs, they were labeled with $v_{i j}$, where $i\in [1,n]$ and $j\in [1,5]$. The vertices on paths connecting cycles graphs were labeled with $u_{i j}$, where $i\in [1,n-1]$ and $j\in [1,2]$. Furthermore, we use $x=\left({\displaystyle \frac{n-1}{2}}\right)$, $y=\left({\displaystyle \frac{n+1}{2}}\right)$, $w=(n-1)$, $p= (n-2)$, $s=\left({\displaystyle \frac{n}{2}}\right)$, and $d=\left({\displaystyle \frac{n}{2}}+1\right)$ in Figure 2 and Figure 3.

In addition, there are seven types of vertices distinguished by location, with different types based on their degrees, the sum of their neighbors’ degrees, the frequencies of the vertices, and the eccentricity values of the vertices.

The degrees of every vertex in $DN{A}_n$ of length n are as follows:

$${\mathrm{deg}}_{DN{A}_n}(v_{11})={\mathrm{deg}}_{DN{A}_n}(v_{n1})=2 \mathrm{and} {\mathrm{deg}}_{DN{A}_n}(v_{i1})=3, \mathrm{where} 2\le i\le n-1;$$

$${\mathrm{deg}}_{DN{A}_n}(v_{i2})=3, \mathrm{where} 1\le i\le n;$$

$${\mathrm{deg}}_{DN{A}_n}(v_{i3})={\mathrm{deg}}_{DN{A}_n}(v_{i4})={\mathrm{deg}}_{DN{A}_n}(v_{i5})=2, \mathrm{where} 1\le i\le n;$$

(6)

$${\mathrm{deg}}_{DN{A}_n}(u_{i1})={\mathrm{deg}}_{DN{A}_n}(u_{i2})=2, \mathrm{where} 1\le i\le n-1.$$

The eccentricity values of the vertices in $DN{A}_n$ of length n can be derived as follows:

$$\left.\begin{array}{l}{\epsilon }_{DN{A}_n}(v_{i1})={\epsilon }_{DN{A}_n}(v_{i3})=4n-4i+2\\ {\epsilon }_{DN{A}_n}(v_{i2})=4n-4i+1\\ {\epsilon }_{DN{A}_n}(v_{i4})={\epsilon }_{DN{A}_n}(v_{i5})=4n-4i+3\end{array}\right\}, where i\le ⌊{\displaystyle \frac{n}{2}}⌋.$$

(7)

Additionally, if n is odd, let $j={\displaystyle \frac{n+1}{2}}$.

Then, we get

$${\epsilon }_{DN{A}_n}(v_{j1})={\epsilon }_{DN{A}_n}(v_{j2})=2n \mathrm{and} {\epsilon }_{DN{A}_n}(v_{j3})={\epsilon }_{DN{A}_n}(v_{j4})={\epsilon }_{DN{A}_n}(v_{j5})=2n+1.$$

(8)

Furthermore,

$$\left.\begin{array}{l}{\epsilon }_{DN{A}_n}(u_{i1})=4n-4i\\ {\epsilon }_{DN{A}_n}(u_{i2})=4n-4i-1\end{array}\right\}, where i\le ⌊{\displaystyle \frac{n-1}{2}}⌋.$$

(9)

Additionally, if n is even, let $j={\displaystyle \frac{n}{2}}$.

Then, we get

$${\epsilon }_{DN{A}_n}(u_{j1})=2n \mathrm{and} {\epsilon }_{DN{A}_n}(u_{j2})=2n.$$

(10)

We proceed by considering the cases n odd and n even.

• Case 1. n is odd.

The eccentricity values of all vertices are given in (7), where $1 \le i\le {\displaystyle \frac{n-1}{2}}$. Clearly, we have ${\epsilon }_{DN{A}_n}(v_{11})={\epsilon }_{DN{A}_n}(v_{n1}), {\epsilon }_{DN{A}_n}(v_{21})={\epsilon }_{DN{A}_n}(v_{(n-1)1}), \dots , {\epsilon }_{DN{A}_n}(v_{\left({\displaystyle \frac{n-1}{2}}\right)1})={\epsilon }_{DN{A}_n}(v_{\left({\displaystyle \frac{n+3}{2}}\right)1})$. This equality holds for the vertices ${\epsilon }_{DN{A}_n}(v_{i2})={\epsilon }_{DN{A}_n}(v_{(n-i+1)2})$, ${\epsilon }_{DN{A}_n}(v_{i3})={\epsilon }_{DN{A}_n}(v_{(n-i+1)3})$, ${\epsilon }_{DN{A}_n}(v_{i4})={\epsilon }_{DN{A}_n}(v_{(n-i+1)4})$, and ${\epsilon }_{DN{A}_n}(v_{i5})={\epsilon }_{DN{A}_n}(v_{(n-i+1)5})$, where $1 \le i\le {\displaystyle \frac{n-1}{2}}$. When n is odd, there is a $\left({\displaystyle \frac{n+1}{2}}\right)$-nth cycle; see Figure 2. The eccentricity values of all vertices in $\left({\displaystyle \frac{n+1}{2}}\right)$-nth cycle is given in (8). Furthermore, the eccentricity values of the vertices $u_{i j}$ are given in (9). Similarly, we have ${\epsilon }_{DN{A}_n}(u_{i1})={\epsilon }_{DN{A}_n}(u_{(n-i)1})$ and ${\epsilon }_{DN{A}_n}(u_{i2})={\epsilon }_{DN{A}_n}(u_{(n-i)2})$, where $1 \le i\le {\displaystyle \frac{n-1}{2}}$. Moreover, all degrees of every vertex in $DN{A}_n$ of length n are given in (6).

Thus, we have:

$$\begin{array}{l}{\xi }^c(DN{A}_n)={\displaystyle \sum _{u\in V(DN{A}_n)}{\epsilon }_{DN{A}_n}(u)} {\mathrm{deg}}_{DN{A}_n}(u)\\ =2\ast \left(\begin{array}{l}{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(v_{i1}){\mathrm{deg}}_{DN{A}_n}(v_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(v_{i2}){\mathrm{deg}}_{DN{A}_n}(v_{i2})}\\ + {\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(v_{i3}){\mathrm{deg}}_{DN{A}_n}(v_{i3})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(v_{i4}){\mathrm{deg}}_{DN{A}_n}(v_{i4})+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(v_{i5}){\mathrm{deg}}_{DN{A}_n}(v_{i5})}}\end{array}\right)\\ +{\displaystyle \sum _{j=1}^n{\epsilon }_{DN{A}_n}(v_{\left(\frac{n+1}{2}\right) j}){\mathrm{deg}}_{DN{A}_n}(v_{\left(\frac{n+1}{2}\right) j})}\\ +2\ast \left( {\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(u_{i1}){\mathrm{deg}}_{DN{A}_n}(u_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(u_{i2}){\mathrm{deg}}_{DN{A}_n}(u_{i2})}\right)\\ =2\ast \left(\begin{array}{l}\left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+2).3}\right)-(4n-2)\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+1).3}\right)\\ + \left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+2).2}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+3).2}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+3).2}\right)\end{array}\right)\\ +\left((2n).3+(2n).3+(2n+1).2+(2n+1).2+(2n+1).2\right)\\ +2\ast \left( \left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i).2}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i-1).2}\right)\right)\\ =48n^2-25n+3.\end{array}$$

• Case 2. n is even.

The eccentricity values of all vertices are given in (7), where $1 \le i\le {\displaystyle \frac{n}{2}}$. Clearly, we have ${\epsilon }_{DN{A}_n}(v_{11})={\epsilon }_{DN{A}_n}(v_{n1}), {\epsilon }_{DN{A}_n}(v_{21})={\epsilon }_{DN{A}_n}(v_{(n-1) 1}), \dots , {\epsilon }_{DN{A}_n}(v_{\left({\displaystyle \frac{n}{2}}\right) 1})={\epsilon }_{DN{A}_n}(v_{\left({\displaystyle \frac{n}{2}}+1\right) 1})$. This equality holds for the vertices ${\epsilon }_{DN{A}_n}(v_{i2})={\epsilon }_{DN{A}_n}(v_{(n-i+1)2})$, ${\epsilon }_{DN{A}_n}(v_{i3})={\epsilon }_{DN{A}_n}(v_{(n-i+1)3})$, ${\epsilon }_{DN{A}_n}(v_{i4})={\epsilon }_{DN{A}_n}(v_{(n-i+1)4})$, and ${\epsilon }_{DN{A}_n}(v_{i5})={\epsilon }_{DN{A}_n}(v_{(n-i+1)5})$, where $1 \le i\le {\displaystyle \frac{n}{2}}$. Furthermore, the eccentricity values of the vertices $u_{i j}$ are given in (9). Similarly, we have ${\epsilon }_{DN{A}_n}(u_{i1})={\epsilon }_{DN{A}_n}(u_{(n-i)1})$ and ${\epsilon }_{DN{A}_n}(u_{i2})={\epsilon }_{DN{A}_n}(u_{(n-i)2})$, where $1 \le i\le {\displaystyle \frac{n}{2}}-1$. When n is even, we have two vertices as $u_{j 1}$ and $u_{j 2}$, where $j={\displaystyle \frac{n}{2}}$. The eccentricity values of these vertices can be obtained as in (10). Moreover, all degrees of every vertex in $DN{A}_n$ of length n are given in (6).

Thus, we have:

$$\begin{array}{l}{\xi }^c(DN{A}_n)={\displaystyle \sum _{u\in V(DN{A}_n)}{\epsilon }_{DN{A}_n}(u)} {\mathrm{deg}}_{DN{A}_n}(u)\\ =2\ast \left(\begin{array}{l}{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(v_{i1}){\mathrm{deg}}_{DN{A}_n}(v_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(v_{i2}){\mathrm{deg}}_{DN{A}_n}(v_{i2})}\\ + {\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(v_{i3}){\mathrm{deg}}_{DN{A}_n}(v_{i3})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(v_{i4}){\mathrm{deg}}_{DN{A}_n}(v_{i4})+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(v_{i5}){\mathrm{deg}}_{DN{A}_n}(v_{i5})}}\end{array}\right)\\ +2\ast \left( {\displaystyle \sum _{i=1}^{\frac{n}{2}-1}{\epsilon }_{DN{A}_n}(u_{i1}){\mathrm{deg}}_{DN{A}_n}(u_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}-1}{\epsilon }_{DN{A}_n}(u_{i2}){\mathrm{deg}}_{DN{A}_n}(u_{i2})}\right)+{\displaystyle \sum _{j=1}^2{\epsilon }_{DN{A}_n}(u_{\left(\frac{n}{2}\right) j}){\mathrm{deg}}_{DN{A}_n}(u_{\left(\frac{n}{2}\right) j})}\\ \\ =2\ast \left(\begin{array}{l}\left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+2).3}\right)-(4n-2)\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+1).3}\right)\\ + \left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+2).2}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+3).2}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+3).2}\right)\end{array}\right)\\ +2\ast \left( \left({\displaystyle \sum _{i=1}^{\frac{n}{2}-1}(4n-4i).2}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}-1}(4n-4i-1).2}\right)\right)+\left((2n).2+(2n).2\right)\\ =48n^2-25n+8.\end{array}$$

Via Cases 1 and 2, the proof is completed. □

Theorem 2.

The modified eccentric connectivity index of backbone $DN{A}_n$ graph of length n is

$${\xi }_c(DN{A}_n)==\left\{\begin{array}{c}114n^2-75n+15 , if n is odd;\\ 114n^2-75n+26 , if n is even.\end{array}\right.$$

Proof. 

We give all degrees of vertices of $DN{A}_n$ in (6) and all eccentricity values of vertices of $DN{A}_n$ in (7)–(10).

Furthermore, the sum of neighbor’s degrees of every vertex in $DN{A}_n$ of length n is as follows:

$${\delta }_{DN{A}_n}(v_{11})={\delta }_{DN{A}_n}(v_{n1})=5 \mathrm{and} {\delta }_{DN{A}_n}(v_{i1})=7, \mathrm{where} 2\le i\le n-1;$$

$${\delta }_{DN{A}_n}(v_{12})={\delta }_{DN{A}_n}(v_{n2})=6 \mathrm{and} {\delta }_{DN{A}_n}(v_{i2})=7, \mathrm{where} 2\le i\le n-1;$$

$${\delta }_{DN{A}_n}(v_{i3})=5 \mathrm{where} 1\le i\le n;$$

$${\delta }_{DN{A}_n}(v_{i4})=4, \mathrm{where} 1\le i\le n;$$

(11)

$${\delta }_{DN{A}_n}(v_{15})={\delta }_{DN{A}_n}(v_{n5})=4 \mathrm{and} {\delta }_{DN{A}_n}(v_{i5})=5, \mathrm{where} 2\le i\le n-1;$$

$${\delta }_{DN{A}_n}(u_{i1})={\delta }_{DN{A}_n}(u_{i2})=5, \mathrm{where} 1\le i\le n-1.$$

Two cases are considered depending on the parity of n.

• Case 1. n is odd.

The eccentricity values of all vertices of $DN{A}_n$ of length n are similar to Case 1 of proof of Theorem 1. Additionally, the sum of neighbor’s degrees of every vertex in $DN{A}_n$ of length n is given in (11).

Thus, we have:

$$\begin{array}{l}{\xi }_c(DN{A}_n)={\displaystyle \sum _{u\in V(DN{A}_n)}{\epsilon }_{DN{A}_n}(u)} {\delta }_{DN{A}_n}(u)\\ =2\ast \left(\begin{array}{l}{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(v_{i1}){\delta }_{DN{A}_n}(v_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(v_{i2}){\delta }_{DN{A}_n}(v_{i2})}\\ + {\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(v_{i3}){\delta }_{DN{A}_n}(v_{i3})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(v_{i4}){\delta }_{DN{A}_n}(v_{i4})+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(v_{i5}){\delta }_{DN{A}_n}(v_{i5})}}\end{array}\right)\\ +{\displaystyle \sum _{j=1}^n{\epsilon }_{DN{A}_n}(v_{\left(\frac{n+1}{2}\right) j}){\delta }_{DN{A}_n}(v_{\left(\frac{n+1}{2}\right) j})}+2\ast \left( {\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(u_{i1}){\delta }_{DN{A}_n}(u_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(u_{i2}){\delta }_{DN{A}_n}(u_{i2})}\right)\end{array}$$

$$\begin{array}{l} =2\ast \left(\begin{array}{l}\left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+2).7}\right)-2\ast (4n-2)\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+1).7}\right)-(4n-3)\right)\\ + \left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+2).5}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+3).4}\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+3).5}\right)-(4n-1)\right)\end{array}\right)\\ +\left((2n).7+(2n).7+(2n+1).5+(2n+1).4+(2n+1).5\right)\\ +2\ast \left( \left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i).5}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i-1).5}\right)\right)\\ =114n^2-75n+15.\end{array}$$

• Case 2. n is even.

The eccentricity values of all vertices of $DN{A}_n$ of length n are similar to Case 2 of proof of Theorem 1. Additionally, the sum of neighbor’s degrees of every vertex in $DN{A}_n$ of length n is given in (11).

Thus, we have:

$$\begin{array}{l}{\xi }_c(DN{A}_n)={\displaystyle \sum _{u\in V(DN{A}_n)}{\epsilon }_{DN{A}_n}(u)} {\delta }_{DN{A}_n}(u)\\ =2\ast \left(\begin{array}{l}{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(v_{i1}){\delta }_{DN{A}_n}(v_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(v_{i2}){\delta }_{DN{A}_n}(v_{i2})}\\ + {\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(v_{i3}){\delta }_{DN{A}_n}(v_{i3})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(v_{i4}){\delta }_{DN{A}_n}(v_{i4})+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(v_{i5}){\delta }_{DN{A}_n}(v_{i5})}}\end{array}\right)\\ +2\ast \left( {\displaystyle \sum _{i=1}^{\frac{n}{2}-1}{\epsilon }_{DN{A}_n}(u_{i1}){\delta }_{DN{A}_n}(u_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}-1}{\epsilon }_{DN{A}_n}(u_{i2}){\delta }_{DN{A}_n}(u_{i2})}\right)+{\displaystyle \sum _{j=1}^n{\epsilon }_{DN{A}_n}(u_{\left(\frac{n}{2}\right) j}){\delta }_{DN{A}_n}(u_{\left(\frac{n}{2}\right) j})}\\ =2\ast \left(\begin{array}{l}\left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+2).7}\right)-2\ast (4n-2)\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+1).7}\right)-(4n-3)\right)\\ + \left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+2).5}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+3).4}\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+3).5}\right)-(4n-1)\right)\end{array}\right)\\ +2\ast \left( \left({\displaystyle \sum _{i=1}^{\frac{n}{2}-1}(4n-4i).5}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}-1}(4n-4i-1).5}\right)\right)+\left((2n).5+(2n).5\right)\\ =114n^2-75n+26.\end{array}$$

Via Cases 1 and 2, the proof is completed. □

Theorem 3.

The edge eccentric connectivity index of backbone $DN{A}_n$ graph of length n is

$${\xi }_e^c(DN{A}_n)=\left\{\begin{array}{c}66n^2-61n+17 , if n is odd;\\ 66n^2-61n+24 , if n is even.\end{array}\right.$$

Proof. 

Considering the definition of backbone $DN{A}_n$ graphs, we have eight types of edges in $DN{A}_n$ according to location of the edges. In Figure 4 and Figure 5, labels of the edges of backbone $DN{A}_n$ graphs are given. When labeling the edges, on cycle graphs, they were labeled with $e_{i j}$, where $i\in [1,n]$ and $j\in [1,5]$. The edges on paths connecting cycles graphs were labeled with $f_{i j}$, where $i\in [1,n-1]$ and $j\in [1,3]$. Furthermore, we use $x=\left({\displaystyle \frac{n-1}{2}}\right)$, $y=\left({\displaystyle \frac{n+1}{2}}\right)$, $w=(n-1)$, $p= (n-2)$, $s=\left({\displaystyle \frac{n}{2}}\right)$, and $d=\left({\displaystyle \frac{n}{2}}+1\right)$ in Figure 4 and Figure 5.

In addition, there are eight types of edges distinguished by location, with different types based on their degrees, the sum of their neighbors’ degrees, the frequencies of the edges, and the eccentricity values of the edges.

The degrees of every edge in $DN{A}_n$ of length n are as follows:

$${\mathrm{deg}}_{DN{A}_n}(e_{11})={\mathrm{deg}}_{DN{A}_n}(e_{n1})=3 \mathrm{and} {\mathrm{deg}}_{DN{A}_n}(e_{i1})=4, \mathrm{where} 2\le i\le n-1;$$

$${\mathrm{deg}}_{DN{A}_n}(e_{i2})=3, \mathrm{where} 1\le i\le n;$$

$${\mathrm{deg}}_{DN{A}_n}(e_{i3})={\mathrm{deg}}_{DN{A}_n}(e_{i4})=2, \mathrm{where} 1\le i\le n;$$

$${\mathrm{deg}}_{DN{A}_n}(e_{15})={\mathrm{deg}}_{DN{A}_n}(e_{n5})=2 \mathrm{and} {\mathrm{deg}}_{DN{A}_n}(e_{i5})=3, \mathrm{where} 2\le i\le n-1;$$

(12)

$${\mathrm{deg}}_{DN{A}_n}(f_{i1})={\mathrm{deg}}_{DN{A}_n}(f_{i3})=3, \mathrm{where} 1\le i\le n-1;$$

$${\mathrm{deg}}_{DN{A}_n}(f_{i2})=2, \mathrm{where} 1\le i\le n-1.$$

The eccentricity values of the edge in $DN{A}_n$ of length n can be derived as follows:

$$\left.\begin{array}{l}{\epsilon }_{DN{A}_n}(e_{i1})={\epsilon }_{DN{A}_n}(e_{i2})=4n-4i+1\\ {\epsilon }_{DN{A}_n}(e_{i3})=4n-4i+2\\ {\epsilon }_{DN{A}_n}(e_{i4})={\epsilon }_{DN{A}_n}(e_{i5})=4n-4i+3\end{array}\right\}, where i\le ⌊{\displaystyle \frac{n}{2}}⌋.$$

(13)

Additionally, if n is odd, let $j={\displaystyle \frac{n+1}{2}}$.

Then, we get

$${\epsilon }_{DN{A}_n}(e_{j1})=2n-1;$$

$${\epsilon }_{DN{A}_n}(e_{j2})={\epsilon }_{DN{A}_n}(e_{j5})=2n;$$

(14)

$${\epsilon }_{DN{A}_n}(e_{j3})={\epsilon }_{DN{A}_n}(e_{j4})=2n+1.$$

Furthermore,

$$\left.\begin{array}{l}{\epsilon }_{DN{A}_n}(f_{i1})=4n-4i\\ {\epsilon }_{DN{A}_n}(f_{i2})=4n-4i-1\\ {\epsilon }_{DN{A}_n}(f_{i3})=4n-4i-2\end{array}\right\}, where i\le ⌊{\displaystyle \frac{n-1}{2}}⌋.$$

(15)

Additionally, if n is even, let $j={\displaystyle \frac{n}{2}}$.

Then, we get

$${\epsilon }_{DN{A}_n}(f_{j1})={\epsilon }_{DN{A}_n}(f_{j3})=2n \mathrm{and} {\epsilon }_{DN{A}_n}(f_{j2})=2n-1.$$

(16)

The proof will continue according to whether the results (a) and (b) are odd or even.

• Case 1. n is odd.

The eccentricity values of all edges are given in (13), where $1 \le i\le {\displaystyle \frac{n-1}{2}}$. Clearly, we have ${\epsilon }_{DN{A}_n}(e_{11})={\epsilon }_{DN{A}_n}(e_{n1}), {\epsilon }_{DN{A}_n}(e_{21})={\epsilon }_{DN{A}_n}(e_{(n-1) 1}), \dots , {\epsilon }_{DN{A}_n}(e_{\left({\displaystyle \frac{n-1}{2}}\right) 1})={\epsilon }_{DN{A}_n}(e_{\left({\displaystyle \frac{n+3}{2}}\right) 1})$. This equality holds for the edges ${\epsilon }_{DN{A}_n}(e_{i2})={\epsilon }_{DN{A}_n}(e_{(n-i+1)2})$, ${\epsilon }_{DN{A}_n}(e_{i3})={\epsilon }_{DN{A}_n}(e_{(n-i+1)3})$, ${\epsilon }_{DN{A}_n}(e_{i4})={\epsilon }_{DN{A}_n}(e_{(n-i+1)4})$, and ${\epsilon }_{DN{A}_n}(e_{i5})={\epsilon }_{DN{A}_n}(e_{(n-i+1)5})$, where $1\le i\le {\displaystyle \frac{n-1}{2}}$. When n is odd, there is a $\left({\displaystyle \frac{n+1}{2}}\right)$-nth cycle; see Figure 4. The eccentricity values of all edges in $\left({\displaystyle \frac{n+1}{2}}\right)$-nth cycle is given in (14). Furthermore, the eccentricity values of the edges $f_{i j}$ are given in (15). Similarly, we have ${\epsilon }_{DN{A}_n}(f_{i1})={\epsilon }_{DN{A}_n}(f_{(n-i)1})$, ${\epsilon }_{DN{A}_n}(f_{i2})={\epsilon }_{DN{A}_n}(f_{(n-i)2})$, and ${\epsilon }_{DN{A}_n}(f_{i3})={\epsilon }_{DN{A}_n}(f_{(n-i)3})$, where $1 \le i\le {\displaystyle \frac{n-1}{2}}$. Moreover, all degrees of every edge in $DN{A}_n$ of length n are given in (12).

Thus, we have:

$$\begin{array}{l}{\xi }_e^c(DN{A}_n)={\displaystyle \sum _{e\in E(DN{A}_n)}{\epsilon }_{DN{A}_n}(e)} {\mathrm{deg}}_{DN{A}_n}(e)\\ =2\ast \left(\begin{array}{l}{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(e_{i1}){\mathrm{deg}}_{DN{A}_n}(e_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(e_{i2}){\mathrm{deg}}_{DN{A}_n}(e_{i2})}\\ + {\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(e_{i3}){\mathrm{deg}}_{DN{A}_n}(e_{i3})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(e_{i4}){\mathrm{deg}}_{DN{A}_n}(e_{i4})+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(e_{i5}){\mathrm{deg}}_{DN{A}_n}(e_{i5})}}\end{array}\right)\\ +{\displaystyle \sum _{j=1}^n{\epsilon }_{DN{A}_n}(e_{\left(\frac{n+1}{2}\right) j}){\mathrm{deg}}_{DN{A}_n}(e_{\left(\frac{n+1}{2}\right) j})}\\ +2\ast \left( {\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(f_{i1}){\mathrm{deg}}_{DN{A}_n}(f_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(f_{i2}){\mathrm{deg}}_{DN{A}_n}(f_{i2})+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(f_{i3}){\mathrm{deg}}_{DN{A}_n}(f_{i3})}}\right)\\ =2\ast \left(\begin{array}{l}\left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+1).4}\right)-(4n-3)\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+1).3}\right)\\ + \left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+2).2}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+3).2}\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+2).3}\right)-(4n-2)\right)\end{array}\right)\\ +\left((2n-1).4+(2n).3+(2n+1).2+(2n+1).2+(2n).3\right)\\ +2\ast \left( \left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i).3}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i-1).2}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i-2).3}\right)\right)\\ =66n^2-61n+17.\end{array}$$

• Case 2. n is even.

The eccentricity values of all edges are given in (13), where $1 \le i\le {\displaystyle \frac{n}{2}}$. Clearly, we have ${\epsilon }_{DN{A}_n}(e_{11})={\epsilon }_{DN{A}_n}(e_{n1}), {\epsilon }_{DN{A}_n}(e_{21})={\epsilon }_{DN{A}_n}(e_{(n-1) 1}), \dots , {\epsilon }_{DN{A}_n}(e_{\left({\displaystyle \frac{n}{2}}\right) 1})={\epsilon }_{DN{A}_n}(e_{\left({\displaystyle \frac{n}{2}}+1\right) 1})$. This equality holds for the edges ${\epsilon }_{DN{A}_n}(e_{i2})={\epsilon }_{DN{A}_n}(e_{(n-i+1)2})$, ${\epsilon }_{DN{A}_n}(e_{i3})={\epsilon }_{DN{A}_n}(e_{(n-i+1)3})$, ${\epsilon }_{DN{A}_n}(e_{i4})={\epsilon }_{DN{A}_n}(e_{(n-i+1)4})$, and ${\epsilon }_{DN{A}_n}(e_{i5})={\epsilon }_{DN{A}_n}(e_{(n-i+1)5})$, where $1 \le i\le {\displaystyle \frac{n}{2}}$. Furthermore, the eccentricity values of the edges $f_{i j}$ are given in (15). Similarly, we have ${\epsilon }_{DN{A}_n}(f_{i1})={\epsilon }_{DN{A}_n}(f_{(n-i)1})$, ${\epsilon }_{DN{A}_n}(f_{i2})={\epsilon }_{DN{A}_n}(f_{(n-i)2})$, and ${\epsilon }_{DN{A}_n}(f_{i3})={\epsilon }_{DN{A}_n}(f_{(n-i)3})$, where $1 \le i\le {\displaystyle \frac{n}{2}}-1$. When n is even, we have three edges as $f_{j 1}$, $f_{j 2}$, and $f_{j 3}$, where $j={\displaystyle \frac{n}{2}}$. The eccentricity values of these edges can be obtained as in (16). Moreover, all degrees of every edge in $DN{A}_n$ of length n are given in (12).

Thus, we have:

$$\begin{array}{l}{\xi }_e^c(DN{A}_n)={\displaystyle \sum _{e\in E(DN{A}_n)}{\epsilon }_{DN{A}_n}(e)} {\mathrm{deg}}_{DN{A}_n}(e)\\ =2\ast \left(\begin{array}{l}{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(e_{i1}){\mathrm{deg}}_{DN{A}_n}(e_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(e_{i2}){\mathrm{deg}}_{DN{A}_n}(e_{i2})}\\ + {\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(e_{i3}){\mathrm{deg}}_{DN{A}_n}(e_{i3})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(e_{i4}){\mathrm{deg}}_{DN{A}_n}(e_{i4})+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(e_{i5}){\mathrm{deg}}_{DN{A}_n}(e_{i5})}}\end{array}\right)\\ +2\ast \left( {\displaystyle \sum _{i=1}^{\frac{n}{2}-1}{\epsilon }_{DN{A}_n}(f_{i1}){\mathrm{deg}}_{DN{A}_n}(f_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}-1}{\epsilon }_{DN{A}_n}(f_{i2}){\mathrm{deg}}_{DN{A}_n}(f_{i2})+{\displaystyle \sum _{i=1}^{\frac{n}{2}-1}{\epsilon }_{DN{A}_n}(f_{i3}){\mathrm{deg}}_{DN{A}_n}(f_{i3})}}\right)\\ +{\displaystyle \sum _{j=1}^3{\epsilon }_{DN{A}_n}(f_{\left(\frac{n}{2}\right) j}){\mathrm{deg}}_{DN{A}_n}(f_{\left(\frac{n}{2}\right) j})}\\ =2\ast \left(\begin{array}{l}\left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+1).3}\right)-(4n-3)\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+1).3}\right)\\ + \left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+2).2}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+3).2}\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+2).3}\right)-(4n-2)\right)\end{array}\right)\\ +2\ast \left( \left({\displaystyle \sum _{i=1}^{\frac{n}{2}-1}(4n-4i).3}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}-1}(4n-4i-1).2}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}-1}(4n-4i-2).3}\right)\right)\\ +\left((2n).3+(2n-1).2+(2n).3\right)\\ =66n^2-61n+24.\end{array}$$

Via Cases 1 and 2, the proof is completed. □

Theorem 4.

The modified edge eccentric connectivity index of backbone $DN{A}_n$ graph of length n is

$${\wedge }_e^c(DN{A}_n)=\left\{\begin{array}{c}192n^2-224n+78 , if n is odd;\\ 192n^2-224n+98 , if n is even.\end{array}\right.$$

Proof. 

We give all degrees of edges of $DN{A}_n$ in (12) and all eccentricity values of vertices of $DN{A}_n$ in (13)–(16). Furthermore, the sum of neighbor’s degrees of every edge in $DN{A}_n$ of length n is as follows:

$$S_{DN{A}_n}(e_{11})=S_{DN{A}_n}(e_{n1})=8 \mathrm{and} S_{DN{A}_n}(e_{i1})=12, \mathrm{where} 2\le i\le n-1;$$

$$S_{DN{A}_n}(e_{12})=S_{DN{A}_n}(e_{n2})=8 \mathrm{and} S_{DN{A}_n}(e_{i2})=9, \mathrm{where} 2\le i\le n-1;$$

$$S_{DN{A}_n}(e_{i3})=5 \mathrm{where} 1\le i\le n;$$

$$S_{DN{A}_n}(e_{14})=S_{DN{A}_n}(e_{n4})=4 \mathrm{and} S_{DN{A}_n}(e_{i4})=5, \mathrm{where} 2\le i\le n-1;$$

(17)

$$S_{DN{A}_n}(e_{15})=S_{DN{A}_n}(e_{n5})=5 \mathrm{and} S_{DN{A}_n}(e_{i5})=9, \mathrm{where} 2\le i\le n-1;$$

$$S_{DN{A}_n}(f_{11})=S_{DN{A}_n}(f_{(n-1)1})=8 \mathrm{and} S_{DN{A}_n}(f_{i1})=9, \mathrm{where} 2\le i\le n-2;$$

$$S_{DN{A}_n}(f_{i2})=6, \mathrm{where} 1\le i\le n-1;$$

$$S_{DN{A}_n}(f_{i3})=9, \mathrm{where} 1\le i\le n-1.$$

The proof will continue according to whether the results (a) and (b) are odd or even.

• Case 1. n is odd.

The eccentricity values of all edges of backbone $DN{A}_n$ graph of length n are similar to Case 1 of proof of Theorem 3. Additionally, the sum of neighbor’s edges of every edge in $DN{A}_n$ of length n is given in (17).

Thus, we have:

$$\begin{array}{l}{\wedge }_e^c(DN{A}_n)={\displaystyle \sum _{e\in E(DN{A}_n)}{\epsilon }_{DN{A}_n}(e)} S_{DN{A}_n}(e)\\ =2\ast \left(\begin{array}{l}{\displaystyle \sum _{ i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(e_{i1})S_{DN{A}_n}(e_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(e_{i2})S_{DN{A}_n}(e_{i2})}\\ + {\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(e_{i3})S_{DN{A}_n}(e_{i3})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(e_{i4})S_{DN{A}_n}(e_{i4})+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(e_{i5})S_{DN{A}_n}(e_{i5})}}\end{array}\right)\\ +{\displaystyle \sum _{j=1}^n{\epsilon }_{DN{A}_n}(e_{\left(\frac{n+1}{2}\right) j})S_{DN{A}_n}(e_{\left(\frac{n+1}{2}\right) j})}\\ +2\ast \left( {\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(f_{i1})S_{DN{A}_n}(f_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(f_{i2})S_{DN{A}_n}(f_{i2})+{\displaystyle \sum _{i=1}^{\frac{n-1}{2}}{\epsilon }_{DN{A}_n}(f_{i3})S_{DN{A}_n}(f_{i3})}}\right)\\ =2\ast \left(\begin{array}{l}\left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+2).12}\right)-4\ast (4n-3)\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+1).9}\right)-(4n-3)\right)\\ + \left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+2).5}\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+3).4}\right)-(4n-1)\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i+3).9}\right)-4\ast (4n-2)\right)\end{array}\right)\\ +\left((2n-1).12+(2n).9+(2n+1).5+(2n+1).5+(2n).9\right)\\ +2\ast \left( \left(\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i).9}\right)-(4n-4)\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i-1).6}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n-1}{2}}(4n-4i-2).9}\right)\right)\\ =192n^2-224n+78.\end{array}$$

• Case 2. n is even.

The eccentricity values of all edges of backbone $DN{A}_n$ graph of length n are similar to Case 2 of proof of Theorem 3. Additionally, the sum of neighbor’s edges of every edge in $DN{A}_n$ of length n is given in (17).

Thus, we have:

$$\begin{array}{l}{\wedge }_e^c(DN{A}_n)={\displaystyle \sum _{e\in E(DN{A}_n)}{\epsilon }_{DN{A}_n}(e)} S_{DN{A}_n}(e)\\ =2\ast \left(\begin{array}{l}{\displaystyle \sum _{ i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(e_{i1})S_{DN{A}_n}(e_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(e_{i2})S_{DN{A}_n}(e_{i2})}\\ + {\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(e_{i3})S_{DN{A}_n}(e_{i3})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(e_{i4})S_{DN{A}_n}(e_{i4})+{\displaystyle \sum _{i=1}^{\frac{n}{2}}{\epsilon }_{DN{A}_n}(e_{i5})S_{DN{A}_n}(e_{i5})}}\end{array}\right)\\ +2\ast \left( {\displaystyle \sum _{i=1}^{\frac{n}{2}-1}{\epsilon }_{DN{A}_n}(f_{i1})S_{DN{A}_n}(f_{i1})}+{\displaystyle \sum _{i=1}^{\frac{n}{2}-1}{\epsilon }_{DN{A}_n}(f_{i2})S_{DN{A}_n}(f_{i2})+{\displaystyle \sum _{i=1}^{\frac{n}{2}-1}{\epsilon }_{DN{A}_n}(f_{i3})S_{DN{A}_n}(f_{i3})}}\right)\\ +{\displaystyle \sum _{j=1}^n{\epsilon }_{DN{A}_n}(f_{\left(\frac{n}{2}\right) j})S_{DN{A}_n}(f_{\left(\frac{n}{2}\right) j})}\end{array}$$

$$\begin{array}{l} =2\ast \left(\begin{array}{l}\left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+1).12}\right)-4\ast (4n-3)\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+1).9}\right)-(4n-3)\right)\\ + \left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+2).5}\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+3).5}\right)-(4n-1)\right)+\left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}}(4n-4i+3).9}\right)-4\ast (4n-2)\right)\end{array}\right)\\ +2\ast \left( \left(\left({\displaystyle \sum _{i=1}^{\frac{n}{2}-1}(4n-4i).9}\right)-(4n-4)\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}-1}(4n-4i-1).6}\right)+\left({\displaystyle \sum _{i=1}^{\frac{n}{2}-1}(4n-4i-2).9}\right)\right)\\ +\left((2n).9+(2n-1).6+(2n).9\right)\\ =192n^2-224n+98.\end{array}$$

Via cases 1 and 2, the proof is completed. □

In order to help us know the properties of some eccentric connectivity indices based on vertices and edges of backbone DNA graphs ($DN{A}_n$) in Figure 6 and Figure 7, the results of the vertex eccentric connectivity index (ver-eccon) and modified vertex eccentric connectivity index (modver-eccon) obtained in Theorems 1 and 2, as well as the edge eccentric connectivity index (edge-eccon) and modified edge eccentric connectivity index (modedge-eccon) obtained in Theorems 3 and 4, are compared in a two-dimensional plane. Additionally, the results of the values of ${\xi }^c(DN{A}_n)$, ${\xi }_c(DN{A}_n)$, ${\xi }_e^c(DN{A}_n)$, and ${\wedge }_e^c(DN{A}_n)$ for the interval $n\in [3,20]$ are given in

Table 1. The results of the values of ${\xi }^c(DN{A}_n)$, ${\xi }_c(DN{A}_n)$, ${\xi }_e^c(DN{A}_n)$, and ${\wedge }_e^c(DN{A}_n)$ for the interval $n\in [3,20]$.

n	${\mathit{\xi }}^{\mathit{c}}(\mathit{D}\mathit{N}{\mathit{A}}_{\mathit{n}})$	${\mathit{\xi }}_{\mathit{e}}^{\mathit{c}}(\mathit{D}\mathit{N}{\mathit{A}}_{\mathit{n}})$	${\mathit{\xi }}_{\mathit{c}}(\mathit{D}\mathit{N}{\mathit{A}}_{\mathit{n}})$	${\wedge }_{\mathit{e}}^{\mathit{c}}(\mathit{D}\mathit{N}{\mathit{A}}_{\mathit{n}})$
3	360	428	816	1134
4	676	836	1550	2274
5	1078	1362	2490	3758
6	1586	2034	3680	5666
7	2180	2824	5076	7918
8	2880	3760	6722	10,594
9	3666	4814	8574	13,614
10	4558	6014	10,676	17,058
11	5536	7332	12,984	20,846
12	6620	8796	15,542	25,058
13	7790	10,378	18,306	29,614
14	9066	12,106	21,320	34,594
15	10,428	13,952	24,540	39,918
16	11,896	15,944	28,010	45,666
17	13,450	18,054	31,686	51,758
18	15,110	20,310	35,612	58,274
19	16,856	22,684	39,744	65,134
20	18,708	25,204	44,126	72,418

.

3. Conclusions

In this paper, the eccentric connectivity index, the edge eccentric connectivity index, the modified eccentric connectivity index, and the modified edge eccentric connectivity index have been computed for the backbone DNA graphs. These indices exhibit strong correlations with the fundamental physical and chemical characteristics of compounds, making the obtained results particularly valuable for QSAR/QSPR analyses of chemical or biological structures represented by the investigated graphs. Moreover, the outcomes of this study provide insights not only from a chemical perspective but also in the broader context of pharmaceutical chemistry. To further illustrate the behavior and variation in the computed topological indices, two-dimensional plots of ${\xi }^c(DN{A}_n)$, ${\xi }_c(DN{A}_n)$, ${\xi }_e^c(DN{A}_n)$, and ${\wedge }_e^c(DN{A}_n)$ have been generated using a Cartesian coordinate framework. This graphical representation allows for a clearer understanding of their spatial relationships and numerical trends. The present research may also be extended in future studies by deriving novel molecular architectures based on backbone DNA graph models, contributing to the development of new classes of chemical graph.