We use graphs to define sets of Salem and Pisot numbers and prove that the union of these sets is closed, supporting a conjecture of Boyd that the set of all Salem and Pisot numbers is closed. We find all trees that define Salem numbers. We show that for all integers $n$ the smallest known element of the {\small$n$}th derived set of the set of Pisot numbers comes from a graph. We define the Mahler measure of a graph and find all graphs of Mahler measure less than {\small $\frac12(1+\sqrt{5})$}. Finally, we list all small Salem numbers known to be definable using a graph.