Graph homomorphisms between trees

In this paper we study several problems concerning the number of homomorphisms of trees. We give an algorithm for the number of homomorphisms from a tree to any graph by the Transfer-matrix method. 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 of Bollob\'as and Tyomkyn's result concerning the number of walks in trees. Some other highlights 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 $T_m$ on $m$ vertices we give a general lower bound for $\hom(T_m,G)$ by certain entropies of Markov chains defined on the graph $G$. As a particular case, we show that for any graph $G$, $$\exp(H_{\lambda}(G))\lambda^{m-1}\leq\hom(T_m,G),$$ where $\lambda$ is the largest eigenvalue of the adjacency matrix of $G$ and $H_{\lambda}(G)$ is a certain constant depending only on $G$ which we call the spectral entropy of $G$. In the particular case when $G$ is the path $P_n$ on $n$ vertices, we prove that $$\hom(P_m,P_n)\leq \hom(T_m,P_n)\leq \hom(S_m,P_n),$$ where $T_m$ is any tree on $m$ vertices, and $P_m$ and $S_m$ denote the path and star on $m$ vertices, respectively. We also show that if $T_m$ is any fixed tree and $$\hom(T_m,P_n)>\hom(T_m,T_n),$$ for some tree $T_n$ on $n$ vertices, then $T_n$ must be the tree obtained from a path $P_{n-1}$ by attaching a pendant vertex to the second vertex of $P_{n-1}$. All the results together enable us to show that $$|\End(P_m)|\leq|\End(T_m)|\leq|\End(S_m)|,$$ where $\End(T_m)$ is the set of all endomorphisms of $T_m$ (homomorphisms from $T_m$ to itself).