** Lupine Publishers| Current Trends in Computer Sciences & Applications (CTCSA)**

## Abstract

For a graph G, the second Zagreb eccentricity index E_{2}(G) and eccentric connectivity index ∈^{c}(G) are two eccentricity-based
invariants of graph G. In this paper we prove some results on the comparison between and of connected graphs G of order
n and with m edges.

The authors demonstrated how a combination of both techniques and human interventions enhances control, decision-making and data analysis systems.

**Keywords:** Graph; Eccentricity (of vertex); Second Zagreb eccentricity index; Eccentric connectivity index

## Introduction

Throughout this paper we only consider the note, undirected,
simple and connected graphs. The degree of v∈ V(G), denoted by
deg_{G}(v), is the number of vertices in G adjacent to v. For any two
vertices u; v in a graph G, the distance between them, denoted by
d_{G}(u; v), is the length of a shortest path connecting them in G. As
usual, let Sn, Pn, Cn, Kn be the star graph, path graph, cycle graph
and complete graph, respectively, on n vertices. Other undefined
notations and terminology on the graph theory can be found in [1].
For any vertex of graph G, the eccentricity ∈_{G} (v) (or ∈(v) for short)
is the maximum distance from v to other vertices of G, i.e., ∈_{G} (v)=
max_{u≠v} d_{G}(u,v). The eccentricity of a vertex is an important parameter
in pure graph theory. The radius of a graph G is denoted by r(G) and
defined by . Also, the diameter
of G, denoted by d(G), is the maximum distance between vertices
of a graph G and hence . A vertex
v with ∈_{G}(v)= r(G) is called a central vertex in G. A graph G with
d(G) = r(G) is called a self-centered graph. A graph which contains
only two non-central vertices is called almost self-centered graph
[2] (ASC graph for short). Moreover, the eccentricity is also applied
in chemical graph theory. There are several eccentricity-based
topological indices, including the second Zagreb eccentricity index
E2(G) [3] and eccentric connectivity index ∈^{c} (G) [4], of graphs G
where

In particular, we have or any graph G. In this paper we prove some comparison results between and of connected graphs G of order n with m edges. Main results In this we prove several results on the comparison between and of graphs G. Firstly we present two useful lemmas.

**Lemma 2.1:** [5] Let G be a connected graph of order n with
maximum degree Ξ . If Ξ= *n −1* then E_{2}(G) =ΞΎ^{c}(G) .Otherwise,
E_{2}(G) ≥ΞΎ^{c}(G)with equality holds if and only if G is a 2-SC graph.

**Lemma 2.1:** [6] If u and v are two adjacent vertices of a
connected graph G, then ∈(π°)−∈(π±) |≤1.

Denote by G_{n}(m; d) the set of connected graphs of order n with
m edges and diameter d.

**Theorem 2.3.** Let G∈ΞΆ(*π,πΉ*) with n>5 and πΉ≤2. Then <. Proof. If d = 1, G∈ΞΆ(*π,πΉ*) contains a single graph Kn
with and . Then our result follows. Next
it suffices to consider the case when d = 2. If G has maximum
degree Ξ = n −1by Lemma 2.1, we have E_{2}(G) <ΞΎ^{c}(G) for any graph
G∈ΞΆ(*π,πΉ*) .Moreover, we have π≥π−1 If π=n-1, then G≅S_{n} with for any n ≥ 5. Moreover,<holds clearly
form ≥ n. If Ξ ≤ n − 2 then G is a 2-SC graph. By Lemma 2.2, G is never
a tree. Therefore m ≥ n with equality holding if and only if G ≅ C_{4}
or G ≅ C_{5} . Consider that n > 5, m > n holds immediately. It follows
that . This completes the proof of the
theorem.

In the following we consider the graphs G∈ΞΆ(*π,πΉ*) with
diameter d ≥ 3.

**Theorem 2.4:** Let G∈ΞΆ(*π,πΉ*). with d ≥ 3, n > 5 be a tree or a
unicyclic graph. Then >Proof. If d ≥ 3, then Ξ(G) ≤ n − 2 .
From Lemma 2:1, we have E_{2}(G) <ΞΎ^{c}(G). Note that m ≥ n for any tree
or unicyclic graph G. Thus, it follows . finishing the
proof of the theorem. Next we consider the case

m > n. In the following theorem we give a sufficient condition for the graph G of order n with ≥ .

**Theorem 2.5.** Let G∈ΞΆ(*π,πΉ*) with d ≥ 3, m = n + t and . If
r(G) ≥ 3, then
≥ ,

Proof. Making a difference, we have

Set Ξ_{1}= nE_{2}(G)−(n + t)ΞΎ^{c}(G) . From Lemma 2.2, we have

Since r(G) ≥3 and , we have

Therefore, Ξ_{1} ≥ 0 with equality holding if and only if ∈(π) = 3for
each vertex π∈*V(G)* that is, G is a self-centered graph with radius 3.
This completes the proof of the theorem.

F or, G∈ΞΆ(*π,πΉ*) with d ≥ 3, r = 2 and considering that
r(G)≤d(G)≤2r(G) we have d(G) = 3 or d(G) = 4. In this case, the value of Ξ_{1} may be negative, zero or positive. Let

Denote by mi the cardinality of *i*∈{1, 2,3, 4,5}. Then

In the following result we present some comparison results for ASC graphs.

**Theorem 2.6:** Let G∈ΞΆ(*π,πΉ*) with d = 3, r = 2, m = n + t, t ≥ 1
where .

If G is an ASC graph, then < Proof. If G is an ASC
graph with d = 3, r = 2, from the structure of ASC graph, we have
π_{3}≤ π − 2,π_{3}=0, that is, 1 π ≥ t + 2 . If π ≤ 5t, clearly, we have Ξ_{1} ≤ 0 .For n >

5t, we have

holds if and only if Note thatThus Ξ_{1} < 0 is equivalent that
with t ≥1. Therefore the result holds immediately. It is much
interesting to search more generalized graphs G with different
comparison results between
and which can be a topic for
further research in the future.

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