There are five simple cubic graphs of order 8 (listed here) and there are 6 connected graphs of order 4 (listed here). But before we get started, I have a conjecture.

Let be a simple graph on n1 vertices, a simple graph on n2 vertices, and assume there is a harmonic morphism . Call an n1-tuple of “colors” a *harmonic color list* (HCL) if it’s attached to a harmonic morphism in the usual way (the ith coordinate is j if sends vertex i of to vertex j of ). Let S be the set of all such HCLs. The automorphism group of acts on S (by permuting coordinates associated to the vertices of , as does the automorphism group of (by permuting the “colors” associated to the vertices of ). These actions commute. Clearly S decomposes as a disjoint union of distinct orbits. The **conjecture** is that there is only one such orbit.

Onto the topic of the post! The 6 connected graphs of order 4 are called P4 (the path graph), D3 (the star graph, also ), C4 (the cycle graph), K4 (the complete graph), Paw (C3 with a “tail”), and Diamond (K4 but missing an edge). All these terms are used on graphclasses.org. The results below were obtained using SageMath.

- We start with the graph listed 1st on wikipedia’s Table of simple cubic graphs and defined using the sage code
`sage: Gamma1 = graphs.LCFGraph(8, [2, 2, -2, -2], 2)`

. This graph has diameter 3, girth 3, and its automorphism group G is generated by (5,6), (1,2), (0,3)(4,7), (0,4)(1,5)(2,6)(3,7), . This graph is not vertex transitive. Its characteristic polynomial is . Its edge connectivity and vertex connectivity are both 2. This graph has*no non-trivial harmonic morphisms*to D3 or P4 or C4 or Paw. However, there are 48 non-trivial harmonic morphisms to . For example,

(the automorphism group of K4, ie the symmetric group of degree 4, acts on the colors {0,1,2,3} and produces 24 total plots), and (again, the automorphism group of K4, ie the symmetric group of degree 4, acts on the colors {0,1,2,3} and produces 24 total plots). There are 8 non-trivial harmonic morphisms to . For example, and Here the automorphism group of K4, ie the symmetric group of degree 4, acts on the colors {0,1,2,3}, while the automorphism group of the graph acts by permuting some of the coordinates, for example, it can swap the 5th and 6th coordinates.Next, we take for the graph listed 2nd on wikipedia’s Table of simple cubic graphs and defined using the sage code`sage: Gamma1 = graphs.LCFGraph(8, [4, -2, 4, 2], 2)`

. This graph has diameter 3, girth 3, and its automorphism group G is generated by (1,7)(2,6)(3,5), (0,4)(1,3)(5,7), (obviously too small to act transitively on the vertices). Its characteristic polynomial is , its edge connectivity and vertex connectivity are both 3. This graph has*no non-trivial harmonic morphisms*to D3 or P4 or C4 or Paw or K4. However, it has 4 non-trivial harmonic morphisms to Diamond. They are:

Let denote the graph listed 3rd on wikipedia’s Table of simple cubic graphs and defined using the sage code`sage: Gamma1 = graphs.LCFGraph(8, [2, 4, -2, 3, 3, 4, -3, -3], 1)`

. This graph has diameter 2, girth 3, and its automorphism group G is generated by (4,6), (1,2)(3,5), (0,1)(5,7), . It does not act transitively on the vertices. Its characteristic polynomial is and its edge connectivity and vertex connectivity are both 3.

This graph has*no non-trivial harmonic morphisms*to P4 or C4 or Paw or K4 or Diamond. However, it has 6 non-trivial harmonic morphisms to D3, for example,

The automorphism group of D3 (the symmetric group of degree 3) acts by permuting the colors {0,1,2,3} and so yields a total of 6=3! such harmonic color plots.Let denote the graph listed 4th on wikipedia’s Table of simple cubic graphs and defined using the sage code`sage: Gamma1 = graphs.LCFGraph(8, [4, -3, 3, 4], 2)`

. This example is especially interesting. Otherwise known as the “cube graph” , this graph has diameter 3, girth 4, and its automorphism group G is generated by ((2,4)(5,7), (1,7)(4,6), (0,1,4,5)(2,3,6,7), . It is vertex transitive. Its characteristic polynomial is and its edge connectivity and vertex connectivity are both 3.

This graph has*no non-trivial harmonic morphisms*to D3 or P4 or Paw. However, it has 24 non-trivial harmonic morphisms to C4, 24 non-trivial harmonic morphisms to K4, and 24 non-trivial harmonic morphisms to Diamond. An example of a non-trivial harmonic morphism to K4:

A few examples of a non-trivial harmonic morphism to Diamond:

and

A few examples of a non-trivial harmonic morphism to C4:

The automorphism group of C4 acts by permuting the colors {0,1,2,3} cyclically, while the automorphism group G acts by permuting coordinates. These yield more harmonic color plots.