In the 1980's, 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 FG(x) of a graph G to be k Fk(G)x k where for each k, Fk( 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.
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