Article
Kyungpook Mathematical Journal 2021; 61(1): 6174
Published online March 31, 2021
Copyright © Kyungpook Mathematical Journal.
Distance Eccentric Connectivity Index of Graphs
Akram Alqesmah, Anwar Saleh, R. Rangarajan, Aysun Yurttas Gunes and Ismail Naci Cangul∗
Department of Studies in Mathematics, University of Mysore, Mysore 570006, India
email: aalqesmah@gmail.com
Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia
email: asaleh1@uj.edu.sa
Department of Studies in Mathematics, University of Mysore, Mysore 570006, India
email: rajra63@gmail.com
Bursa Uludag University, Mathematics, Gorukle 16059 BursaTurkey
email: ayurttas@uludag.edu.tr and cangul@uludag.edu.tr
Received: July 6, 2019; Revised: April 14, 2020; Accepted: May 18, 2020
Abstract
Let
Keywords: eccentric connectivity index, distance eccentric connectivity index, topological graph index, graph operation.
1. Introduction
In this paper, we are concerned with only connected simple graphs
In chemical graph theory, various graph invariants are used for establishing correlations of chemical structures with various physical properties, chemical reactivity, or biological activity. These graph invariants are called topological indices of (molecular) graphs. Although most of the topological graph indices are degree, distance or matrix based, there are some eccentricitybased topological indices in chemical graph theory.
In [11], the authors introduced a topological descriptor called the connectivity eccentricity index when investigating the antihypertensive activity of derivatives of Nbenzylimidazole. They showed that the results obtained using the connectivity eccentricity index were better than the corresponding values obtained using Balaban's mean square distance index, [5, 6], and the accuracy of prediction was found to be about 80 percent of the active range, [11]. In [22], the authors introduced the eccentric connectivity index of a given graph, which has been employed successfully for the development of numerous mathematical models for the prediction of biological activities of diverse nature, [12, 21, 22]. The eccentric connectivity index
Let
The cardinality of
The maximum and minimum distance eccentricity degrees of a vertex in
see [2]. Also, we denote the set of vertices of
The applications of vertex eccentricity and the eccentric connectivity index of graphs motivated us to use the distance eccentricity degree instead of the normal degree which is very closed to the vertex eccentricity.
In this paper, we study this topological index by investigating some of its properties and its relations with
2. Some General Results on
${\xi}^{De}$
${\xi}^{De}$
In this section, we define the distance eccentric connectivity index
Definition 2.1.
Let
where
The distance eccentric connectivity index
Proposition 2.2.

(i) For
$p\ge 2$ ,${\xi}^{De}({K}_{p})=p(p1)$ . 
(ii) For
$p\ge 2$ ,${\xi}^{De}({P}_{p})=\u2308\frac{p}{2}\u2309(2p\u2308\frac{p}{2}\u23091).$ 
(iii) For
$p\ge 3$ ,${\xi}^{De}({C}_{p})=\left\{\begin{array}{ll}\frac{{p}^{2}}{2},\hfill & \text{}p\text{\hspace{0.17em}}is\text{\hspace{0.17em}}even;\hfill \\ 2p\u2308\frac{p}{2}\u2309,\hfill & \text{}p\text{\hspace{0.17em}}is\text{\hspace{0.17em}}odd.\hfill \end{array}\right.$ 
(iv) For
$r,s\ge 2$ ,${\xi}^{De}({K}_{r,s})=2(r(r1)+s(s1))$ . 
(v) For
$p\ge 3$ ,${\xi}^{De}({S}_{p})=(p1)(2p3)$ . 
(vi) For
$p\ge 5$ ,${\xi}^{De}({W}_{p})=(p1)(2p7)$ .
In the following, we try to determine some general bounds on
Proposition 2.3.
where
For the equality, it is clear that
Corollary 2.4.
Let
Theorem 2.5.
Let
Proposition 2.6.
For any
Furthermore, the equality is attained if and only if
for all
Suppose now
which is the required result.
For the equality, it is clear that
The first Zagreb eccentricity and the first distance eccentricity Zagreb indices of a connected graph
and
[2, 23]. In the following two propositions we give relations between the distance eccentric connectivity index
Proposition 2.7.
Let
with the equality if and only if
Therefore,
Suppose the equality holds. Then
Proposition 2.8.
Let
3. Distance Eccentric Connectivity Index of Some Graph Operations
In this section, we compute the distance eccentric connectivity index of some graph operations.
The cartesian product of two graphs
The cartesian product of more than two graphs is similarly denoted by
in which any two vertices
Lemma 3.1.
Let
Lemma 3.2.
Let
As a result we have
Theorem 3.3.
Let
The composition
The degree of a vertex
Lemma 3.4.
Let
Lemma 3.5.
Let
Theorem 3.6.
Let
Corollary 3.7.
Let
The disjunction
The degree of a vertex
Also, the symmetric difference
The degree of a vertex
The distance between any two vertices of a disjunction or a symmetric difference cannot exceed two. Thus, if
Lemma 3.8.

(i)
$de{g}_{{G}_{1}\vee {G}_{2}}^{De}(u,{u}^{\prime})=de{g}_{\overline{{G}_{1}\vee {G}_{2}}}(u,{u}^{\prime})$ 
(ii)
$de{g}_{{G}_{1}\oplus {G}_{2}}^{De}(u,{u}^{\prime})=de{g}_{\overline{{G}_{1}\oplus {G}_{2}}}(u,{u}^{\prime})$ .
Proposition 3.9.
Let

(i)
${\xi}^{De}({G}_{1}\vee {G}_{2})=4{q}_{\overline{{G}_{1}\vee {G}_{2}}}$ , 
(ii)
${\xi}^{De}({G}_{1}\oplus {G}_{2})=4{q}_{\overline{{G}_{1}\oplus {G}_{2}}}$ .
The join
Actually, by using the definition of the join graph
Lemma 3.10.
Let
Theorem 3.11.
Let
Corollary 3.12.
If
The corona product
and
It follows from the definition of the corona product
Lemma 3.13.
Let
where
Lemma 3.14.
Let
where
Theorem 3.15.
Let
Example 3.16.

(i) For any cycle
${C}_{{p}_{1}}$ and any path${P}_{{p}_{2}}$ with${p}_{1}\ge 3$ and${p}_{2}\ge 1$ ,$${\xi}^{De}({C}_{{p}_{1}}\circ {P}_{{p}_{2}})=\left\{\begin{array}{ll}2{p}_{1}{p}_{2}[({p}_{2}+1)\u230a\frac{{p}_{1}}{2}\u230b+2{p}_{2}+1],\hfill & if\text{\hspace{0.17em}}{p}_{1}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}odd;\hfill \\ \frac{{p}_{1}{p}_{2}}{2}[{p}_{1}({p}_{2}+1)+4{p}_{2}+2)],\hfill & if\text{\hspace{0.17em}}{p}_{1}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}even.\hfill \end{array}\right.$$ 
(ii) For any two cycles
${C}_{{p}_{1}}$ and${C}_{{p}_{2}}$ with${p}_{1},{p}_{2}\ge 3$ ,$${\xi}^{De}({C}_{{p}_{1}}\circ {C}_{{p}_{2}})=\left\{\begin{array}{ll}2{p}_{1}{p}_{2}[({p}_{2}+1)\u230a\frac{{p}_{1}}{2}\u230b+2{p}_{2}+1],\hfill & if\text{\hspace{0.17em}}{p}_{1}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}odd;\hfill \\ \frac{{p}_{1}{p}_{2}}{2}[{p}_{1}({p}_{2}+1)+4{p}_{2}+2)],\hfill & if\text{\hspace{0.17em}}{p}_{1}\text{\hspace{0.17em}}is\text{\hspace{0.17em}}even.\hfill \end{array}\right.$$
4. Declarations
The authors declare that they have no competing interests.
Funding
The authors declare that they have no financial or nonfinancial support.
Authors' contributions
AA and AS has proposed the problem and constructed the propositions' statements. RR has done calculations with indices. AYG and INC has obtained the inequalities and also made calculations and proofs of the results on graph operations.
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