Eccentric Connectivity Research Articles

Ordering graphs with large eccentricity-based topological indices

For a connected graph, the first Zagreb eccentricity index xi _{1} is defined as the sum of the squares of the eccentricities of all vertices, and the second Zagreb eccentricity index xi _{2} is defined as the sum of the products of the eccentricities of pairs of adjacent vertices. In this paper, we mainly present a different and universal approach to determine the upper bounds respectively on the Zagreb eccentricity indices of trees, unicyclic graphs and bicyclic graphs, and characterize these corresponding extremal graphs, which extend the ordering results of trees, unicyclic graphs and bicyclic graphs in (Du et al. in Croat. Chem. Acta 85:359–362, 2012; Qi et al. in Discrete Appl. Math. 233:166–174, 2017; Li and Zhang in Appl. Math. Comput. 352:180–187, 2019). Specifically, we determine the n-vertex trees with the i-th largest indices xi _{1} and xi _{2} for i up to lfloor n/2+1 rfloor compared with the first three largest results of xi _{1} and xi _{2} in (Du et al. in Croat. Chem. Acta 85:359–362, 2012), the n-vertex unicyclic graphs with respectively the i-th and the j-th largest indices xi _{1} and xi _{2} for i up to lfloor n/2-1 rfloor and j up to lfloor 2n/5+1 rfloor compared with respectively the first two and the first three largest results of xi _{1} and xi _{2} in (Qi et al. in Discrete Appl. Math. 233:166–174, 2017), and the n-vertex bicyclic graphs with respectively the i-th and the j-th largest indices xi _{1} and xi _{2} for i up to lfloor n/2-2rfloor and j up to lfloor 2n/15+1rfloor compared with the first two largest results of xi _{2} in (Li and Zhang in Appl. Math. Comput. 352:180–187, 2019), where nge 6. More importantly, we propose two kinds of index functions for the eccentricity-based topological indices, which can yield more general extremal results simultaneously for some classes of indices. As applications, we obtain and extend some ordering results about the average eccentricity of bicyclic graphs, and the eccentric connectivity index of trees, unicyclic graphs and bicyclic graphs.

On sufficient topological indices conditions for properties of graphs

In this paper, we present sufficient conditions on eccentric connectivity index, eccentric distance sum and connective eccentricity index for graphs to be k-hamiltonian, k-edge-hamiltonian or k-path-coverable.

On Connected Graphs Having the Maximum Connective Eccentricity Index

The connective eccentricity index (CEI) of a connected graph G is defined as $$\xi ^{ee}(G)=\sum _{u\in V_G}[d_G(u)/\varepsilon _G(u)]$$ , where $$d_G(u)$$ and $$\varepsilon _G(u)$$ are the degree and eccentricity, respectively, of the vertex $$u\in V_G$$ of G. In this paper, graphs with the maximum CEI are characterized from the class of all connected graphs of a fixed order and size. Graphs having maximum CEI are also determined from some other well-known classes of connected graphs of a given order; namely, the Halin graphs, triangle-free graphs, planar graphs and outer-planar graphs.

On intersection graph of dihedral group

Let $G$ be a finite group. The intersection graph of $G$ is a graph whose vertex set is the set of all proper non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K \neq \{e\}$, where $e$ is the identity of the group $G$. In this paper, we investigate some properties and exploring some topological indices such as Wiener, Hyper-Wiener, first and second Zagreb, Schultz, Gutman and eccentric connectivity indices of the intersection graph of $D_{2n}$ for $n=p^2$, $p$ is prime. We also find the metric dimension and the resolving polynomial of the intersection graph of $D_{2p^2}$.

Network Similarity Measure and Ediz Eccentric Connectivity Index

Network similarity measures have proven essential in the field of network analysis. Also, topological indices have been used to quantify the topology of networks and have been well studied. In this paper, we employ a new topological index which we call the Ediz eccentric connectivity index. We use this quantity to define network similarity measures as well. First, we determine the extremal value of the Ediz eccentric connectivity index on some network classes. Second, we compare the network similarity measure based on the Ediz eccentric connectivity index with other well-known topological indices such as Wiener index, graph energy, Randić index, the largest eigenvalue, the largest Laplacian eigenvalue, and connectivity eccentric index. Numerical results underpin the usefulness of the chosen measures. They show that our new measure outperforms all others, except the one based on Wiener index. This means that the measure based on Wiener index is still the best, but the new one has certain advantage to some extent.

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 $G$.

On the non-commuting graph of dihedral group

For a nonabelian group G, the non-commuting graph Γ of G is defined as the graph with vertex-set G - Z ( G ), where Z ( G ) is the center of G , and two distinct vertices of Γ are adjacent if they do not commute in G . In this paper, we investigate the detour index, eccentric connectivity and total eccentricity polynomials of the non-commuting graph on D 2 n . We also find the mean distance of the non-commuting graph on D 2 n .

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.

Critical Evaluation of the Shear Lag Factor Provisions for W-Sections

Shear lagging resulting from non-uniformity of stress distribution around the connection, is one of the important design considerations in steel construction since it reduces the load capacity of tension members. In this paper, the rationality of the AISC provisions for calculating the shear lag factor (U) for W and WT-sections was assessed. In the current AISC provisions, there is an enormous difference in the calculated value of U using two allowed for W and WT sections; yet the AISC allows using the largest value for U. Accordingly, this study was conducted to investigate the reasonable value of U in W and WT-sections based on finite element analysis (FEA). Two criteria were used to calculate this lagging for 50 sizes of W sections and 50 sizes of WT sections. For each section size, three models having different connection length were created, which results in a 150 models for W sections and 150 models for WT-sections. For each section type, the influence of the bolt diameter, the connection eccentricity, flange width, depth, flange area, and gross-sectional area were evaluated individually through a parametric study. It was shown that, the calculated values for U based on the two cases allowed by the AISC provisions were comparable for shallow W and WT sections, but differ enormously for larger sections. Accordingly, a new equation was developed for more reasonable calculation of U for any W and WT standard size.

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.