This post expands on a previous post and gives more examples of harmonic morphisms to the path graph .

If and are graphs then a map (that is, ) is a **morphism** provided

- if sends an edge to an edge then the edges vertices must also map to each other: and then is an edge in having vertices and , where , and
- if sends an edge to a vertex then the edges vertices must also map to that vertex: if and then .

As a *non-example*, if is a planar graph, if is its dual graph, and if is the dual map and , then is *not a morphism*.

Given a map , an edge is called **horizontal** if and is called **vertical** if . We say that a graph morphism is a **graph homomorphism** if . Thus, a graph morphism is a homomorphism if it has no vertical edges.

Suppose that has at least one edge. Let denote the star subgraph centered at the vertex v. A graph morphism is called **harmonic** if for all vertices , the quantity

(the number of edges in adjacent to and mapping to the edge in ) is independent of the choice of edge in .

An example of a harmonic morphism can be described in the plot below as follows: sends the red vertices in to the red vertex of , the green vertices in to the green vertex of , and the white vertices in to the white vertex of .

Example 1:

Example 2:

Example 3: