Convergence of Graphs and Trees

Abstract

We follow the treatment in [10] to present some results regarding graphons and their application to the convergence of sequences of graphs. This theory is meaningful when dealing with dense graphs. Bubeck and Linial introduce the local profiles of trees as an alternative for trees [4]. We solve an open problem regarding the local profiles which states that as the proportion of paths in a sequence of trees goes to zero the proportion of stars goes to one.