169 research outputs found
Counting MSTD Sets in Finite Abelian Groups
In an abelian group G, a more sums than differences (MSTD) set is a subset A
of G such that |A+A|>|A-A|. We provide asymptotics for the number of MSTD sets
in finite abelian groups, extending previous results of Nathanson. The proof
contains an application of a recently resolved conjecture of Alon and Kahn on
the number of independent sets in a regular graph.Comment: 17 page
Independence Polynomials of Molecular Graphs
In the 1980\u27s, it was noticed by molecular chemists that the stability and boiling point of certain molecules were related to the number of independent vertex sets in the molecular graphs of those chemicals. This led to the definition of the Merrifield-Simmons index of a graph G as the number of independent vertex sets in G. This parameter was extended by graph theorists, who counted independent sets of different sizes and defined the independence polynomial F_G(x) of a graph G to be \sum_k F_k(G)x^k where for each k, F_k(G) is the number of independent sets of k vertices. This thesis is an investigation of independence polynomials of several classes of graphs, some directly related to molecules of hydrocarbons. In particular, for the graphs of alkanes, alkenes, and cycloalkanes, we have determined the Merrifield-Simmons index, the independence polynomial, and, in some cases, the generating function for the independence polynomial. These parameters are also determined in several classes of graphs which are natural generalizations of the hydrocarbons. The proof techniques used in studying the hydrocarbons have led to some possibly interesting results concerning the coefficients of independence polynomials of regular graphs with large girth
The number of maximum matchings in a tree
We determine upper and lower bounds for the number of maximum matchings
(i.e., matchings of maximum cardinality) of a tree of given order.
While the trees that attain the lower bound are easily characterised, the trees
with largest number of maximum matchings show a very subtle structure. We give
a complete characterisation of these trees and derive that the number of
maximum matchings in a tree of order is at most (the
precise constant being an algebraic number of degree 14). As a corollary, we
improve on a recent result by G\'orska and Skupie\'n on the number of maximal
matchings (maximal with respect to set inclusion).Comment: 38 page
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