May | 2020 | Yet Another Mathblog

I’m going to start off with two big caveats:

This is not Duursma‘s definition, it’s mine.
I’m not convinced (yet?) that it’s a useful idea to examine such a zeta function.

So that’s your warning – you may be wasting your time reading this!

The Duursma zeta function of a linear block (error-correcting) code is due to Iwan Duursma and is a fascinating object of study. (More precisely, it’s defined for “formal” linear block codes, ie, defined via a weight enumerator polynomial with a suitable MacWilliams-like functional equation.) Sometimes it satisfies an analog of the Riemann hypothesis and sometimes it doesn’t. And sometimes that analog is still an open question.

Duursma zeta function of a code

Before we define the Duursma zeta function of a graph, we introduce the Duursma zeta function of a code.

Let $C$ be an $[n,k,d]_q$ code, ie a linear code over $GF(q)$ of length $n$ , dimension $k$ , and minimum distance $d$ . In general, if $C$ is an $[n,k,d]_q$ -code then we use $[n,k^\perp,d^\perp]_q$ for the parameters of the dual code, $C^\perp$ . It is a consequence of Singleton’s bound that $n+2-d-d^\perp\geq$ , with equality when $C$ is an MDS code. Motivated by analogies with local class field theory, in [Du] Iwan Duursma introduced the (Duursma) zeta function $\zeta=\zeta_C$ :

$\zeta_C(T)=\frac{P(T)}{(1-T)(1-qT)},$
where $P(T)=P_C(T)$ is a polynomial of degree $n+2-d-d^\perp$ , called the zeta polynomial of $C$ . We next sketch the definition of the zeta polynomial.

If $C^\perp$ denotes the dual code of $C$ , with parameters $[n,n-k,d^\perp]$ then the MacWilliams identity relates the weight enumerator $A_{C^\perp}$ of $C^\perp$ to the weight enumerator $A_{C}$ of $C$ :

$A_{C^\perp}(x,y) = |C|^{-1}A_C(x+(q-1)y,x-y).$
A polynomial $P(T)=P_C(T)$ for which

$\frac{(xT+(1-T)y)^n}{(1-T)(1-qT)}P(T)=\dots +\frac{A_C(x,y)-x^n}{q-1}T^{n-d}+\dots \ .$
is a (Duursma) zeta polynomial of $C$ .

Theorem (Duursma): If $C$ be an $[n,k,d]_q$ code with $d\geq 2$ and $d^\perp\geq 2$ then the Duursma zeta polynomial $P=P_C$ exists and is unique.

See the papers of Duursma for interesting properties and conjectures.

Duursma zeta function of a graph

Let $\Gamma=(V,E)$ be a graph having $|V(\Gamma)|$ vertices and $|E(\Gamma)|$ edges. We define the zeta function of $\Gamma$ via the Duursma zeta function of the binary linear code defined by the cycle space of $\Gamma$ .

Theorem (see [DKR], [HB], [JV]): The binary code $B=B_\Gamma$ generated by the rows of the incidence matrix of $\Gamma$ is the cocycle space of $\Gamma$ over $GF(2)$ , and the dual code $B^\perp$ is the cycle space $Z=Z_\Gamma$ of $\Gamma$ :

$B_\Gamma^\perp = Z_\Gamma.$
Moreover,
(a) the length of $B_\Gamma$ is $|E|$ , dimension of $B_\Gamma$ is $|V|-1$ , and the minimum distance of $B_\Gamma$ is the edge-connectivity of $\Gamma$ ,
(b) length of $Z_\Gamma$ is $|E|$ , dimension of $Z_\Gamma$ is $|E|-|V|+1$ , and the minimum distance of $Z_\Gamma$ is the girth of $\Gamma$ .

Call $Z_\Gamma$ the cycle code and $B_\Gamma$ the cocycle code.

Finally, we can introduce the (Duursma) zeta function $\Gamma$ :

$\zeta_\Gamma(T)=\zeta_{Z_\Gamma} =\frac{P(T)}{(1-T)(1-qT)},$
where $P=P_\Gamma=P_{Z_\Gamma}$ is the Duursma polynomial of $\Gamma$ .

Example: Using SageMath, when $\Gamma = W_5$ , the wheel graph on 5 vertices, we have

$P_\Gamma(T) = \frac{2}{7}T^4 + \frac{2}{7}T^3 + \frac{3}{14}T^2 + \frac{1}{7}T + \frac{1}{14}.$
All its zeros are of absolute value $1/\sqrt{2}$ .

References

[Du] I. Duursma, Combinatorics of the two-variable zeta function, in Finite fields and applications, 109–136, Lecture Notes in Comput. Sci., 2948, Springer, Berlin, 2004.

[DKR] P. Dankelmann, J. Key, B. Rodrigues, Codes from incidence matrices of graphs, Designs, Codes and Cryptography 68 (2013) 373-393.

[HB] S. Hakimi and J. Bredeson, Graph theoretic error-correcting codes, IEEE Trans. Info. Theory 14(1968)584-591.

[JV] D. Jungnickel and S. Vanstone, Graphical codes revisited, IEEE Trans. Info. Theory 43(1997)136-146.

This post expands on a previous post and gives more examples of harmonic morphisms to the tree $\Gamma_2=D_3$ . This graph is also called a star graph $Star_3$ on 3+1=4 vertices, or the bipartite graph $K_{1,3}$ .
D3-0123

We indicate a harmonic morphism by a vertex coloring. An example of a harmonic morphism can be described in the plot below as follows: $\phi:\Gamma_1\to \Gamma_2=D_3$ sends the red vertices in $\Gamma_1$ to the red vertex of $\Gamma_2=D_3$ (we let 3 be the numerical notation for the color red), the blue vertices in $\Gamma_1$ to the blue vertex of $\Gamma_2=D_3$ (we let 2 be the numerical notation for the color blue), the green vertices in $\Gamma_1$ to the green vertex of $\Gamma_2=D_3$ (we let 1 be the numerical notation for the color green), and the white vertices in $\Gamma_1$ to the white vertex of $\Gamma_2=D_3$ (we let 0 be the numerical notation for the color white).

First, a simple remark about harmonic morphisms in general: roughly speaking, they preserve adjacency. Suppose $\phi:\Gamma_1\to \Gamma_2$ is a harmonic morphism. Let $v,w\in V_1$ be adjacent vertices of $\Gamma_1$ . Then either (a) $\phi(v)=\phi(w)$ and $\phi$ “collapses” the edge (vertical) $(v,w)$ or (b) $\phi(v)\not= \phi(w)$ and the vertices $\phi(v)$ and $\phi(w)$ are adjacent in $\Gamma_2$ . In the particular case of this post (ie, the case of $\Gamma_2=D_3$ ), this remark has the following consequence: since in $D_3$ the green vertex is not adjacent to the blue or red vertex, none of the harmonic colored graphs below can have a green vertex adjacent to a blue or red vertex. In fact, any colored vertex can only be connected to a white vertex or a vertex of like color.

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

Example 1: There are only the 4 trivial harmonic morphisms $Star_4 \to D_3$ , plus the “obvious” ones obtained from that below and those induced by permutations of the vertices:
Star4-D3-00321 .

My guess is that the harmonic morphisms $Star_5\to D_3$ can be described in a similar manner. Likewise for the higher $Star_n$ graphs. Given a star graph $\Gamma$ with a harmonic morphism to $D_3$ , a leaf (connected to the center vertex 0) can be added to $\Gamma$ and preserve “harmonicity” if its degree 1 vertex is colored 0. You can try to “collapse” such leafs, without ruining the harmonicity property.

Example 2: For graphs like $\Gamma_1=$
C3LeafLeafLeaf-D3-000321
there are only the 4 trivial harmonic morphisms $\Gamma_1 \to D_3$ , plus the “obvious” ones obtained from that above and those induced by permutations of the vertices with a non-zero color.

Example 2.5: Likewise, for graphs like $\Gamma_1=$
3C3-D3-0332211
there are only the 4 trivial harmonic morphisms $\Gamma_1 \to D_3$ , plus the “obvious” ones obtained from that above and those induced by permutations of the vertices with a non-zero color.

Example 3: This is really a non-example. There are no harmonic morphisms from the (3-dimensional) cube graph (whose vertices are those of the unit cube) to $D_3$ .
More generally, take two copies of a cyclic graph on n vertices, $C_n$ , one hovering over the other. Now, connect each vertex of the top copy to the corresponding vertex of the bottom copy. This is a cubic graph that can be visualized as a “thick” regular polygon. (The cube graph is the case $n=4$ .) I conjecture that there is no harmonic morphism from such a graph to $D_3$ .

Example 4: There are 30 non-trivial harmonic morphisms $\Gamma_1 \to D_3$ for the Peterson graph (the last of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page). Here is an example:
petersen-D3-0330120021
Another interesting fact is that this graph has an automorphism group (isomorphic to the symmetric group on 5 letters) which acts transitively on the vertices.

Example 5: There are 12 non-trivial harmonic morphisms $\Gamma_1=K_{3,3} \to D_3$ for the complete bipartite (“utility”) graph $K_{3,3}$ . They are all obtained from either
K_3_3-D3-321000
or
K_3_3-D3-000231
by permutations of the vertices with a non-zero color (3!+3!=12).

Example 6: There are 6 non-trivial harmonic morphisms $\Gamma_1 \to D_3$ for the cubic graph $\Gamma_1=(V,E)$ , where $V=\{0,1,\dots, 9\}$ and $E = \{(0, 3), (0, 4), (0, 6), (1, 2), (1, 5), (1, 9), (2, 3), (2, 7), (3, 6), (4, 5), (4, 9), (5, 8), (6, 7), (7, 8), (8, 9)\}$ . This graph has diameter 3, girth 3, and edge-connectivity 3. It’s automorphism group is size 4, generated by (5,9) and (1,8)(2,7)(3,6). The harmonic morphisms are all obtained from
random3regular10e-D3-1011031102
by permutations of the vertices with a non-zero color (3!=6). This graph might be hard to visualize but it is isomorphic to the simple cubic graph having LCF notation [−4, 3, 3, 5, −3, −3, 4, 2, 5, −2]:
random3regular10e2
which has a nice picture. This is the ninth of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page.

Example 7: (a) The first of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10f
This graph has diameter 5, automorphism group generated by (7,8), (6,9), (3,4), (2,5), (0,1)(2,6)(3,7)(4,8)(5,9). There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .
(b) The second of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10g
This graph has diameter 4, girth 3, automorphism group generated by (7,8), (0,5)(1,2)(6,9). There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .
(c) The third of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10h
This graph has diameter 3, girth 3, automorphism group generated by (4,5), (0,1)(8,9), (0,8)(1,9)(2,7)(3,6). There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .

Example 8: The fourth of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10i
This graph has diameter 3, girth 3, automorphism group generated by (4,6), (3,5), (1,8)(2,7)(3,4)(5,6), (0,9). There are 12 non-trivial harmonic morphisms $\Gamma_1\to D_3$ . For example,
3regular10i-D3-2220301022
and the remaining (3!=6 total) colorings obtained by permutating the non-zero colors. Another example is
3regular10i-D3-1103020111
and the remaining (3!=6 total) colorings obtained by permutating the non-zero colors.

Example 9: (a) The fifth of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10j
Its SageMath command is Gamma1 = graphs.LCFGraph(10,[2,2,-2,-2,5],2) There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .
(b) The sixth of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10k
Its SageMath command is Gamma1 = graphs.LCFGraph(10,[2,3,-2,5,-3],2) There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .

Example 10: The seventh of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10l-D3-3330222010
Its SageMath command is Gamma1 = graphs.LCFGraph(10,[2,3,-2,5,-3],2). Its automorphism group is order 12, generated by (1,2)(3,7)(4,6), (0,1)(5,6)(7,9), (0,4)(1,6)(2,5)(3,9). There are 6 non-trivial harmonic morphisms $\Gamma_1\to D_3$ , each obtained from the one above by permuting the non-zero colors.

Example 11: The eighth of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10m
Its SageMath command is Gamma1 = graphs.LCFGraph(10,[5, 3, 5, -4, -3, 5, 2, 5, -2, 4],1). Its automorphism group is order 2, generated by (0,3)(1,4)(2,5)(6,7). There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .

Example 12: (a) The tenth (recall the 9th was mentioned above) of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10o
Its SageMath command is Gamma1 = graphs.LCFGraph(10,[3, -3, 5, -3, 2, 4, -2, 5, 3, -4],1). Its automorphism group is order 6, generated by (2,8)(3,9)(4,5), (0,2)(5,6)(7,9). There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .
(b) The 11th of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10p
Its SageMath command is Gamma1 = graphs.LCFGraph(10,[-4, 4, 2, 5, -2],2). Its automorphism group is order 4, generated by (0,1)(2,9)(3,8)(4,7)(5,6), (0,5)(1,6)(2,7)(3,8)(4,9). There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .
(c) The 12th of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10q
Its SageMath command is Gamma1 = graphs.LCFGraph(10,[5, -2, 2, 4, -2, 5, 2, -4, -2, 2],1). Its automorphism group is order 6, generated by (1,9)(2,8)(3,7)(4,6), (0,4,6)(1,3,8)(2,7,9). There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .
(d) The 13th of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10r
Its SageMath command is Gamma1 = graphs.LCFGraph(10,[2, 5, -2, 5, 5],2). Its automorphism group is order 8, generated by (4,8)(5,7), (0,2)(3,9), (0,5)(1,6)(2,7)(3,8)(4,9). There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .

Example 13: The 14th of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10s-D3-2033020110
By permuting the non-zero colors, we obtain 3!=6 harmonic morphisms from this one. Another harmonic morphism $\Gamma_1\to D_3$ is depicted as:
3regular10s-D3-0302222201
By permuting the non-zero colors, we obtain 3!=6 harmonic morphisms from this one. And another harmonic morphism $\Gamma_1\to D_3$ is depicted as:
3regular10s-D3-1110302011
By permuting the non-zero colors, we obtain 3!=6 harmonic morphisms from this one. Its SageMath command is Gamma1 = graphs.LCFGraph(10,[5, -3, -3, 3, 3],2). Its automorphism group is order 48, generated by (4,6), (2,8)(3,7), (1,9), (0,2)(3,5), (0,3)(1,4)(2,5)(6,9)(7,8). There are a total of 18=3!+3!+3! non-trivial harmonic morphisms $\Gamma_1\to D_3$ .

Example 14: The 15th of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10t-D3-2033020110
By permuting the non-zero colors, we obtain 3!=6 harmonic morphisms from this one. Its SageMath command is Gamma1 = graphs.LCFGraph(10,[5, -4, 4, -4, 4],2). Its automorphism group is order 8, generated by (2,7)(3,8), (1,9)(2,3)(4,6)(7,8), (0,5)(1,4)(2,3)(6,9)(7,8). There are a total of 6=3! non-trivial harmonic morphisms $\Gamma_1\to D_3$ .

Example 15: (a) The 16th of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10u
Its SageMath command is Gamma1 = graphs.LCFGraph(10,[5, -4, 4, 5, 5],2). Its automorphism group is order 4, generated by (0,3)(1,2)(4,9)(5,8)(6,7), (0,5)(1,6)(2,7)(3,8)(4,9). There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .
(b) The 17th of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10v
Its SageMath command is Gamma1 = graphs.LCFGraph(10,[5, 5, -3, 5, 3],2). Its automorphism group is order 20, generated by (2,6)(3,7)(4,8)(5,9), (0,1)(2,5)(3,4)(6,9)(7,8), (0,2)(1,9)(3,5)(6,8). This group acts transitively on the vertices. There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .
(c) The 18th of the 19 simple cubic graphs on 10 vertices listed on this wikipedia page is the graph $\Gamma_1$ depicted as:
3regular10w
This is an example of a “thick polygon” graph, already mentioned in Example 3 above. Its SageMath command is Gamma1 = graphs.LCFGraph(10,[-4, 4, -3, 5, 3],2). Its automorphism group is order 20, generated by (2,5)(3,4)(6,9)(7,8), (0,1)(2,6)(3,7)(4,8)(5,9), (0,2)(1,9)(3,6)(4,7)(5,8). This group acts transitively on the vertices. There are no non-trivial harmonic morphisms $\Gamma_1\to D_3$ .
(d) The 19th in the list of 19 is the Petersen graph, already in Example 4 above.

We now consider some examples of the cubic graphs having 12 vertices. According to the House of Graphs there are 109 of these, but we use the list on this wikipedia page.

Example 16: I wrote a SageMath program that looked for harmonic morphisms on a case-by-case basis. If there is no harmonic morphism $\Gamma_1\to D_3$ then, instead of showing a graph, I’ll list the edges (of course, the vertices are 0,1,…,11) and the SageMath command for it.

$\Gamma_1=(V_1,E_1)$ , where $E_1=\{ (0, 1), (0, 2), (0, 11), (1, 2), (1, 6), (2, 3), (3, 4), (3, 5), (4, 5), (4, 6), (5, 6), (7, 8), (7, 9), (7, 11), (8, 9), (8, 10), (9, 10), (10, 11)\}$ .
SageMath command:
V1 = [0,1,2,3,4,5,6,7,8,9,10,11] E1 = [(0, 1), (0, 2), (0, 11), (1, 2), (1, 6), (2, 3), (3, 4), (3, 5), (4, 5), (4, 6), (5, 6), (7, 8), (7, 9), (7, 11), (8, 9), (8, 10), (9, 10), (10, 11)] Gamma1 = Graph([V1,E1])
(Not in LCF notation since it doesn’t have a Hamiltonian cycle.)
$\Gamma_1=(V_1,E_1)$ , where $E_1=\{(0, 1), (0, 6), (0, 11), (1, 2), (1, 3), (2, 3), (2, 5), (3, 4), (4, 5), (4, 6), (5, 6), (7, 8), (7, 9), (7, 11), (8, 9), (8, 10), (9, 10), (10, 11)\}$ .
SageMath command:
V1 = [0,1,2,3,4,5,6,7,8,9,10,11] E1 = [(0, 1), (0, 6), (0, 11), (1, 2), (1, 3), (2, 3), (2, 5), (3, 4), (4, 5), (4, 6), (5, 6), (7, 8), (7, 9), (7, 11), (8, 9), (8, 10), (9, 10), (10, 11)] Gamma1 = Graph([V1,E1])
(Not in LCF notation since it doesn’t have a Hamiltonian cycle.)
$\Gamma_1=(V_1,E_1)$ , where $E_1=\{(0,1),(0,3),(0,11),(1,2),(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(10,11)\}$ .
SageMath command:
V1 = [0,1,2,3,4,5,6,7,8,9,10,11] E1 = [(0,1),(0,3),(0,11),(1,2),(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(10,11)] Gamma1 = Graph([V1,E1])
(Not in LCF notation since it doesn’t have a Hamiltonian cycle.)
This example has 12 non-trivial harmonic morphisms.
SageMath command:
V1 = [0,1,2,3,4,5,6,7,8,9,10,11] E1 = [(0,1),(0,3),(0,11),(1,2),(1,6),(2,3),(2,5),(3,4),(4,5),(4,6),(5,6),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(10,11)] Gamma1 = Graph([V1,E1])
(Not in LCF notation since it doesn’t have a Hamiltonian cycle.) We show two such morphisms:

The other non-trivial harmonic morphisms are obtained by permuting the non-zero colors. There are 3!=6 for each graph above, so the total number of harmonic morphisms (including the trivial ones) is 6+6+4=16.
$\Gamma_1=(V_1,E_1)$ , where $E_1=\{(0, 1), (0, 3), (0, 11), (1, 2), (1, 11), (2, 3), (2, 10), (3, 4), (4, 5), (4, 8), (5, 6), (5, 7), (6, 7), (6, 9), (7, 8), (8, 9), (9, 10), (10, 11)\}$ .
SageMath command:
Gamma1 = graphs.LCFGraph(12, [3, -2, -4, -3, 4, 2], 2)
This example has 12 non-trivial harmonic morphisms. $\Gamma_1=(V_1,E_1)$ , where $E_1=\{(0, 1), (0, 3), (0, 11), (1, 2), (1, 11), (2, 3), (2, 10), (3, 4), (4, 5), (4, 7), (5, 6), (5, 8), (6, 7), (6, 9), (7, 8), (8, 9), (9, 10), (10, 11)\}$ . (This only differs by one edge from the one above.)
SageMath command:
Gamma1 = graphs.LCFGraph(12, [3, -2, -4, -3, 3, 3, 3, -3, -3, -3, 4, 2], 1)
We show two such morphisms:

And here is another plot of the last colored graph:

The other non-trivial harmonic morphisms are obtained by permuting the non-zero colors. There are 3!=6 for each graph above, so the total number of harmonic morphisms (including the trivial ones) is 6+6+4=16.
$\Gamma_1=(V_1,E_1)$ , where $E_1=\{(0, 1), (0, 4), (0, 11), (1, 2), (1, 3), (2, 3), (2, 5), (3, 4), (4, 5), (5, 6), (6, 7), (6, 8), (7, 8), (7, 10), (8, 9), (9, 10), (9, 11), (10, 11)\}$ .
SageMath command:
Gamma1 = graphs.LCFGraph(12, [4, 2, 3, -2, -4, -3, 2, 3, -2, 2, -3, -2], 1)
This example has 48 non-trivial harmonic morphisms. $\Gamma_1=(V_1,E_1)$ , where $E_1=\{(0, 1), (0, 3), (0, 11), (1, 2), (1, 4), (2, 3), (2, 5), (3, 4), (4, 5), (5, 6), (6, 7), (6, 9), (7, 8), (7, 10), (8, 9), (8, 11), (9, 10), (10, 11)\}$ .
SageMath command:
Gamma1 = graphs.LCFGraph(12, [3, 3, 3, -3, -3, -3], 2)
This example is also interesting as it has a large number of automorphisms – its automorphism group is size 64, generated by (8,10), (7,9), (2,4), (1,3), (0,5)(1,2)(3,4)(6,11)(7,8)(9,10), (0,6)(1,7)(2,8)(3,9)(4,10)(5,11). Here are examples of some of the harmonic morphisms using vertex-colored graphs:

I think all the other non-trivial harmonic morphisms are obtained by (a) permuting the non-zero colors, or (b) applying a element of the automorphism group of the graph.
(list under construction)