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 α\alpha-spectral radius of hypergraphs

For real $\alpha\in [0,1)$ and a hypergraph $G$, the $\alpha$-spectral radius of $G$ is the largest eigenvalue of the matrix $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ is the adjacency matrix of $G$, which is a symmetric matrix with zero diagonal such that for distinct vertices $u,v$ of $G$, the $(u,v)$-entry of $A(G)$ is exactly the number of edges containing both $u$ and $v$, and $D(G)$ is the diagonal matrix of row sums of $A(G)$. We study the $\alpha$-spectral radius of a hypergraph that is uniform or not necessarily uniform. We propose some local grafting operations that increase or decrease the $\alpha$-spectral radius of a hypergraph. We determine the unique hypergraphs with maximum $\alpha$-spectral radius among $k$-uniform hypertrees, among $k$-uniform unicyclic hypergraphs, and among $k$-uniform hypergraphs with fixed number of pendant edges. We also determine the unique hypertrees with maximum $\alpha$-spectral radius among hypertrees with given number of vertices and edges, the unique hypertrees with the first three largest (two smallest, respectively) $\alpha$-spectral radii among hypertrees with given number of vertices, the unique hypertrees with minimum $\alpha$-spectral radius among the hypertrees that are not $2$-uniform, the unique hypergraphs with the first two largest (smallest, respectively) $\alpha$-spectral radii among unicyclic hypergraphs with given number of vertices, and the unique hypergraphs with maximum $\alpha$-spectral radius among hypergraphs with fixed number of pendant edges