On the distance spectrum and distance-based topological indices of central vertex-edge join of three graphs

Topological indices are molecular descriptors that describe the properties of chemical compounds. These topological indices correlate specific physico-chemical properties like boiling point, enthalpy of vaporization, strain energy, and stability of chemical compounds. This article introduces a new graph operation based on central graph called central vertex-edge join and provides its results related to graph invariants like eccentric-connectivity index, connective eccentricity index, total-eccentricity index, average eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic index, eccentric atom-bond connectivity index, and Wiener index. Also, we discuss the distance spectrum of the central vertex-edge join of three regular graphs. Furthermore, we obtain new families of $D$-equienergetic graphs, which are non $D$-cospectral

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

Comparison Between Two Eccentricity-based Topological Indices of Graphs

For a connected graph (G), the eccentric connectivity index (ECI) and the first Zagreb eccentricity index of (G) are defined as ( xi ^{c}(G)= sum_{v_i in V(G)}deg_G(v_i)varepsilon_G(v_i)) and (E_1(G)=sum_{v_iin V(G)}varepsilon_{G}(v_i)^{2}), respectively, where (deg_G(v_i)) is the degree of (v_i) in (G) and (varepsilon_G(v_i)) denotes the eccentricity of vertex (v_i )in (G). In this paper we compare the eccentric connectivity index and the first Zagreb eccentricity index of graphs. It is proved that (E_1(T)>xi^c(T)) for any tree (T). This improves a result by Das[25] for the chemical trees. Moreover, we also show that there are infinite number of chemical graphs (G) with (E_1(G)>xi^c(G)). We also present an example in which infinite graphs (G) are constructed with (E_1(G)=xi^c(G)) and give some results on the graphs (G) with (E_1(G)<xi^c(G)). Finally, an effective construction is proposed for generating infinite graphs with each comparative inequality possibility between these two topological indices

Some Topological Indices of Subgroup Graph of Symmetric Group

The concept of the topological index of a graph is increasingly diverse because researchers continue to introduce new concepts of topological indices. Researches on the topological indices of a graph which initially only examines graphs related to chemical structures begin to examine graphs in general. On the other hand, the concept of graphs obtained from an algebraic structure is also increasingly being introduced. Thus, studying the topological indices of a graph obtained from an algebraic structure such as a group is very interesting to do. One concept of graph obtained from a group is subgroup graph introduced by Anderson et al in 2012 and there is no research on the topology index of the subgroup graph of the symmetric group until now. This article examines several topological indices of the subgroup graphs of the symmetric group for trivial normal subgroups. This article focuses on determining the formulae of various Zagreb indices such as first and second Zagreb indices and co-indices, reduced second Zagreb index and first and second multiplicatively Zagreb indices and several eccentricity-based topological indices such as first and second Zagreb eccentricity indices, eccentric connectivity, connective eccentricity, eccentric distance sum and adjacent eccentric distance sum indices of these graphs

Eccentric connectivity index

The eccentric connectivity index $\xi^c$ 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 $\xi^c (G) = \sum_{v \in V (G)} deg (v) \cdot \epsilon (v)$\,, where $deg (v)$ and $\epsilon (v)$ denote the vertex degree and eccentricity of $v$\,, 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

On the extremal properties of the average eccentricity

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $ecc (G)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The average eccentricity is deeply connected with a topological descriptor called the eccentric connectivity index, defined as a sum of products of vertex degrees and eccentricities. In this paper we analyze extremal properties of the average eccentricity, introducing two graph transformations that increase or decrease $ecc (G)$. Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX, about the average eccentricity and other graph parameters (the clique number, the Randi\' c index and the independence number), refute one AutoGraphiX conjecture about the average eccentricity and the minimum vertex degree and correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure