Eccentric Connectivity Index Research Articles

Some Properties of the Leap Eccentric Connectivity Index of Graphs

The leap eccentric connectivity index of $G$ is defined as $$Lxi^{C}(G)=sum_{vin V(G)}d_{2}(v|G)e(v|G)$$ where $d_{2}(v|G) $ be the second degree of the vertex $v$ and $e(v|G)$ be the eccentricity of the vertex $v$ in $G$. In this paper, we give some properties of the leap eccentric connectivity index of the graph $G$.

Eccentric topological properties of a graph associated to a finite dimensional vector space

Abstract A topological index is actually designed by transforming a chemical structure into a number. Topological index is a graph invariant which characterizes the topology of the graph and remains invariant under graph automorphism. Eccentricity based topological indices are of great importance and play a vital role in chemical graph theory. In this article, we consider a graph (non-zero component graph) associated to a finite dimensional vector space over a finite filed in the context of the following eleven eccentricity based topological indices: total eccentricity index; average eccentricity index; eccentric connectivity index; eccentric distance sum index; adjacent distance sum index; connective eccentricity index; geometric arithmetic index; atom bond connectivity index; and three versions of Zagreb indices. Relationship of the investigated indices and their dependency with respect to the involved parameters are also visualized by evaluating them numerically and by plotting their results.

On the Difference Between the Eccentric Connectivity Index and Eccentric Distance Sum of Graphs

The eccentric connectivity index of a graph G is $$\xi ^c(G) = \sum _{v \in V(G)}\varepsilon (v)\deg (v)$$ , and the eccentric distance sum is $$\xi ^d(G) = \sum _{v \in V(G)}\varepsilon (v)D(v)$$ , where $$\varepsilon (v)$$ is the eccentricity of v, and D(v) the sum of distances between v and the other vertices. A lower and an upper bound on $$\xi ^d(G) - \xi ^c(G)$$ is given for an arbitrary graph G. Regular graphs with diameter at most 2 and joins of cocktail-party graphs with complete graphs form the graphs that attain the two equalities, respectively. Sharp lower and upper bounds on $$\xi ^d(T) - \xi ^c(T)$$ are given for arbitrary trees. Sharp lower and upper bounds on $$\xi ^d(G)+\xi ^c(G)$$ for arbitrary graphs G are also given, and a sharp lower bound on $$\xi ^d(G)$$ for graphs G with a given radius is proved.

On the maximum connective eccentricity index among k-connected graphs

The connective eccentricity index (CEI for short) of a graph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the degree of [Formula: see text] and [Formula: see text] is the eccentricity of [Formula: see text] in [Formula: see text]. In this paper, we characterize the unique graphs with maximum CEI from three classes of graphs: the [Formula: see text]-vertex graphs with fixed connectivity and diameter, the [Formula: see text]-vertex graphs with fixed connectivity and independence number, and the [Formula: see text]-vertex graphs with fixed connectivity and minimum degree.

Further results on Zagreb eccentricity coindices

The eccentric connectivity index and second Zagreb eccentricity index are well-known graph invariants defined as the sums of contributions dependent on the eccentricities of adjacent vertices over all edges of a connected graph. The coindices of these invariants have recently been proposed by considering analogous contributions from the pairs of non-adjacent vertices. Here, we obtain several lower and upper bounds on the eccentric connectivity coindex and second Zagreb eccentricity coindex in terms of some graph parameters such as order, size, number of non-adjacent vertex pairs, radius, and diameter, and relate these invariants to some well-known graph invariants such as Zagreb indices and coindices, status connectivity indices and coindices, ordinary and multiplicative Zagreb eccentricity indices, Wiener index, degree distance, total eccentricity, eccentric connectivity index, second eccentric connectivity index, and eccentric-distance sum. Moreover, we compute the values of these coindices for two graph constructions, namely, double graphs and extended double graphs.

Distance and eccentricity based polynomials and indices of m−level Wheel graph

Distance and degree based topological polynomial and indices of molecular graphs have various applications in chemistry, computer networking and pharmacy. In this paper, we give hosoya polynomial, Harary polynomial, Schultz polynomial, modified Schultz polynomial, eccentric connectivity polynomial, modified Wiener index, modified hyper Wiener index, generalized Harary index, multiplicative Wiener index, Schultz index, modified Schultz index, eccentric connectivity index of generalized wheel networks Wn,m. We also give pictorial representation of computed topological polynomials and indices on the involved parameters m and n.

The connective eccentricity index of graphs and its applications to octane isomers and benzenoid hydrocarbons

AbstractThe connective eccentricity index (CEI) of a graph G is defined as , where εG(.) denotes the eccentricity of the corresponding vertex. The CEI obligates an influential ability, which is due to its estimating pharmaceutical properties. In this paper, we first characterize the extremal graphs with respect to the CEI among k‐connected graphs (k‐connected bipartite graphs) with a given diameter. Then, the sharp upper bound on the CEI of graphs with given connectivity and minimum degree (independence number) is determined. Finally, we calculate the CEI of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons. We compare their CEI with some other distance‐based topological indices through their correlations with the chemical properties. The linear model for the CEI is better than or as good as the models corresponding to the other distance‐based indices.

Eccentric connectivity polynomial [ECP] of some standard graphs

Eccentric connectivity index of a graph G denoted by ξc (G), can be defined as here deqG (v) represents the vertex degree of v in G and and ecc(v) = max[(x, v)//x ∈ V(G)] eccentricity of v, is the distance to a farthest vertex from v and it is denoted by ξc (G). Also corresponding Eccentric connectivity polynomial [ECP] is denoted by ECP(G, x) is given by, . This paper consists of Eccentric Connectivity polynomial [ECP] of some standard graphs and for some class of graphs.

Computing Eccentricity-Based Topological Indices of 2-Power Interconnection Networks

In a connected graph G with a vertex v, the eccentricity εv of v is the distance between v and a vertex farthest from v in the graph G. Among eccentricity-based topological indices, the eccentric connectivity index, the total eccentricity index, and the Zagreb index are of vital importance. The eccentric connectivity index of G is defined by ξG = ∑v∈VGdvεv, where dv is the degree of the vertex v and εv is the eccentricity of v in G. The topological structure of an interconnected network can be modeled by using graph explanation as a tool. This fact has been universally accepted and used by computer scientists and engineers. More than that, practically, it has been shown that graph theory is a very powerful tool for designing and analyzing the topological structure of interconnection networks. The topological properties of the interconnection network have been computed by Hayat and Imran (2014), Haynes et al. (2002), and Imran et al. (2015). In this paper, we compute the close results for eccentricity-based topological indices such as the eccentric connectivity index, the total eccentricity index, and the first, second, and third Zagreb eccentricity index of a hypertree, sibling tree, and X-tree for k-level by using the edge partition method.

On the quotients between the eccentric connectivity index and the eccentric distance sum of graphs with diameter 2

For a connected graph G and a vertex v in G, let dG(v), εG(v) and DG(v) be the degree, eccentricity and distance sum of v, respectively. The eccentric connectivity index of G, denoted by ξc(G), is defined to be ξc(G)=∑v∈V(G)dG(v)εG(v), and the eccentric distance sum of G, denoted by ξd(G), is defined to be ξd(G)=∑v∈V(G)εG(v)DG(v). Denote by Gn2 the set of connected graphs of order n and diameter two. More recently, Zhang et al. (2019) investigated the relationship between the eccentric connectivity index and eccentric distance sum, and posed the problem to determine sharp upper and lower bounds on ξc(G)ξd(G) for graph in Gn2. In this short paper, we solve this problem. Sharp upper and lower bounds on ξc(G)ξd(G) for graph in Gn2 are determined, and the corresponding extremal graphs are characterized as well.

On the Eccentric Connectivity Polynomial ofℱ-Sum of Connected Graphs

The eccentric connectivity polynomial (ECP) of a connected graphG=VG,EGis described asξcG,y=∑a∈VGdegGayecGa, whereecGaanddegGarepresent the eccentricity and the degree of the vertexa, respectively. The eccentric connectivity index (ECI) can also be acquired fromξcG,yby taking its first derivatives aty=1. The ECI has been widely used for analyzing both the boiling point and melting point for chemical compounds and medicinal drugs in QSPR/QSAR studies. As the extension of ECI, the ECP also performs a pivotal role in pharmaceutical science and chemical engineering. Graph products conveniently play an important role in many combinatorial applications, graph decompositions, pure mathematics, and applied mathematics. In this article, we work out the ECP ofℱ-sum of graphs. Moreover, we derive the explicit expressions of ECP for well-known graph products such as generalized hierarchical, cluster, and corona products of graphs. We also apply these outcomes to deduce the ECP of some classes of chemical graphs.

Comparative study of distance-based graph invariants

The investigation on relationships between various graph invariants has received much attention over the past few decades, and some of these research are associated with Graffiti conjectures (Fajtlowicz and Waller in Congr Numer 60:187–197, 1987) or AutoGraphiX conjectures (Aouchiche et al. in: Liberti, Maculan (eds) Global optimization: from theory to implementation, Springer, New York, 2006). The reciprocal degree distance (RDD), the adjacent eccentric distance sum (AEDS), the average distance (AD) and the connective eccentricity index (CEI) are all distance-based graph invariants or topological indices, some of which found applications in Chemistry. In this paper, we investigate the relationship between RDD and other three graph invariants AEDS, CEI and AD. First, we prove that AEDS > RDD for any tree with at least three vertices. Then, we prove that RDD > CEI for all connected graphs with at least three vertices. Moreover, we prove that RDD > AD for all connected graphs with at least three vertices. As a consequence, we prove that AEDS > CEI and AEDS > AD for any tree with at least three vertices.

A new approach to find eccentric indices of some graphs

Let G be a simple and connected graph. The eccentricity connectivity index ECI(G) and the eccentric version of atom-bond connectivity index ABC 5(G) of a graph G are and respectively. In this paper, we discuss a new approach to calculate the exact values of ECI of butterfly network, bene network graph and toroidal grid graph. We also find the ABC5 (G) indices of these graphs.

On modified eccentric connectivity index of nanotube

Eccentricity based topological indices have attracted large attraction in the field of chemical graph theory. Eccentricity based topological indices are prominent due to their wide range applications. With the help of eccentricity based topological indices we can predict excellent accuracy rate in certain biological activities of diverse nature as com-pared to other indices. Carbon nanotubes are the cylindrical molecules that consist of rolled up sheets of carbon atoms. In this paper we consider the chemical graph of NAn nanotube and compute an important eccentricity based topological index called Modified eccentric connectivity index.

General eccentric connectivity index of trees and unicyclic graphs

We introduce the general eccentric connectivity index of a graph G, ECIα(G)=∑v∈V(G)eccG(v)dGα(v) for α∈R, where V(G) is the vertex set of G, eccG(v) is the eccentricity of a vertex v and dG(v) is the degree of v in G. We present lower and upper bounds on the general eccentric connectivity index for trees of given order, trees of given order and diameter, and trees of given order and number of pendant vertices. Then we give lower and upper bounds on the general eccentric connectivity index for unicyclic graphs of given order, and unicyclic graphs of given order and girth. The upper bounds for trees of given order and diameter, and trees of given order and number of pendant vertices hold for α>1. All the other bounds are valid for 0<α≤1 or 0<α<1. We present all the extremal graphs, which means that our bounds are best possible.

Eccentric connectivity indices of titania nanotubes TiO2[m;n

The eccentric connectivity index ECI is a chemical structure descriptor that is currently being used for modeling of biological activities of a chemical compound. This index has been proved to provide a high degree of predictability compared to some other well-known indices in case of anticonvulsant, anti-inflammatory, and diuretic activities. The ECI of an infinite class of 1-polyacenic (phenylenic) nanotubes has been recently studied. In this study, we computed Ediz eccentric index and augmented eccentric connectivity index of Titania nanotube TiO2[m;n].

Extremal trees of given segment sequence with respect to some eccentricity-based invariants

A path P is a segment of a tree if the endpoints of P are of degree 1 or at least 3, and each of the rest vertices are of degree 2 in the tree. The lengths of all the segments of this tree form its segment sequence. Denote by Tl the set of all trees on n vertices with the segment sequence l=(l1,l2,…,lm), where l1⩾l2⩾⋯⩾lm. In this paper, the extremal structures of trees among Tl, which minimize or maximize the first (resp. second) Zagreb eccentricity index, the eccentric connectivity index, and the eccentric distance sum, are characterized. Furthermore, we consider similar extremal problems for trees with fixed number of segments.

The eccentric connectivity index of polycyclic aromatic hydrocarbons (PAHs)

Mathematical chemistry is the area of research engaged in new application of Mathematics in Chemistry. Major areas of research in mathematical chemistry include chemical graph theory. Chemical graph theory applies graph theory to mathematical modeling of chemical phenomena. If G=(V(G),E(G)) is a connected graph,where V(G) is a non-empty set of vertices and E(G) is a set of edges, then the eccentric connectivity index of G (denoted by ξ(G)) was defined as ζ(G)= where dv is the degree of a vertex v and e(v) is its eccentricity. In this study, we investigated the eccentric connectivity index of polycyclic aromatic hydrocarbons (PAHs).

ECCENTRIC INDICES OF CRYSTAL CUBIC CARBON STRUCTURE

Chemical graph theory helps to understand the structural properties of a molecular graph. The molecular graphs are the graphs that consists of atoms called vertices and the covalent bond between them called edges. The eccentricity _u of vertex u in a connected graph G, is the distance between u and a vertex far- thermost from u. In this article, we study the modified eccentric connectivity index _c(G), Ediz eccentric connectivity index E_c(G), Augmented Eccentric Connectivity index A_(G), superaugmented eccentric connectivity index-1, index-2, index-3 and modi_ed eccentric connectivity polynomial _c(G; x) of CCC(n) of crystal structure of cubic carbon for n-levels.

New results on eccentric connectivity indices of V-Phenylenic nanotube

Topological index is a type of molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices are used for developing the quantitative structure activity relationships (QSARs) in which the biological activity or other properties of the molecules are correlated with their chemical structure. Eccentric connectivity indices are the well-known topological indices in this regards. In this research study, we computed some eccentric connectivity indices of the V-Phenylenic nanotube VPHX[p;q], these are our results.