# Harmonic morphisms to P_3 – examples

This post expands on a previous post and gives more examples of harmonic morphisms to the path graph $\Gamma_2=P_3$.

If $\Gamma_1 = (V_1, E_1)$ and $\Gamma_2 = (V_2, E_2)$ are graphs then a map $\phi:\Gamma_1\to \Gamma_2$ (that is, $\phi: V_1\cup E_1\to V_2\cup E_2$) is a morphism provided

1. if $\phi$ sends an edge to an edge then the edges vertices must also map to each other: $e=(v,w)\in E_1$ and $\phi(e)\in E_2$ then $\phi(e)$ is an edge in $\Gamma_2$ having vertices $\phi(v)\in V_2$ and $\phi(w)\in V_2$, where $\phi(v)\not= \phi(w)$, and
2. if $\phi$ sends an edge to a vertex then the edges vertices must also map to that vertex: if $e=(v,w)\in E_1$ and $\phi(e)\in V_2$ then $\phi(e) = \phi(v) = \phi(w)$.

As a non-example, if $\Gamma_1$ is a planar graph, if $\Gamma_2$ is its dual graph, and if $\phi:\Gamma_1\to\Gamma_2$ is the dual map $V_1\to E_2$ and $E_1\to V_2$, then $\phi$ is not a morphism.

Given a map $\phi_E : E_1 \rightarrow E_2 \cup V_2$, an edge $e_1$ is called horizontal if $\phi_E(e_1) \in E_2$ and is called vertical if $\phi_E(e_1) \in V_2$. We say that a graph morphism $\phi: \Gamma_1 \rightarrow \Gamma_2$ is a graph homomorphism if $\phi_E (E_1) \subset E_2$. Thus, a graph morphism is a homomorphism if it has no vertical edges.

Suppose that $\Gamma_2$ has at least one edge. Let $Star_{\Gamma_1}(v)$ denote the star subgraph centered at the vertex v. A graph morphism $\phi : \Gamma_1 \to \Gamma_2$ is called harmonic if for all vertices $v \in V(\Gamma_1)$, the quantity
$\mu_\phi(v,f)= |\phi^{-1}(f) \cap Star_{\Gamma_1}(v)|$
(the number of edges in $\Gamma_1$ adjacent to $v$ and mapping to the edge $f$ in $\Gamma_2$) is independent of the choice of edge $f$ in $Star_{\Gamma_2}(\phi(v))$.

An example of a harmonic morphism can be described in the plot below as follows: $\phi:\Gamma_1\to \Gamma_2=P_3$ sends the red vertices in $\Gamma_1$ to the red vertex of $\Gamma_2=P_3$, the green vertices in $\Gamma_1$ to the green vertex of $\Gamma_2=P_3$, and the white vertices in $\Gamma_1$ to the white vertex of $\Gamma_2=P_3$.

Example 1:

Example 2:

Example 3: