Reciprocal version of product degree distance of cactus graphs

The reciprocal version of product degree distance is a product degree weighted version of Harary index defined for a connected graph G as RDD*(G) = Sigma({x, y}subset of V(G) )(d(G)(x).d(G)(y))/d(G)(x,y), where d(G)(x) is the degree of the vertex x and d(G)(x, y) is the distance from x to y in G. This article is attain the value of RDD* of different types of cactus such as triangular, square and hexagonal chain cactus graphs.Publisher's Versio

The modified Schultz index of graph operations

Given a simple and connected graph G with vertex set V , denoting by dG(u) the degree of a vertex u and dG(u, v) the distance of two vertices, the modified Schultz index of G is given by S ∗ P (G) = {u,v}⊆V dG(u) dG(v) dG(u, v), where the summation goes over all non ordered pairs of vertices of G. In this paper we consider some graph operations, namely cartesian product, complete product, composition and subdivision, and we obtain explicit formulae for the modified Schultz index of a graph in terms of the number of vertices and edges as well as some other topological invariants such as the Wiener index, the Schultz index and the first and second Zagreb indices

Splices, Links, and their Edge-Degree Distances

The edge-degree distance of a simple connected graph G is defined as the sum of the terms (d(e|G)+d(f|G))d(e,f|G) over all unordered pairs {e,f} of edges of G, where d(e|G) and d(e,f|G) denote the degree of the edge e in G and the distance between the edges e and f in G, respectively. In this paper, we study the behavior of two versions of the edge-degree distance under two graph products called splice and link

Graphical Indices and their Applications

The biochemical community has been using graphical (topological, chemical) indices in the study of Quantitative Structure-Activity Relationships (QSAR) and Quantitative Structure-Property Relationships (QSPR), as they have been shown to have strong correlations with the chemical properties of certain chemical compounds (i.e. boiling point, surface area, etc.). We examine some of these chemical indices and closely related pure graph theoretical indices: the Randić index, the Wiener index, the degree distance, and the number of subtrees. We find which structure will maximize the Randić index of a class of graphs known as cacti, and we find a functional relationship between the Wiener index and the degree distance for several types of graphs. We also develop an algorithm to find the structure that maximizes the number of subtrees of trees, a characterization of the second maximal tree may also follow as an immediate result of this algorithm

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