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

(SI10-130) On Regular Inverse Eccentric Fuzzy Graphs

Two new concepts of regular inverse eccentric fuzzy graphs and totally regular inverse eccentric fuzzy graphs are established in this article. By illustrations, these two graphs are compared and the results are derived. Equivalent condition for the existence of these two graphs are found. The exact values of Order and Size for some standard inverse eccentric graphs are also derived

UNIQUE ECCENTRIC CLIQUE GRAPHS

Let $G$ be a connected graph and $\zeta$ the set of all cliques in $G$. In this paper we introduce the concepts of unique $(\zeta, \zeta)$-eccentric clique graphs and self $(\zeta, \zeta)$-centered graphs. Certain standard classes of graphs are shown to be self $(\zeta, \zeta)$-centered, and we characterize unique $(\zeta, \zeta)$-eccentric clique graphs which are self $(\zeta, \zeta)$-centered

Equitable eccentric domination in graphs

In this paper, we define equitable eccentric domination in graphs. An eccentric dominating set S ⊆ V (G) of a graph G(V, E) is called an equitable eccentric dominating set if for every v ∈ V − S there exist at least one vertex u ∈ V such that |d(v) − d(u)| ≤ 1 where vu ∈ E(G). We find equitable eccentric domination number γeqed(G) for most popular known graphs. Theorems related to γeqed(G) have been stated and proved

Distance based indices of generalized transformation graphs

In this paper, the expressions for the Wiener index, Gutman index, degree distance, eccentric connectivity index and eccentric distance sum of the generalized transformation graphs G+-and G -+ are obtained in terms of the parameters of underline graphs