On Rank Functions of Graphs
نویسنده
چکیده
We study rank functions (also known as graph homomorphisms onto Z), ways of imposing graded poset structures on graphs. We first look at a variation on rank functions called discrete Lipschitz functions. We relate the number of Lipschitz functions of a graph G to the number of rank functions of both G and G× E . We then find generating functions that enable us to compute the number of rank or Lipschitz functions of a given graph. We look at a subset of graphs called squarely generated graphs, which are graphs whose cycle space has a basis consisting only of 4-cycles. We show that the number of rank functions of such a graph is proportional to the number of 3-colorings of the same graph, thereby connecting rank functions to the Potts model of statistical mechanics. Lastly, we look at some asymptotics of rank and Lipschitz functions for various types of graphs.
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