The quotients between the (revised) Szeged index and Wiener index of graphs

Let $Sz(G),Sz^*(G)$ and $W(G)$ be the Szeged index, revised Szeged index and Wiener index of a graph $G.$ In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order $n\geqslant 10$ are characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on $Sz(G)/W(G)$ is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on $Sz^*(G)/W(G)$ is identified for $G$ containing at least one cycle.Comment: 25 pages, 5 figure

The Wiener polarity index of benzenoid systems and nanotubes

In this paper, we consider a molecular descriptor called the Wiener polarity index, which is defined as the number of unordered pairs of vertices at distance three in a graph. Molecular descriptors play a fundamental role in chemistry, materials engineering, and in drug design since they can be correlated with a large number of physico-chemical properties of molecules. As the main result, we develop a method for computing the Wiener polarity index for two basic and most commonly studied families of molecular graphs, benzenoid systems and carbon nanotubes. The obtained method is then used to find a closed formula for the Wiener polarity index of any benzenoid system. Moreover, we also compute this index for zig-zag and armchair nanotubes

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

Recognition of generalized network matrices

In this PhD thesis, we deal with binet matrices, an extension of network matrices. The main result of this thesis is the following. A rational matrix A of size n times m can be tested for being binet in time O(n^6 m). If A is binet, our algorithm outputs a nonsingular matrix B and a matrix N such that [B N] is the node-edge incidence matrix of a bidirected graph (of full row rank) and A=B^{-1} N. Furthermore, we provide some results about Camion bases. For a matrix M of size n times m', we present a new characterization of Camion bases of M, whenever M is the node-edge incidence matrix of a connected digraph (with one row removed). Then, a general characterization of Camion bases as well as a recognition procedure which runs in O(n^2m') are given. An algorithm which finds a Camion basis is also presented. For totally unimodular matrices, it is proven to run in time O((nm)^2) where m=m'-n. The last result concerns specific network matrices. We give a characterization of nonnegative {r,s}-noncorelated network matrices, where r and s are two given row indexes. It also results a polynomial recognition algorithm for these matrices.Comment: 183 page

On oriented graphs with minimal skew energy

Let $S(G^\sigma)$ be the skew-adjacency matrix of an oriented graph $G^\sigma$. The skew energy of $G^\sigma$ is defined as the sum of all singular values of its skew-adjacency matrix $S(G^\sigma)$. In this paper, we first deduce an integral formula for the skew energy of an oriented graph. Then we determine all oriented graphs with minimal skew energy among all connected oriented graphs on $n$ vertices with $m \ (n\le m < 2(n-2))$ arcs, which is an analogy to the conjecture for the energy of undirected graphs proposed by Caporossi {\it et al.} [G. Caporossi, D. Cvetkovi$\acute{c}$, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996.]Comment: 15 pages. Actually, this paper was finished in June 2011. This is an updated versio