The smallest Hosoya index of unicyclic graphs with given diameter

The Hosoya index of a (molecular) graph is defined as the total number of the matchings, including the empty edge set, of this graph. Let ${cal{U}}_{n,d}$ be the set of connected unicyclic (molecular) graphs of order n with diameter d. In this paper we completely characterize the graphs from ${cal{U}}_{n,d}$ minimizing the Hosoya index and determine the values of corresponding indices. Moreover, the third smallest Hosoya index of unicyclic graphs is determined

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

MAXIMISING THE NUMBER OF CONNECTED INDUCED SUBGRAPHS OF UNICYCLIC GRAPHS

Denote by G(n, d, g, k) the set of all connected graphs of order n, having d > 0 cycles, girth g and k pendent vertices. In this paper, we give a partial characterisation of the structure of all maximal graphs in G(n, d, g, k) for the number of connected induced subgraphs. For the special case d = 1, we find a complete characterisation of all maximal unicyclic graphs. We also derive a precise formula for the maximum number of connected induced subgraphs given: (1) order, girth, and number of pendent vertices; (2) order and girth; (3) order