Graph homomorphisms between trees

Abstract

In this paper we study several problems concerning the number of homomorphisms of trees. We begin with an algorithm for the number of homomorphisms from a tree to any graph. By using this algorithm and some transformations on trees, we study various extremal problems about the number of homomorphisms of trees. These applications include a far reaching generalization and a dual of Bollobás and Tyomkyn's result concerning the number of walks in trees. Some other main results of the paper are the following. Denote by hom(H, G) the number of homomorphisms from a graph H to a graph G. For any tree Tm on m vertices we give a general lower bound for hom(Tm, G) by certain entropies of Markov chains defined on the graph G. As a particular case, we show that for any graph G exp(Hλ(G))λm−1 ≤ hom(Tm,G) where λ is the largest eigenvalue of the adjacency matrix of G and Hλ(G) is a certain constant depending only on G which we call the spectral entropy of G. We also show that if Tm is any fixed tree and hom(Tm, Pn) > hom(Tm, Tn) for some tree Tn on n vertices, then Tn must be the tree obtained from a path Pn−1 by attaching a pendant vertex to the second vertex of Pn−1. All the results together enable us to show that among all trees with fixed number of vertices, the path graph has the fewest number of endomorphisms while the star graph has the most.