11 research outputs found
Eccentric connectivity index
The eccentric connectivity index is a novel distance--based molecular
structure descriptor that was recently used for mathematical modeling of
biological activities of diverse nature. It is defined as \,, where and
denote the vertex degree and eccentricity of \,, respectively. We survey
some mathematical properties of this index and furthermore support the use of
eccentric connectivity index as topological structure descriptor. We present
the extremal trees and unicyclic graphs with maximum and minimum eccentric
connectivity index subject to the certain graph constraints. Sharp lower and
asymptotic upper bound for all graphs are given and various connections with
other important graph invariants are established. In addition, we present
explicit formulae for the values of eccentric connectivity index for several
families of composite graphs and designed a linear algorithm for calculating
the eccentric connectivity index of trees. Some open problems and related
indices for further study are also listed.Comment: 25 pages, 5 figure
Hyper-Zagreb indices of graphs and its applications
The first and second Hyper-Zagreb index of a connected graph is defined by and . In this paper, the first and second Hyper-Zagreb indices of certain graphs are computed. Also the bounds for the first and second Hyper-Zagreb indices are determined. Further linear regression analysis of the degree based indices with the boiling points of benzenoid hydrocarbons is carried out. The linear model, based on the Hyper-Zagreb index, is better than the models corresponding to the other distance based indices
A lower bound on the eccentric connectivity index of a graph
AbstractIn pharmaceutical drug design, an important tool is the prediction of physicochemical, pharmacological and toxicological properties of compounds directly from their structure. In this regard, the Wiener index, first defined in 1947, has been widely researched, both for its chemical applications and mathematical properties. Many other indices have since been considered, and in 1997, Sharma, Goswami and Madan introduced the eccentric connectivity index, which has been identified to give a high degree of predictability. If G is a connected graph with vertex set V, then the eccentric connectivity index of G, ξC(G), is defined as ∑v∈Vdeg(v)ec(v), where deg(v) is the degree of vertex v and ec(v) is its eccentricity. Several authors have determined extremal graphs, for various classes of graphs, for this index. We show that a known tight lower bound on the eccentric connectivity index for a tree T, in terms of order and diameter, is also valid for a general graph G, of given order and diameter
Bounds on distance-based topological indices in graphs.
Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.This thesis details the results of investigations into bounds on some distance-based
topological indices.
The thesis consists of six chapters. In the first chapter we define the standard
graph theory concepts, and introduce the distance-based graph invariants called
topological indices. We give some background to these mathematical models, and
show their applications, which are largely in chemistry and pharmacology. To complete
the chapter we present some known results which will be relevant to the work.
Chapter 2 focuses on the topological index called the eccentric connectivity index.
We obtain an exact lower bound on this index, in terms of order, and show that this
bound is sharp. An asymptotically sharp upper bound is also derived. In addition,
for trees of given order, when the diameter is also prescribed, tight upper and lower
bounds are provided.
Our investigation into the eccentric connectivity index continues in Chapter 3.
We generalize a result on trees from the previous chapter, proving that the known
tight lower bound on the index for a tree in terms of order and diameter, is also
valid for a graph of given order and diameter.
In Chapter 4, we turn to bounds on the eccentric connectivity index in terms of
order and minimum degree. We first consider graphs with constant degree (regular
graphs). Došlić, Saheli & Vukičević, and Ilić posed the problem of determining
extremal graphs with respect to our index, for regular (and more specifically,
cubic) graphs. In addressing this open problem, we find upper and lower bounds
for the index. We also provide an extremal graph for the upper bound. Thereafter,
the chapter continues with a consideration of minimum degree. For given order and
minimum degree, an asymptotically sharp upper bound on the index is derived.
In Chapter 5, we turn our focus to the well-studied Wiener index. For trees
of given order, we determine a sharp upper bound on this index, in terms of the
eccentric connectivity index. With the use of spanning trees, this bound is then
generalized to graphs.
Yet another distance-based topological index, the degree distance, is considered
in Chapter 6. We find an asymptotically sharp upper bound on this index, for a
graph of given order. This proof definitively settles a conjecture posed by Tomescu
in 1999