Lupine Publishers| Current Trends in Computer Sciences & Applications (CTCSA)
Abstract
For a graph G, the second Zagreb eccentricity index E2(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
degG(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
dG(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)=
maxu≠v dG(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 E2(G) =ΞΎc(G) .Otherwise, E2(G) ≥ΞΎc(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 Gn(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 E2(G) <ΞΎc(G) for any graph
G∈ΞΆ(π,πΉ) .Moreover, we have π≥π−1 If π=n-1, then G≅Sn 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 ≅ C4
or G ≅ C5 . 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 E2(G) <ΞΎc(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 Ξ1= nE2(G)−(n + t)ΞΎc(G) . From Lemma 2.2, we have
Since r(G) ≥3 and , we have
Therefore, Ξ1 ≥ 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 Ξ1 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
π3≤ π − 2,π3=0, that is, 1 π ≥ t + 2 . If π ≤ 5t, clearly, we have Ξ1 ≤ 0 .For n >
5t, we have
holds if and only if 
Note that
Thus Ξ1 < 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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