Article
Kyungpook Mathematical Journal 2018; 58(2): 221-229
Published online June 23, 2018
Copyright © Kyungpook Mathematical Journal.
Bounds for the First Zagreb Eccentricity Index and First Zagreb Degree Eccentricity Index
P. Padmapriya* and Veena Mathad
Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore - 570 006, India, e-mail: padmapriyap7@gmail.com and veena_mathad@rediffmail.com
Received: August 28, 2017; Accepted: June 8, 2018
The first Zagreb eccentricity index
Keywords: eccentricity, diameter, radius, Zagreb eccentricity indices, total eccentricity of a graph
A systematic study of topological indices is one of the most striking aspects in many branches of mathematics with its applications and various other fields of science and technology. A topological index is a numeric quantity from the structural graph of a molecule. According to the IUPAC definition,[15] a topological index (or molecular structure descriptor) is a numerical value associated with chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity.
All the graphs
The Zagreb indices were introduced by Gutman and Trinajsti
The invariants based on vertex eccentricities attracted some attention in Chemistry. In an analogy with the first and the second Zagreb indices, M. Ghorbani et al. and D. Vuki
The Zagreb degree eccentricity indices are introduced in [13]. First Zagreb degree eccentricity index (
The total eccentricity index of G is defined as
In this paper we obtain some bounds for the first Zagreb eccentricity index and first Zagreb degree eccentricity index.
Theorem 2.1
([1])
We can see the appearance of Theorem 2.1, in [10].
Theorem 2.2
Let
where
We choose
Since equality in (
Corollary 2.3
Since
Theorem 2.4
Let
Equality of (
We choose
If
Theorem 2.5
Let
Equality holds if and only if
Since equality in 2.4 holds if and only if
Corollary 2.6
Since
Theorem 2.7
Let
Equality of (
If
Lemma 2.8
([2])
The Lagrange identity is as follows.
Lemma 2.9
([11])
Theorem 2.10
(i) If we set
If
Now,
By power-mean inequality [3], we have
with equality if and only if
for any 2 ≤
From the above, we get
with equality if and only if
Using (
Using the above result in (
Conversely, suppose
that is,
Using the above result we get
(ii) If we set
But,
using the above result in (
The equality holds in (
In this paper we have established some bounds of the first Zagreb eccentricity index and first Zagreb degree eccentricity index in terms of some graph parameters such as order, size, maximum and minimum degree, radius, diameter and total eccentricity index. It may be useful to give the bounds for
The first author is thankful to the University Grants Commission, Government of India, for the financial support under the Basic Science Research Fellowship. UGC vide No.F.25 – 1/2014 – 15(BSR)/7 – 349/2012(BSR), January 2015.
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