143 research outputs found
The Expected Values of Hosoya Index and Merrifield-Simmons Index of Random Hexagonal Cacti
Hosoya index and Merrifield-Simmons index are two well-known topological
descriptors that reflex some physical properties, boiling point or heat of
formation for instance, of bezenoid hydrocarbon compounds. In this paper, we
establish the generating functions of the expected values of these two indices
of random hexagonal cacti. This generalizes the results of Doslic and Maloy,
published in Discrete Mathemaics, in 2010. By applying the ideas on meromorphic
functions and the growth of power series coefficients, the asymptotic behaviors
of these indices on the random cacti have been established.Comment: 27 pages, 4 figures, author's name spelling in references revise
Counting Independent Sets of a Fixed Size in Graphs with Given Minimum Degree
Galvin showed that for all fixed δ and sufficiently large n, the n-vertex graph with minimum degree δ that admits the most independent sets is the complete bipartite graph . He conjectured that except perhaps for some small values of t, the same graph yields the maximum count of independent sets of size t for each possible t. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples with , no n-vertex bipartite graph with minimum degree δ admits more independent sets of size t than . Here, we make further progress. We show that for all triples with and , no n-vertex graph with minimum degree δ admits more independent sets of size t than , and we obtain the same conclusion for and . Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree δ whose minimum degree drops on deletion of an edge or a vertex
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