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eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2021; 61(1): 61-74

Published online March 31, 2021

### 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
e-mail: aalqesmah@gmail.com

Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia
e-mail: asaleh1@uj.edu.sa

Department of Studies in Mathematics, University of Mysore, Mysore 570006, India
e-mail: rajra63@gmail.com

Bursa Uludag University, Mathematics, Gorukle 16059 Bursa-Turkey
e-mail: ayurttas@uludag.edu.tr and cangul@uludag.edu.tr

Received: July 6, 2019; Revised: April 14, 2020; Accepted: May 18, 2020

Let $G=(V,E)$ be a connected graph. The eccentric connectivity index of G is defined by $ξC(G)=∑ u∈V(G)deg(u)e(u)$, where $deg(u)$ and $e(u)$ denote the degree and eccentricity of the vertex u in G, respectively. In this paper, we introduce a new formulation of $ξC$ that will be called the distance eccentric connectivity index of G and defined by $ξDe(G)=∑ u∈V(G)degDe(u)e(u)$ where $degDe(u)$ denotes the distance eccentricity degree of the vertex u in G. The aim of this paper is to introduce and study this new topological index. The values of the eccentric connectivity index is calculated for some fundamental graph classes and also for some graph operations. Some inequalities giving upper and lower bounds for this index are obtained.

Keywords: eccentric connectivity index, distance eccentric connectivity index, topological graph index, graph operation.

In this paper, we are concerned with only connected simple graphs G=(V,E) which are finite, undirected with no loops nor multiple edges and for any two vertices u and v in G, there exists a uv-path starting from u and ending in v. Throughout this paper, we take p=|V(G)| and q=|E(G)|. The complement of G, denoted by $G¯$, is a simple graph on the same set of vertices V(G) in which two vertices u and v are adjacent in $G¯$ if and only if they are not adjacent in G. The distance between any two vertices u and v in G denoted by d(u,v) is the number of edges on a shortest path joining u and v. There are many interesting research areas in graph theory concentrate on the distance. Many well known graph topological indices, the most famous one being Wiener index, are defined in terms of distances between the vertices of a given graph. The eccentricity e(u) of a vertex u in G is another distance based concept and defined as the maximum distance between u and any other vertex v in G, that is $e(u)=max{d(u,v):v∈V(G)}$. This notion plays an important role in the study of graphs and has many applications. The maximum and minimum eccentricity over all vertices of G are called the diameter diam(G) and the radius rad(G) of G, respectively. All the definitions and terminologies about graphs used in this paper are available in [13]. The path, wheel, cycle, star and complete graphs with p vertices are denoted by Pp, Wp, Cp, Sp and Kp, respectively, and Kr,s is the complete bipartite graph on r+s vertices.

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 eccentricity-based 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 N-benzylimidazole. 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 ξC(G) of a graph G is defined as $ξC(G)=∑ u∈V(G)deg(u)e(u)$, [22]. When the vertex degrees are not taken into account, we obtain the total eccentricity of the graph G as $ζ(G)=∑ u∈V(G)e(u)$. We refer the reader to [3, 4, 10, 15, 17, 19, 18, 24] for explicit explanations of the related notions to eccentric connectivity index of graphs. The relation between the eccentricity indices and other topological indices has also been investigated. In [14], the relationship between eccentric connectivity index and Zagreb indices were studied and in [7], the Zagreb and multiplicative Zagreb indices of some graphs were calculated. In [20], the relation of the notion of eccentricity and other notions related to distance in graphs are correlated with the edge fixed geodomination number of a graph.

Let $u∈V(G)$. The distance eccentricity neighborhood of u denoted by $NDe(u)$ is defined as

$NDe(u)={v∈V(G):d(u,v)=e(u)}.$

The cardinality of $NDe(u)$ is called the distance eccentricity degree of the vertex u in G and denoted by $degDe(u)$, and $NDe[u]=NDe(u)∪{u}$ will be called as the closed distance eccentricity neighborhood of u. Note that if u has a full degree in G, then $deg(u)=degDe(u)$ and if e(u)=2, then $degG¯(u)=degDe(u)$. We use the symbol $Q(G)$ to denote the sum of distance eccentricity degrees of all vertices in a connected graph G, that is

$Q(G)=∑ u∈V(G)degDe(u).$

The maximum and minimum distance eccentricity degrees of a vertex in G are denoted respectively by $ΔDe(G)$ and $δDe(G)$:

$ΔDe(G)=maxu∈V|NDe(u)| and δDe(G)=minu∈V|NDe(u)|,$

see [2]. Also, we denote the set of vertices of G with eccentricity equal to α by $Veα(G)⊆V(G)$, where $α=1,2,…,diam(G)$.

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$ξC(G)$ and $ζ(G)$ of a connected graph G. One of the ways of studying graphs is graph operations. There is a number of such operations which help us to determine the properties of a large graph in terms of similar properties of the relatively smaller component graphs. For some examples, see [1, 8, 9, 16]. We compute the exact values of the distance eccentric connectivity index for some specific graphs and graph operations. Some upper and lower bounds and interesting results are obtained.

### 2. Some General Results on $ξ D e$

In this section, we define the distance eccentric connectivity index $ξDe(G)$ of a connected graph G and compute its exact values for some specific graphs. Also, we give some bounds on $ξDe$ and its relation with some other topological indices of graph.

### Definition 2.1.

Let G=(V,E) be a connected graph. The distance eccentric connectivity index of G is defined by

$ξDe(G)=∑ u∈V(G)degDe(u)e(u),$

where $degDe(u)$ denotes to the distance eccentricity degree of the vertex u in G.

The distance eccentric connectivity index $ξDe(G)$ of some well-known graph classes are as follows:

### Proposition 2.2.

• (i) For $p≥2$, $ξDe(Kp)=p(p−1)$.

• (ii) For $p≥2$, $ξDe(Pp)=p2(2p−p2−1).$

• (iii) For $p≥3$,

• (iv) For $r,s≥2$, $ξDe(Kr,s)=2(r(r−1)+s(s−1))$.

• (v) For $p≥3$, $ξDe(Sp)=(p−1)(2p−3)$.

• (vi) For $p≥5$, $ξDe(Wp)=(p−1)(2p−7)$.

In the following, we try to determine some general bounds on $ξDe$ for a graph G and its complement $G¯$.

### Proposition 2.3.

Let G be a connected graph on p vertices and q edges. Then

$ξDe(G)≤|Ve1(G)|(p−1)+2q¯ diam(G),$

where $Ve1(G)⊆V(G)$ is the set of vertices of eccentricity one in G and $q¯$ is the number of edges in the complement $G¯$ of G. The equality holds if and only if $diam(G)≤2$.

Proof. Let G be a connected graph and u ∈ V(G). If $u∈Ve1(G)$, then $degDe(u)=deg(u)=p−1$ and if $u∈V=V(G)−Ve1(G)$, then $degDe(u)≤degG¯(u)$. Thus

$ξDe(G)=∑ u∈V(G)degDe(u)e(u) =|Ve1(G)|(p−1)+∑ u∈VdegDe(u)e(u) ≤|Ve1(G)|(p−1)+diam(G)∑ u∈VdegG¯(u) =|Ve1(G)|(p−1)+2q¯ diam(G).$

For the equality, it is clear that $degDe(u)=degG¯(u)$, for every u ∈ V if and only if $diam(G)≤2$. Hence, the result holds.

### Corollary 2.4.

Let G be a (p,q)-connected graph such that $Ve1(G)=ϕ$ (G has no vertex of full degree). Then

$p⋅rad(G)≤ξDe(G)≤2q¯⋅diam(G).$

### Theorem 2.5.

Let G be a (p,q)-connected graph with $diam(G)≥4$ and $G¯$ be also connected. Then

$ξDe(G¯)=4q.$

Proof. It is well known that if a graph G is connected with $diam(G)≥4$ and its complement $G¯$ is also connected, then $diam(G¯)≤2$. Also, since G and $G¯$ are both connected, then $eG¯(u)=2$, for every $u∈V(G)$. Therefore, $degG¯De(u)=degG(u)$, for every $u∈V(G)$. Hence,

### Proposition 2.6.

For any (p,q)-connected graph G, we have

$ξDe(G)≤(p−1)ζ(G)−ξc(G)+|Ve1(G)|(p−1).$

Furthermore, the equality is attained if and only if $diam(G)≤2$.

Proof. Suppose first G has no vertices of full degree ($Ve1(G)=ϕ$). Then

$degGDe(u)≤deg G¯(u),$

for all $u∈V(G)$. Therefore,

$ξDe(G)≤∑ u∈V(G)(p−1−degG(u))e(u)=(p−1)ζ(G)−ξc(G).$

Suppose now $Ve1(G)≠ϕ$. Then

$ξDe(G)−|Ve1(G)|(p−1)≤∑ u∈V(G)(p−1−degG(u))e(u)=(p−1)ζ(G)−ξc(G),$

which is the required result.

For the equality, it is clear that $degDe(u)=degG¯(u)$, for all u ∈ V if and only if $diam(G)≤2$.

The first Zagreb eccentricity and the first distance eccentricity Zagreb indices of a connected graph G were defined respectively as

$ζ1(G)=∑ u∈V(G)(e(u))2$

and

$M1De(G)=∑ u∈V(G)(degDe(u))2,$

[2, 23]. In the following two propositions we give relations between the distance eccentric connectivity index $ξDe(G)$, the first Zagreb eccentricity index $ζ1(G)$ and the first distance eccentricity Zagreb index $M1De(G)$ of a connected graph G.

### Proposition 2.7.

Let G be a connected graph on $p≥2$ vertices. Then

$ξDe(G)≤p⋅ζ(G)−ζ1(G),$

with the equality if and only if $G≅Kp$ or $G≅Sp$.

Proof. For u ∈ V(G), let d(u;i) be the number of vertices that have distance i from u in G, where $i=1,2,⋯,e(u)$. Then we have

$p−1=degDe(u)+∑ i=1 e(u)−1d(u;i) ≥degDe(u)+∑ i=1 e(u)−11=degDe(u)+e(u)−1.$

Therefore, $degDe(u)≤p−e(u)$ with the equality if and only if e(u)=1 or $e(u)≥2$ with $d(u;1)=d(u;2)=…=d(u;e(u)−1)=1$. Then

$ξDe(G)=∑ u∈V(G)degDe(u)e(u) ≤∑ u∈V(G)(p−e(u))e(u)=p⋅ζ(G)−ζ1(G).$

Suppose the equality holds. Then $degDe(u)=p−e(u)$ and hence either e(u)=1 or $e(u)≥2$ with $d(u;1)=d(u;2)=…=d(u;e(u)−1)=1$.

Case 1. Suppose e(u)=1 for some u ∈ V(G). Then $degDe(u)=p−1$ and thus e(v)=1 or 2 for every $v≠u$ in G. Now, if e(v)=1 for all $v≠u$, then $G≅Kp$ and if e(v)=2 for some $v≠u$ with d(v;1)=1 and thus the vertex u is unique for all $v≠u$ and hence $G≅Sp$.

Case 2. Suppose now $e(u)≥2$ with $d(u;1)=d(u;2)=…=d(u;e(u)−1)=1$ for all u ∈ V(G). Let diam(G)=r and $P(G)=u1u2…ur$ be the diametral path in G. Clearly $d(u2;1)>1$, a contradiction. Hence there is no connected graph G satisfying the condition $e(u)≥2$ with $d(u;1)=d(u;2)=…=d(u;e(u)−1)=1$ for all $u∈V(G)$. The converse is clear.

### Proposition 2.8.

Let G, $G¯$ be connected graphs on p vertices and q, $q¯$ edges, respectively. Then

$ξDe(G)≤2pq¯−M1De(G).$

Proof. From the proof of Proposition 2.7, we have for any u ∈ V(G), $e(u)≤p−degDe(u)$ with the equality if and only if e(u)=1 or $e(u)≥2$ with $d(u;1)=d(u;2)=…=d(u;e(u)−1)=1$. Since G and $G¯$ are both connected, then $degDe(u)≤degG¯(u)$ for all u ∈ V(G). Therefore,

$ξDe(G)≤∑ u∈V(G)degDe(u)(p−degDe(u)) =p∑ u∈V(G)degDe(u)−M1De(G) ≤2pq¯−M1De(G)$

### 3. Distance Eccentric Connectivity Index of Some Graph Operations

In this section, we compute the distance eccentric connectivity index of some graph operations.

Cartesian Product

The cartesian product of two graphs G1 and G2 where $|V(G1)|=p1$, $|V(G2)|=p2$ and $|E(G1)|=q1$, $|E(G2)|=q2$ is denoted by $G1□G2$. It has the vertex set $V(G1)×V(G2)$ and two vertices $(u,u′)$ and $(v,v′)$ are connected by an edge if and only if either ([u=v and $u′v′∈E(G2)$]) or ([u'=v' and $uv∈E(G1)$]). In other words, $|E(G1□G2)|=q1p2+q2p1$. Therefore the degree of a vertex $(u,u′)$ of $G1□G2$ is as follows:

$degG1□G2(u,u′)=degG1(u)+degG2(u′).$

The cartesian product of more than two graphs is similarly denoted by

$∏ i=1nGi=G1□G2□…□Gn=(G1□G2□…□Gn−1)□Gn,$

in which any two vertices $u=(u1,u2,…,un)$ and $v=(v1,v2,…,vn)$ are adjacent in $∏ i=1nGi$ if and only if $ui=vi$, $∀i≠j$ and $ujvj∈E(Gj)$, where $i,j=1,2,…,n$. If $G1=G2=…=Gn=G$, we have the n-th cartesian power of G and denote it by Gn.

### Lemma 3.1.

Let $G=∏ i=1nGi$ and let $u=(u1,u2,…,un)$ be a vertex in V(G). Then

$e(u)=∑ i=1ne(ui).$

### Lemma 3.2.

Let $G=∏ i=1nGi$ and let $u=(u1,u2,…,un)$ be a vertex in G. Then

$degGDe(u)=∏ i=1ndegGiDe(ui).$

As a result we have

### Theorem 3.3.

Let $G=∏ i=1nGi$. Then

$ξDe(G)=∑ i=1n∏j=1j≠inξDe(Gi)Q(Gj).$

Proof. Let $u=(u1,u2,…,un)$be a vertex in V(G). Then by Lemmas 3.1 and 3.2, we have

$ξDe(G)=∑ u∈V(G)degGDe(u)eG(u) =∑ (u1 ,…,un )∈V(G)∏ i=1ndegGiDe(ui)∑ i=1ne(ui) =∑ u1 ∈V( G1 )∑ u2 ∈V( G2 )…∑ un ∈V( Gn )∏ i=1ndegGiDe(ui)(e(u1)+…+e(un)) =∑ i=1n∏ j=1 j≠i nξDe(Gi)Q(Gj).$

Composition

The composition $G=G1[G2]$ of two graphs G1 and G2 with disjoint vertex sets V(G1) and V(G2) and edge sets E(G1) and E(G2), where $|V(G1)|=p1$, $|E(G1)|=q1$ and $|V(G2)|=p2$, $|E(G2)|=q2$ is the graph with vertex set $V(G1)×V(G2)$ where any two vertices (u,u') and (v,v') are adjacent whenever u is adjacent to v in G1 or u=v and u' is adjacent to v' in G2. Thus

$|E(G1[G2])|=q1p22+q2p1.$

The degree of a vertex (u,u') of G1[G2] is as follows:

$degG1[G2](u,u′)=p2⋅degG1(u)+degG2(u′).$

### Lemma 3.4.

Let $G=G1[G2]$ and $e(v)≠1$ for all $v∈V(G1)$. Then $eG((u,u′))=eG1(u)$.

### Lemma 3.5.

Let $G=G1[G2]$ and $e(v)≠1$ for all $v∈V(G1)$. Then

### Theorem 3.6.

Let $G=G1[G2]$ and $e(v)≠1$ for all $v∈V(G1)$. Then

$ξDe(G)=p22⋅ξDe(G1)+4q2¯⋅|Ve2(G1)|.$

Proof. By Lemmas 3.4 and 3.5, we have

$ξDe(G)=∑ (u,u′ )∈V(G)degGDe(u,u′)eG(u,u′) =∑ u∈V( G1 )∑ u′∈V( G2)degGDe(u,u′)eG(u,u′) =∑ u∈V e2 ( G1 )∑ u′∈V( G2)(p2⋅degG 1De(u)+degG 2 ¯(u′))eG 1(u) +∑ u∈V( G1 )−V e2 ( G1 )∑ u′∈V( G2)(p2⋅degG 1De(u))eG 1(u) =∑ u∈V( G1 )∑ u′∈V( G2)(p2⋅degG 1De(u))eG 1(u) +∑ u∈V e2 ( G1 )∑ u′∈V( G2)degG 2 ¯(u′)eG 1(u) =p22⋅ξDe(G1)+4q2¯⋅|Ve2(G1)|.$

### Corollary 3.7.

Let $G=G1[G2]$ and $e(v)≠1$ or 2 for all $v∈V(G1)$. Then

$ξDe(G)=p22⋅ξDe(G1).$

Disjunction and Symmetric Difference

The disjunction $G1∨G2$ of two graphs G1 and G2 with $|V(G1)|=p1$, $|E(G1)|=q1$ and $|V(G2)|=p2$, $|E(G2)|=q2$ is the graph with vertex set $V(G1)×V(G2)$ in which (u,u') is adjacent to (v,v') whenever u is adjacent to v in G1 or u' is adjacent to v' in G2. So,

$|E(G1∨G2)|=q1p22+q2p12−2q1q2.$

The degree of a vertex (u,u') of $G1∨G2$ is as follows:

$degG1∨G2(u,u′)=p2⋅degG1(u)+p1⋅degG2(u′)−degG1(u)degG2(u′).$

Also, the symmetric difference $G1⊕G2$ of G1 and G2 is the graph with vertex set $V(G1)×V(G2)$ in which (u,u') is adjacent to (v,v') whenever u is adjacent to v in G1 or u' is adjacent to v' in G2, but not both. From definition one can see that,

$|E(G1⊕G2)|=q1p22+q2p12−4q1q2.$

The degree of a vertex (u,u') of $G1⊕G2$ is as follows:

$degG1⊕G2(u,u′)=p2⋅degG1(u)+p1⋅degG2(u′)−2degG1(u)degG2(u′).$

The distance between any two vertices of a disjunction or a symmetric difference cannot exceed two. Thus, if $e(v)≠1$ for all $v∈V(G1)∪V(G2)$, the eccentricity of all vertices is constant and equal to two.

### Lemma 3.8.

• (i) $degG1∨G2De(u,u′)=degG1∨G2¯(u,u′)$

• (ii) $degG1⊕G2De(u,u′)=degG1⊕G2¯(u,u′)$.

### Proposition 3.9.

Let G1 and G2 be two graphs with $e(v)≠1$ for all $v∈V(G1)∪V(G2)$. Then

• (i) $ξDe(G1∨G2)=4qG1 ∨G2 ¯$,

• (ii) $ξDe(G1⊕G2)=4qG1 ⊕G2 ¯$.

Proof. The proof is straightforward.

Join

The join $G1+G2$ of two graphs G1 and G2 with disjoint vertex sets V(G1) and V(G2) and edge sets E(G1) and E(G2) such that $|V(G1)|=p1$, $|V(G2)|=p2$, $|E(G1)|=q1$ and $|E(G2)|=q2$ is the graph on the vertex set $V(G1)∪V(G2)$ and the edge set $E(G1)∪E(G2)∪{u1u2:u1∈V(G1);u2∈V(G2)}$. Hence, the join of two graphs is obtained by connecting each vertex of one graph to each vertex of the other graph, while keeping all edges of both graphs. The degree of any vertex $u∈G1+G2$ is given by

Actually, by using the definition of the join graph $G=∑ i=1nGi$, one can see that the eccentricity of any vertex u∈ G does not exceed two. Therefore, if $Ve1(Gi)=ϕ$ for all $Gi⊂G$, then e(u)=2 for all u ∈ G.

### Lemma 3.10.

Let $G=∑ i=1nGi$ and $u∈V(G)$. Then

### Theorem 3.11.

Let $G=∑ i=1nGi$. Then

$ξDe(G)=(|V(G)|−1)∑ i=1n|Ve1(Gi)|+4∑ i=1nqi¯.$

Proof. By Lemma 3.10, we get

$ξDe(G)=∑ u∈V(G)degGDe(u)eG(u) =∑ i=1n∑ u∈V( Gi)degGDe(u)eG(u) =∑ i=1n∑ u∈V e1( Gi)(|V(G)|−1) +2∑ i=1n∑ u∈V( Gi)−V e1( Gi)(pi−1−degGi(u)) =(|V(G)|−1)∑ i=1n|Ve1(Gi)|+4∑ i=1nqi¯.$

### Corollary 3.12.

If Gi has no vertices of full degree ($Ve1(Gi)=ϕ$) for all $i=1,2,…,n$, then

$ξDe(∑ i=1nGi)=4∑ i=1nqi¯.$

Corona Product

The corona product $G1∘G2$ of two graphs G1 and G2 where $|V(G1)|=p1$, $|V(G2)|=p2$ and $|E(G1)|=q1$, $|E(G2)|=q2$ is the graph obtained by taking |V(G1)| copies of G2 and joining each vertex of the i-th copy with vertex u ∈ V(G1). Obviously,

$|V(G1∘G2)|=p1(p2+1)$

and

$|E(G1∘G2)|=q1+p1(q2+p2).$

It follows from the definition of the corona product $G1∘G2$ that the degree of each vertex $u∈G1∘G2$ is given by

### Lemma 3.13.

Let $G=G1∘G2$ be a connected graph such that $G1≠K1$ and let $u∈V(G)$. Then

$degGDe(u)=p2degG1De(u),u∈V(G1);p2degG1De(v),u∈V(G)−V(G1),$

where v ∈ V(G1) is adjacent to u.

### Lemma 3.14.

Let $G=G1∘G2$ be a connected graph such that $G1≠K1$ and let $u∈V(G)$. Then

where v ∈ V(G1) is adjacent to u.

### Theorem 3.15.

Let $G=G1∘G2$ be a connected graph such that $G1≠K1$. Then

$ξDe(G)(G)=p2[(p2+1)ξDe(G1)+(2p2+1)Q(G1)].$

Proof. By Lemmas 3.13 and 3.14, we have

$ξDe(G)(G)=∑ u∈V(G)degGDe(u)eG(u) =∑ u∈V( G1 )degGDe(u)eG(u)+∑ v∈V( G1 )∑ u∈V( G2)degGDe(u)eG(u) =∑ u∈V( G1 )p2degG1De(u)(eG1(u)+1) +∑ v∈V( G1 )∑ u∈V( G2)p2degG 1De(v)(eG 1(v)+2) =p2[(p2+1)ξDe(G1)+(2p2+1)Q(G1)].$

### Example 3.16.

• (i) For any cycle $Cp1$ and any path $Pp2$ with $p1≥3$ and $p2≥1$, $ξDe(Cp1∘Pp2)=2p1 p2 [(p2 +1) p1 2 +2p2 +1],if p1 is odd; p1 p2 2[p1 (p2 +1)+4p2 +2)],if p1 is even.$

• (ii) For any two cycles $Cp1$ and $Cp2$ with $p1,p2≥3$, $ξDe(Cp1∘Cp2)=2p1 p2 [(p2 +1) p1 2 +2p2 +1],if p1 is odd; p1 p2 2[p1 (p2 +1)+4p2 +2)],if p1 is even.$

Competing interests

The authors declare that they have no competing interests.

The authors declare that they have no financial or non-financial support.

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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