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 .