Two problems on independent sets in graphs

Let $i_t(G)$ denote the number of independent sets of size $t$ in a graph $G$. Levit and Mandrescu have conjectured that for all bipartite $G$ the sequence $(i_t(G))_{t \geq 0}$ (the {\em independent set sequence} of $G$) is unimodal. We provide evidence for this conjecture by showing that is true for almost all equibipartite graphs. Specifically, we consider the random equibipartite graph $G(n,n,p)$, and show that for any fixed $p\in(0,1]$ its independent set sequence is almost surely unimodal, and moreover almost surely log-concave except perhaps for a vanishingly small initial segment of the sequence. We obtain similar results for $p=\tilde{\Omega}(n^{-1/2})$. We also consider the problem of estimating $i(G)=\sum_{t \geq 0} i_t(G)$ for $G$ in various families. We give a sharp upper bound on the number of independent sets in an $n$-vertex graph with minimum degree $\delta$, for all fixed $\delta$ and sufficiently large $n$. Specifically, we show that the maximum is achieved uniquely by $K_{\delta, n-\delta}$, the complete bipartite graph with $\delta$ vertices in one partition class and $n-\delta$ in the other. We also present a weighted generalization: for all fixed $x>0$ and $\delta >0$, as long as $n=n(x,\delta)$ is large enough, if $G$ is a graph on $n$ vertices with minimum degree $\delta$ then $\sum_{t \geq 0} i_t(G)x^t \leq \sum_{t \geq 0} i_t(K_{\delta, n-\delta})x^t$ with equality if and only if $G=K_{\delta, n-\delta}$.Comment: 15 pages. Appeared in Discrete Mathematics in 201

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