Eccentric Connectivity Index Of Graph 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 $G$.

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.

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.

Comparison Between Second Zagreb Eccentricity Index and Eccentric Connectivity Index of Graphs

For a graph G, the second Zagreb eccentricity index E2(G) and eccentric connectivity index ∈c(G) are two eccentricity-based invariants of graph G. In this paper we prove some results on the comparison between and of connected graphs G of order n and with m edges. The authors demonstrated how a combination of both techniques and human interventions enhances control, decision-making and data analysis systems.

On the extremal eccentric connectivity index of graphs

For a graph G=(V,E), the eccentric connectivity index of G, denoted by ξc(G), is defined as ξc(G)=∑v∈Vɛ(v)d(v), where ɛ(v) and d(v) are the eccentricity and the degree of v in G, respectively. In this paper, we first establish the sharp lower bound for the eccentric connectivity index in terms of the order and the minimum degree of a connected G, and characterize some extremal graphs, which generalize some known results. Secondly, we characterize the extremal trees having the maximum or minimum eccentric connectivity index for trees of order n with given degree sequence. Finally, we give a sharp lower bound for the eccentric connectivity index in terms of the order and the radius of a unicyclic G, and characterize all extremal graphs.

Eccentric connectivity index of graphs with subdivided edges

We consider four classes of graphs arising from a given graph via different types of edge subdivisions. We present explicit formulas expressing their eccentric connectivity index in terms of the eccentric connectivity index of the original graph and some auxiliary invariants.