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).Comment: 47 pages, 15 figure

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 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, where λ is the largest eigenvalue of the adjacency matrix of G and H λ (G) is a certain constant depending only on G which we call the spectral entropy of G. We also show that if T m is any fixed tree and 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 . the electronic journal of combinatorics 21(4) (2014), #P4.9 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

Homomorphism Tensors and Linear Equations

Lov\'asz (1967) showed that two graphs $G$ and $H$ are isomorphic if and only if they are homomorphism indistinguishable over the class of all graphs, i.e. for every graph $F$, the number of homomorphisms from $F$ to $G$ equals the number of homomorphisms from $F$ to $H$. Recently, homomorphism indistinguishability over restricted classes of graphs such as bounded treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly powerful framework for capturing diverse equivalence relations on graphs arising from logical equivalence and algebraic equation systems. In this paper, we provide a unified algebraic framework for such results by examining the linear-algebraic and representation-theoretic structure of tensors counting homomorphisms from labelled graphs. The existence of certain linear transformations between such homomorphism tensor subspaces can be interpreted both as homomorphism indistinguishability over a graph class and as feasibility of an equational system. Following this framework, we obtain characterisations of homomorphism indistinguishability over two natural graph classes, namely trees of bounded degree and graphs of bounded pathwidth, answering a question of Dell et al. (2018).Comment: 33 pages, accepted for ICALP 202