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

We only consider the *cyclic graph* on k vertices, as the domain in this post. There are no non-trivial harmonic morphisms , so we start with . We indicate a harmonic morphism by a vertex coloring. 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 (we let 3 be the numerical notation for the color red), the blue vertices in to the blue vertex of (we let 2 be the numerical notation for the color blue), the green vertices in to the green vertex of (we let 1 be the numerical notation for the color green), and the white vertices in to the white vertex of (we let 0 be the numerical notation for the color white).

To get the following data, I wrote programs in Python using SageMath.

**Example 1**: There are only the 4 trivial harmonic morphisms , plus that induced by and all of its cyclic permutations (4+6=10). This set of 6 permutations is closed under the automorphism of induced by the transposition (0,3)(1,2) (so total = 10).

**Example 2**: There are only the 4 trivial harmonic morphisms, plus and all of its cyclic permutations (4+7=11). This set of 7 permutations is not closed under the automorphism of induced by the transposition (0,3)(1,2), so one also has and all 7 of its cyclic permutations (total = 7+11 = 18).

**Example 3**: There are only the 4 trivial harmonic morphisms, plus and all of its cyclic permutations (4+8=12). This set of 8 permutations is not closed under the automorphism of induced by the transposition (0,3)(1,2), so one also has and all of its cyclic permutations (12+8=20). In addition, there is and all of its cyclic permutations (20+8 = 28). The latter set of 8 cyclic permutations of is closed under the transposition (0,3)(1,2) (total = 28).

**Example 4**: There are only the 4 trivial harmonic morphisms, plus and all of its cyclic permutations (4+9=13). This set of 9 permutations is not closed under the automorphism of induced by the transposition (0,3)(1,2), so one also has and all 9 of its cyclic permutations (9+13 = 22). This set of 9 permutations is not closed under the automorphism of induced by the transposition (0,3)(1,2), so one also has and all 9 of its cyclic permutations (9+22 = 31). This set of 9 permutations is not closed under the automorphism of induced by the transposition (0,3)(1,2), so one also has and all 9 of its cyclic permutations (total = 9+31 = 40).