Abstract
Atom-bond connectivity (ABC) index has been applied up to now to study the stability of alkanes and the strain energy of cycloalkanes. Graovac defined the second ABC index as and Kinkar studied the upper bounds. In this paper, we define a new index which is called the third ABC index and it is defined as
(1) and we present some lower and upper bounds on
index of graphs.
1 Introduction
Molecular descriptors are playing significant role in chemistry, pharmacology, etc. Among them, topological indices have a prominent place [Citation1]. One of the best known and widely used index is the connectivity index, , introduced in 1975 by Milan Randić [Citation2], who has shown that this index reflects molecular branching. Some novel results about branching can be found in [Citation3–Citation[4]Citation[5]Citation[6]Citation7] and in the references cited therein. However, many physico-chemical properties are dependent on factors other than branching. The lower and upper bounds on ABC index of chemical trees in terms of the number of vertices were obtained in [Citation8]. Also, it has been shown that the star
, has the maximal ABC value of trees.
Let be a simple connected graph with vertex set
and edge set
, where
and
. Let
, then the distance
between
and
is defined as the length of any shortest path in
connecting
and
. For a vertex
of
, its eccentricity
is the largest distance between
and any other vertex
of
, i.e,
. The diameter
of
is defined as the maximum eccentricity of
. Similarly, the radius
is defined as the minimum eccentricity of
. There are some topological indices which are related to the eccentricity like the eccentric connectivity index
where
is the degree of the vertex
.
The Zagreb indices have been introduced more than 30 years ago by Gutman and Trinajstić [Citation9]. They are defined as
Also, there are similar indices called Zagreb eccentricity indices which are also related to eccentricity. Zagreb eccentricity indices are defined as
In order to take this into account but at the same time to keep the spirit of the Randić index, Estrada et al. proposed a new index, nowadays known as the atom-bond connectivity (ABC) index, which is defined as [Citation18]
The ABC index has proven to be a valuable predictive index in the study of the heat formation of alkanes [Citation18,Citation19], and some mathematical properties are obtained in [Citation20,Citation21].
Very recently, Graovac [Citation22] defined the second ABC index as which was given by replacing the vertex degree to
, where
is the number of vertices of
whose distance to the vertex
is smaller than the distance to the vertex
. Also, Kinkar establised some mathematical properties of the second ABC index [Citation23].
Now we define the third ABC index; ,
(2) which was given by replacing the vertex degree to eccentricity,
.
In this paper, we present lower and upper bounds on index of connected, simple graphs in terms of other indices; the first Zagreb index, the second Zagreb index, eccentric connectivity index, the first Zagreb eccentricity index, and the second Zagreb eccentricity index.
2 Lower and upper bounds on 
index
By direct calculation, we can find ,
,
,
, where
,
and
denote the complete graph, star graph and cycle with
vertices, respectively. We can easily see that the complete graph
gives the minimal value for
index, so in this paper we only consider when graph
differs from
.
Theorem 2.1
Let G be a simple connected graph. Then(3) where
is the second Zagreb eccentricity index.
Proof
Since , it is easy to see that for every
in
,
. By definition of
index, we have
□
Theorem 2.2
Let G be a simple connected graph with m edges, radius , diameter
.
(4) Equality holds if and only if G is a self-centered graph.
Proof
For , we have
(5) with equality holding if and only if
.
Similarly, we can easily show that (6) with equality holding if and only if
. □
Lemma 2.3
[Citation24]
Let be a positive n-tuple such that there exist positive numbers A, a satisfying:
. Then
(7) Equality holds if and only if
or
(8) is an integer and
of the numbers
coincide with a and the remaining
of the
’s coincide with A(
).
Theorem 2.4
Let G be a simple connected graph with m edges, radius , diameter
. Then
(9)
Proof
We know that (10) Also by Lemma 2.3, we have
(11) Since
(12) Hence it follows that
(13) □
Theorem 2.5
Let G be a simple connected graph with n vertices and m edges. Then(14) where
and
are the first Zagreb index and the second Zagreb index, respectively.
Proof
From the proof of Theorem 2.1, we already have (15) Since
, we get
(16) This completes the lower bound in (Equation14
(14) ).
Now, since ,
for
, we have
(17) By Cauchy–Schwarz inequality,
(18) since
for every
, we get
(19) □
Theorem 2.6
Let G be a simple connected graph with n vertices, m edges, radius r, and eccentric connectivity index . Then
(20)
Proof
By Cauchy–Schwarz inequality, we have (21) Also,
,
for all
. Hence the theorem follows. □
Notes
Peer review under responsibility of Kalasalingam University.
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