The Eccentric-distance Sum of Some Graphs

Let $G = (V,E)$ be a simple connected graph. Theeccentric-distance sum of $G$ is defined as\xi^{ds}(G) =\ds\sum_{\{u,v\}\subseteq V(G)} [e(u)+e(v)] d(u,v), where $e(u)$ %\dsis the eccentricity of the vertex $u$ in $G$ and $d(u,v)$ is thedistance between $u$ and $v$. In this paper, we establish formulaeto calculate the eccentric-distance sum for some graphs, namelywheel, star, broom, lollipop, double star, friendship, multi-stargraph and the join of $P_{n-2}$ and $P_2$

On the adjacent eccentric distance sum of graphs

In this paper we show bounds for the adjacent eccentric distance sum of graphs in terms of Wiener index, maximum degree and minimum degree. We extend some earlier results of Hua and Yu [Bounds for the Adjacent Eccentric Distance Sum, International Mathematical Forum, Vol. 7 (2002) no. 26, 1289–1294]. The adjacent eccentric distance sum index of the graph $G$ is defined as$$\xi ^{sv} (G)= \sum_{v\in V(G)}{\frac{\varepsilon (v) D(v)}{deg(v)}},$$where $\varepsilon(v)$ is the eccentricity of the vertex $v$, $deg(v)$ is the degree of the vertex $v$ and$$D(v)=\sum_{u\in V(G)}{d(u,v)}$$is the sum of all distances from the vertex $v$

Eccentric distance sum index for some classes of connected graphs

In this paper we show some properties of the eccentric distance sum index which is defined as follows $\xi^{d}(G)=\sum_{v \in V(G)}D(v) \varepsilon(v)$. This index is widely used by chemists and biologists in their researches. We present a lower bound of this index for a new class of 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