40 research outputs found
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 be the set of connected unicyclic (molecular) graphs of order n with diameter d. In this paper we completely characterize the graphs from 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 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 \,, where and
denote the vertex degree and eccentricity of \,, 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