New bounds for Randic and GA indices
© Lokesha et al.; licensee Springer 2013
Received: 9 February 2013
Accepted: 26 March 2013
Published: 16 April 2013
The main goal of this paper is to present some new lower and upper bounds for the Randic and GA indices in terms of Zagreb and modified Zagreb indices.
MSC: 05C05, 05C20, 05C90.
Dedicated to Professor Hari M Srivastava.
1 Introduction and preliminaries
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. Usage of topological indices in chemistry began in 1947 when H. Wiener developed the most widely known topological descriptor, namely the Wiener index, and used it to determine physical properties of types of alkanes known as paraffin (see, for instance, [1–3]).
Let G be a simple graph with the vertex-set and the edge-set . As usual notion, the maximum vertex degree is denoted by , while the minimum vertex degree is denoted by . Moreover, denotes the minimum nonpendant vertex degree in G. A vertex of the graph G is said to be pendant if its neighborhood contains exactly one vertex. On the other hand, an edge of a graph is said to be pendant if one of its vertices is pendant.
In the following, we recall two fundamental indices that will be used to present some new bounds for Randic and GA indices.
This paper is organized as follows. In the forthcoming section, we present lower and upper bounds on Randic index of connected graphs and trees in terms of modified Zagreb indices given in (4). The final section deals with lower and upper bounds on GA index of connected graphs and trees in terms of Zagreb indices given in (3). We note that this paper is motivated from .
2 Lower and upper bounds on Randic index
Throughout this paper, we refer the book  for a classical result, namely the Pólya-Szegó inequality. From this result, we first establish the following theorem, which will be expressed the lower bound on the Randic index.
with equality holding if and only if .
as desired. □
Proof Since the number of edges in a tree having n vertices is , the proof can be done similarly as in the proof of Theorem 1. □
as required. □
Now we prove another form of the upper bound for the Randic index as in the following.
After that, by using (9) in (5), we get the bound in (8), as required. □
3 Lower and upper bounds on GA index
By taking Pólya-Szegó inequality into account, the next result deals with a new lower bound on GA index in terms of Zagreb index as given in (3).
Hence the result. □
Proof For an order n, since the number of edges in a tree T is , the proof can be done quite similar as the proof of Theorem 4. □
Now, for , since and , by (11) and (12) we get the result, as required. □
The following theorem presents another upper bound for GA index.
Hence, the result. □
I.N. Cangul and A.S. Cevik are partially supported by Research Project Offices of Uludag (2012-15, 2012-19 and 2012-20) and Selcuk Universities, respectively.
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