In the works: a book “Exploring Graphs via Harmonic Morphisms”

Posted on 2025/10/07 by wdjoyner

Caroline Melles and I have been working for some years on a 2-volume book in graph theory which investigates harmonic morphisms. These are, roughly speaking, mappings from one graph to another that preserve locally harmonic functions on these graphs. Therefore, this topic fits into the general framework of harmonic analysis on graphs.

This post only concerns the first volume. The intent here is to mention some of the types of results we obtain. Of course, by no means is it intended to be a complete description.

The second volume will be summarized in a separate post.

Graphs in our book are unweighted and, unless stated otherwise, have no loops or multiple edges. The basic idea is this: in chapter 2 we classify harmonic morphisms using a criteria expressed as a matrix identity. For various graph-theoretical constructions (such as edge deletion or join or a graph product or …) that can be performed on a given graph Gamma, we pick a graph morphism associated to the construction (such as sending a vertex in the constructed graph to the given graph). That morphism is associated to a matrix (which we called the vertex map matrix in chapter 3 of our earlier book, Adventures in Graph Theory). When this matrix satisfies the above-mentioned matrix criteria then the associated morphism is harmonic.

Chapter 1 is on Graph Morphisms.

This chapter is devoted to background on graph morphisms and some of the methods we use to study them.

Roughly speaking, a morphism is a mapping between graphs that preserves incidence structure. After defining horizontal and vertical edges, vertical multiplicities, local horizontal multiplicities, it recalls well-known graph families like cycle graphs, path graphs, and complete graphs.
There are a few very useful degree identities. First, there is a fundamental formula relating vertex degrees to multiplicities under morphisms. There is also a formula for the degree
of the morphism in terms of vertical multiplicities and local horizontal multiplicities.
A topic threading through the book is that of matrix-theoretic methods. This first chapter introduces vertex map matrices and edge map matrices that encode morphisms. After establishing key matrix identities and products, reviews adjacency matrices and their spectra, with detailed analysis of cycle graph eigenvalues using Chebyshev polynomials and complex roots of unity.
It recalls signed and unsigned incidence matrices, with and without edge orientations, and establishes the fundamental Graph Homomorphism Identity relating incidence matrices to morphism matrices,
introduces Laplacian matrices as differences of degree and adjacency matrices, connecting to the incidence matrix framework.
Introduced graph blowup morphisms via a blowup construction where vertices are replaced by independent sets, creating natural homomorphisms with specific structural properties.
Some functorial properties of graph morphisms are established, such as how morphisms behave under graph constructions like subdivisions, smoothing, deletions, and substitutions.
The chapter ends with exercises and a chapter summary.

Chapter 2 on Harmonic Morphisms

This chapter is devoted to the basics of harmonic morphisms.

Introduces the core definition: a graph morphism is harmonic if local horizontal multiplicities are constant across edges incident to each vertex’s image.
Cycle space and cocycle space – Develops the algebraic framework using homology and cohomology of graphs. Covers Urakawa’s theorem on pullbacks of harmonic 1-forms and Baker-Norin results on divisors and Jacobians.
Matrix-theoretic methods – Establishes the fundamental matrix characterization: a morphism is harmonic iff there exists a diagonal multiplicity matrix satisfying specific adjacency matrix identities. Proves equivalence with an analogous Laplacian matrix identity and an analogous incidence matrix criteria.
The Riemann-Hurwitz formula – Presents the graph-theoretic analogue relating genera of graphs via harmonic morphisms, with matrix proof and applications to regular graphs.
Some functorial consequences – Demonstrates how harmonic morphisms interact with graph constructions like subdivision, edge substitution, leaf addition, and deletion. Shows these
operations preserve harmonicity under appropriate conditions.
The chapter ends with exercises and a chapter summary.

All harmonic morphisms from this graph to C4 are covers.

Fundamental Problem: Given a graph Gamma1, for which graphs Gamma2 is there a non-trivial harmonic morphism phi from Gamma2 to Gamma1?

Follow-up question: Can the number of such phi be counted?

Chapter 3 on Counting Problems

This chapter looks at various families, such as the path graphs. What is especially remarkable is that, as we will see, the problem of counting harmonic morphisms often boils down to solving certain recurrance relations, some of which arose (in a completely different context of course) in
the work of medieval mathematicians, both in Europe and in India.

Regarding harmonic morphisms between path graphs, we show how to construct and count the harmonic morphisms from longer path graphs to shorter ones.
Regarding harmonic morphisms between cycle graphs, we show how to construct and count the harmonic morphisms from larger cycle graphs (when they exist) to smaller ones. It turns out all such harmonic morphisms are necessarily covers.
Regarding harmonic morphisms between complete graphs, we show how to construct and count the harmonic morphisms from larger complete graphs (when they exist) to smaller ones.
Harmonic morphisms to P2 (arising from the Baker-Norin Theorem) can be counted.
Harmonic morphisms to P3 (the path graph with only 3 vertices) can be counted in special cases.
There are lots of open questions, such as which trees have a harmonic morphism to P3.
The chapter ends with exercises and a chapter summary.

Chapter 4 on Harmonic Quotient Morphisms

This chapter studies quotient graphs arising from group actions and from vertex partitions.

Quotient graphs from group actions. Harmonic actions and transitive actions are studied separately.
Quotient graphs from paritions. Orbit partitions and equitable partitions are studied.
As a nice application of harmonic morphisms with particularly nice structural properties, we consider multicovers and blowup graphs.
The last section provides explicit formulas for the eigenvalue spectra of harmonic blowups of bipartite graphs, connecting the eigenvalues of the source and target graphs through the blowup parameters. The main result is the Godsil-McKay Theorem.
The chapter ends with exercises and a chapter summary.

Chapter 5 on Graph Morphisms and Graph Products

Roughly speaking, a graph product of Gamma1 with Gamma2 is a graph Gamma3 = (V3, E3), where V3 = V1 x V2 is the Cartesian product and there is a rule for the edges E3 based on some conditions on the vertices. The graph products considered in this book are the disjunctive, Cartesian, tensor, lexicographic, and the strong products.

The most basic questions one wants answered are these:
is the projection pr1 : Gamma1 x Gamma2 to Gamma1 harmonic, and
is the projection pr2 : Gamma1 x Gamma2 to Gamma2 harmonic?
If they do turn out to be harmonic morphisms, we also want to know the vertical and horizontal multiplicities as well. If they do not turn out to be harmonic morphisms, we also want (if possible) to establish conditions on the graphs under which the projections are harmonic.

However, we want to not only consider products of graphs but also products of morphisms.
In this case, the most basic question one wants answered is this:
Given harmonic morphisms phi : Gamma2 to Gamma1 and phi’ : Gamma2′ to Gamma1′, is the
product phi x phi’ harmonic?

For example, we show that projection morphisms from tensor products are always harmonic with explicit horizontal multiplicity formulas.
Moreover, we prove that the tensor product of harmonic morphisms (without vertical edges) yields a harmonic morphism with horizontal multiplicity matrix given by the Kronecker product of the original multiplicity matrices.
If Gamma x Gamma’ is a lexicographical product then the projection onto the first factor, pr1, is a harmonic morphism. However, the projection onto the second factor is not in general.
We establish a connection between the balanced blowup graph and a lexicographical product. One corollary of this connection is that the blowdown graph agrees with the first projection of the product, so is a harmonic morphism.

Chapter 6 on More Products and Constructions

This chapter studies graph morphisms associated to Cartesian/strong/disjunctive products of graphs as well as joins and NEPS graphs.
For example, we show that projection morphisms from Cartesian products or from strong products are always harmonic with explicit horizontal multiplicity formulas.
Roughly speaking, one of the results states:
Given two m-quasi-multicovers phi from Gamma2 to Gamma1 and phi’ from Gamma2′ to Gamma1′, the Cartesian product phi x phi’ is also an m-quasi-multicover (hence harmonic).
Another result, roughly speaking, states:
Given two harmonic morphisms phi from Gamma2 to Gamma1 and phi’ from Gamma2′ to Gamma1′, the
strong product phi x phi’ is also harmonic.
Can one classify the graphs for which the disjunctive product projections pr1 or pr2 are graph morphisms?
For example, we show that if phi from Gamma2 to Gamma1 and phi’ from Gamma2′ to Gamma1′ are graph morphisms, then the associated product map from Gamma2 x Gamma2′ to Gamma1 x Gamma1′ (where x is the disjunctive product) is, in general, not a graph morphism.
Given two harmonic morphisms phi from Gamma2 to Gamma1 and phi’ from Gamma2′ to Gamma1′, the join morphism phi wedge phi’ is harmonic if and only if a certain technical condition is true.
A theorem due to Urakawa states that projection morphisms from a NEPS graph to one
of its factors are always harmonic. Moreover, we give explicit horizontal multiplicity formulas.
The chapter ends with exercises and a chapter summary.

Computations are supported throughout using SageMath and Mathematica. The plan is the publish the volume with Birkhauser. We thank the editors there, especially John Benedetto, for their encouragement and guidance.

A simple trace formula for graphs

Posted on 2023/09/09 by wdjoyner

Let $\Gamma=(V,E)$ be a simple, connected graph with vertices $V={0,1,\dots, n-1}$ and $n\times n$ adjacency matrix $A$ . We start with the geometric series identity

$\frac{1}{I-tA} = \sum_{\ell=0}^\infty t^\ell A^\ell,$
where $I=I_n$ is the $n\times n$ identity matrix. Let $P$ denote the orthonormal matrix of normalized eigenvectors, so that

$PAP^{-1} = D_\Gamma$ , $D_\Gamma = {diag}(\lambda_1,\dots,\lambda_n),$
where diag(…) denotes the diagonal matrix with the given entries on the diagonal. Let the multi-set

$Spec(\Gamma)={\lambda_0,\lambda_1\dots,\lambda_{n-1}}$
denote the spectrum of $\Gamma$ .

We can conjugate the above equation by $P$ to write

$\frac{1}{I-tD_\Gamma}= P\cdot [\sum_{\ell=0}^\infty t^\ell A^\ell]\cdot P^{-1}.$
Taking the trace of each side gives

$\sum_{j=0}^{n-1} \frac{1}{I-t\lambda_j} = \sum_{\ell=0}^\infty t^\ell tr(A^\ell).$
If $\Gamma$ has no eigenvalues equal to $0$ (i.e., $A$ is non-singular) then we may also write this as

$\sum_{j=0}^{n-1} \frac{\lambda_j^{-1}}{\lambda_j^{-1}-t} = \sum_{\ell=0}^\infty t^\ell tr(A^\ell).$

If we multiply both sides of the above equation by a fixed $f\in C_c^\infty({\mathbb{R}})$
and integrate over $t$ in ${\mathbb{R}}$ , we get,

$\sum_{j=0}^{n-1} \lambda_j^{-1}H(f)(\lambda_j^{-1}) = {\frac{1}{\pi}}\sum_{\ell=0}^\infty tr(A^\ell) [M(f)(\ell+1)+(-1)^\ell M(f^*)(\ell+1)],$
where $H$ denotes the Hilbert transform

$H(f)(z) = \frac{1}{\pi}P.V.\int_{-\infty}^\infty \frac{f(t)}{z-t}\, dt,$ and $M$ is the Mellin transform

$M(f)(z) = \int_{0}^\infty t^{z-1}f(t)\, dt,$
and where $f^*$ denotes the negation, $f^*(t)=f(-t)$ . Of course, if $f$ is even then $M(f)(\ell+1) = M(f^*)(\ell+1)$ , for all $\ell$ .

Note that $tr(A^\ell)$ can be expressed in terms of the number of walks on the graph: If $\Gamma$ is a connected graph and $W_\ell=W_\ell(\Gamma)$ denotes the total number of walks of length $\ell$ on $\Gamma$ then

$W_\ell = {\rm tr}(A^\ell)=\sum_{\lambda\in Spec(\Gamma)} \lambda^\ell.$

Harmonic quotients of regular graphs – examples

Posted on 2023/01/06 by wdjoyner

Caroline Melles and I have written a preprint that collects numerous examples of harmonic quotient morphisms $\phi : \Gamma_2 \to \Gamma_1$ , where $\Gamma_1=\Gamma_2/G$ is a quotient graph obtained from some subgroup $G \subset Aut(\Gamma_2)$ . The examples are for graphs having a small number of vertices (no more than 12). For the most part, we also focused on regular graphs with small degree (no more than 5). They were all computed using SageMath and a module of special purpose Python functions I’ve written (available on request). I’ve not counted, but the number of examples is relatively large, maybe over one hundred.

I’ll post it to the math arxiv at some point but if you are interested now, here’s a copy: click here for pdf.

A graph_id function in SageMath

Posted on 2022/09/07 by wdjoyner

While GAP has a group_id function which locates a “small” group in a small groups database (see the SageMath page or the GAP page for more info), AFAIK, SageMath doesn’t have something similar. I’ve written one (see below) based on the mountain of hard work done years ago by Emily Kirkman.

def graph_id_graph6(Gamma, verbose=False):
    """
    Returns the graph6 id of Gamma = (V,E).
    If verbose then it also displays the table of all graphs with
    |V| vertices and |E| edges.

    Assumes Gamma is a "small" graph.

    EXAMPLES:
    sage: Gamma = graphs.HouseGraph()
    sage: graph_id_graph6(Gamma, verbose=False)
    'Dbk'
    sage: graph_id_graph6(Gamma, verbose=True)

     graphs with 5 vertices and 6 edges:

    Graph6               Aut Grp Size         Degree Sequence     
    ------------------------------------------------------------
    DB{                  2                    [1, 2, 2, 3, 4]     
    DFw                  12                   [2, 2, 2, 3, 3]     
    DJ[                  24                   [0, 3, 3, 3, 3]     
    DJk                  2                    [1, 2, 3, 3, 3]     
    DK{                  8                    [2, 2, 2, 2, 4]     
    Dbk                  2                    [2, 2, 2, 3, 3]     


    'Dbk'

"""
n = len(Gamma.vertices())
m = len(Gamma.edges())
ds = Gamma.degree_sequence()
Q = GraphQuery(display_cols=['graph6', 'aut_grp_size', 'degree_sequence'], num_edges=['=', m], num_vertices=["=", n])
for g in Q:
    if g.is_isomorphic(Gamma):
        if verbose:
            print("\n graphs with %s vertices"%n+" and %s edges:\n"%m)
            Q.show()
            print("\n")
        return g.graph6_string()

The truncated tetrahedron covers the tetrahedron

Posted on 2021/04/29 by wdjoyner

At first, you might think this is obvious – just “clip” off each corner of the tetrahedron $\Gamma_1$ to create the truncated tetrahedron $\Gamma_2$ (by essentially creating a triangle from each of these clipped corners – see below for the associated graph). Then just map each such triangle to the corresponding vertex of the tetrahedron. No, it’s not obvious because the map just described is not a covering. This post describes one way to think about how to construct any covering.

First, color the vertices of the tetrahedron in some way.

The coloring below corresponds to a harmonic morphism $\phi : \Gamma_2\to \Gamma_1$ :

All others are obtained from this by permuting the colors. They are all covers of $\Gamma_1 = K_4$ – with no vertical multiplicities and all horizontal multiplicities equal to 1. These 24 harmonic morphisms of $\Gamma_2\to\Gamma_1$ are all coverings and there are no other harmonic morphisms.

Quartic graphs with 12 vertices

Posted on 2020/10/13 by wdjoyner

This is a continuation of the post A table of small quartic graphs. As with that post, it’s modeled on the handy wikipedia page Table of simple cubic graphs.

According to SageMath computations, there are 1544 connected, 4-regular graphs. Exactly 2 of these are symmetric (ie, arc transitive), also vertex-transitive and edge-transitive. Exactly 8 of these are vertex-transitive but not edge-transitive. None are distance regular.

Example 1: The first example of such a symmetric graph is the circulant graph with parameters (12, [1,5]), depicted below. It is bipartite, has girth 4, and its automorphism group has order 768, being generated by $(9,11), (5,6), (4,8), (2,10), (1,2)(5,9)(6,11)(7,10), (1,7), (0,1)(2,5)(3,7)(4,9)(6,10)(8,11)$ .

Example 2: The second example of such a symmetric graph is the cuboctahedral graph, depicted below. It has girth 3, chromatic number 3, and its automorphism group has order 48, being generated by $(1,10)(2,7)(3,6)(4,8)(9,11), (1,11)(3,4)(6,8)(9,10), (0,1,9)(2,8,10)(3,7,11)(4,5,6)$ .

The Riemann-Hurwitz formula for regular graphs

Posted on 2020/08/21 by wdjoyner

A little over 10 years ago, M. Baker and S. Norine (I’ve also seen this name spelled Norin) wrote a terrific paper on harmonic morphisms between simple, connected graphs (see “Harmonic morphisms and hyperelliptic graphs” – you can find a downloadable pdf on the internet of you google for it). Roughly speaking, a harmonic function on a graph is a function in the kernel of the graph Laplacian. A harmonic morphism between graphs is, roughly speaking, a map from one graph to another that preserves harmonic functions.

They proved quite a few interesting results but one of the most interesting, I think, is their graph-theoretic analog of the Riemann-Hurwitz formula. We define the genus of a simple connected graph $\Gamma = (V,E)$ to be

${\rm genus}(\Gamma) = |E| - |V | + 1.$

It represents the minimum number of edges that must be removed from the graph to make it into a tree (so, a tree has genus 0).

Riemann-Hurwitz formula (Baker and Norine): Let $\phi:\Gamma_2\to \Gamma_1$ be a harmonic morphism from a graph $\Gamma_2 = (V_2,E_2)$ to a graph $\Gamma_1 = (V_1, E_1)$ . Then

${\rm genus}(\Gamma_2)-1 = {\rm deg}(\phi)({\rm genus}(\Gamma_1)-1)+\sum_{x\in V_2} [m_\phi(x)+\frac{1}{2}\nu_\phi(x)-1].$

I’m not going to define them here but $m_\phi(x)$ denotes the horizontal multiplicity and $\nu_\phi(x)$ denotes the vertical multiplicity.

I simply want to record a very easy corollary to this, assuming $\Gamma_2 = (V_2,E_2)$ is $k_2$ -regular and $\Gamma_1 = (V_1, E_1)$ is $k_1$ -regular.

Corollary: Let $\Gamma_2 \rightarrow \Gamma_1$ be a non-trivial harmonic morphism from a connected $k_2$ -regular graph
to a connected $k_1$ -regular graph.
Then

$\sum_{x\in V_2}\nu_\phi(x) = k_2|V_2| - k_1|V_1|\deg (\phi).$

A table of small quartic graphs

Posted on 2020/07/02 by wdjoyner

This page is modeled after the handy wikipedia page Table of simple cubic graphs of “small” connected 3-regular graphs, where by small I mean at most 11 vertices.

These graphs are obtained using the SageMath command graphs(n, [4]*n), where n = 5,6,7,… .

5 vertices: Let $V=\{0,1,2,3,4\}$ denote the vertex set. There is (up to isomorphism) exactly one 4-regular connected graphs on 5 vertices. By Ore’s Theorem, this graph is Hamiltonian. By Euler’s Theorem, it is Eulerian.
4reg5a: The only such 4-regular graph is the complete graph $\Gamma = K_5$ .
graph4reg5
We have

diameter = 1
girth = 3
If G denotes the automorphism group then G has cardinality 120 and is generated by (3,4), (2,3), (1,2), (0,1). (In this case, clearly, $G = S_5$ .)
edge set: $\{(0, 1), (0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)\}$

6 vertices: Let $V=\{0,1,\dots, 5\}$ denote the vertex set. There is (up to isomorphism) exactly one 4-regular connected graphs on 6 vertices. By Ore’s Theorem, this graph is Hamiltonian. By Euler’s Theorem, it is Eulerian.
4reg6a: The first (and only) such 4-regular graph is the graph $\Gamma$ having edge set: $\{(0, 1), (0, 2), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (2, 3), (2, 5), (3, 4), (3, 5), (4, 5)\}$ .
graph4reg6
We have

diameter = 2
girth = 3
If G denotes the automorphism group then G has cardinality 48 and is generated by (2,4), (1,2)(4,5), (0,1)(3,5).

7 vertices: Let $V=\{0,1,\dots, 6\}$ denote the vertex set. There are (up to isomorphism) exactly 2 4-regular connected graphs on 7 vertices. By Ore’s Theorem, these graphs are Hamiltonian. By Euler’s Theorem, they are Eulerian.
4reg7a: The 1st such 4-regular graph is the graph $\Gamma$ having edge set: $\{(0, 1), (0, 3), (0, 5), (0, 6), (1, 2), (1, 3), (1, 5), (2, 3), (2, 4), (2, 6), (3, 4), (4, 5), (4, 6), (5, 6)\}$ . This is an Eulerian, Hamiltonian (by Ore’s Theorem), vertex transitive (but not edge transitive) graph.
graph4reg7a
We have

diameter = 2
girth = 3
If G denotes the automorphism group then G has cardinality 14 and is generated by (1,5)(2,4)(3,6), (0,1,3,2,4,6,5).

4reg7b: The 2nd such 4-regular graph is the graph $\Gamma$ having edge set: $\{(0, 1), (0, 3), (0, 4), (0, 6), (1, 2), (1, 5), (1, 6), (2, 3), (2, 4), (2, 6), (3, 4), (3, 5), (4, 5), (5, 6)\}$ . This is an Eulerian, Hamiltonian graph (by Ore’s Theorem) which is neither vertex transitive nor edge transitive.
graph4reg7b
We have

diameter = 2
girth = 3
If G denotes the automorphism group then G has cardinality 48 and is generated by (3,4), (2,5), (1,3)(4,6), (0,2)

8 vertices: Let $V=\{0,1,\dots, 7\}$ denote the vertex set. There are (up to isomorphism) exactly six 4-regular connected graphs on 8 vertices. By Ore’s Theorem, these graphs are Hamiltonian. By Euler’s Theorem, they are Eulerian.
4reg8a: The 1st such 4-regular graph is the graph $\Gamma$ having edge set: $\{(0, 1), (0, 5), (0, 6), (0, 7), (1, 3), (1, 6), (1, 7), (2, 3), (2, 4), (2, 5), (2, 7), (3, 4), (3, 6), (4, 5), (4, 6), (5, 7)\}$ . This is vertex transitive but not edge transitive.
graph4reg8a
We have

diameter = 2
girth = 3
If G denotes the automorphism group then G has cardinality 16 and is generated by $(1,7)(2,3)(5,6)$ and $(0,1)(2,4)(3,5)(6,7)$ .

4reg8b: The 2nd such 4-regular graph is the graph $\Gamma$ having edge set: $\{(0, 1), (0, 5), (0, 6), (0, 7), (1, 3), (1, 6), (1, 7), (2, 3), (2, 4), (2, 5), (2, 7), (3, 4), (3, 6), (4, 5), (4, 6), (5, 7)\}$ . This is a vertex transitive (but not edge transitive) graph.
graph4reg8b
We have

diameter = 2
girth = 3
If G denotes the automorphism group then G has cardinality 48 and is generated by $(2,3)(5,7)$ , $(1,3)(4,5)$ , $(0,1,3)(4,5,6)$ , $(0,4)(1,6)(2,5)(3,7)$ .

4reg8c: The 3rd such 4-regular graph is the graph $\Gamma$ having edge set: $\{(0, 1), (0, 2), (0, 5), (0, 6), (1, 3), (1, 4), (1, 7), (2, 3), (2, 4), (2, 7), (3, 5), (3, 6), (4, 5), (4, 6), (5, 7), (6, 7)\}$ . This is a strongly regular (with “trivial” parameters (8, 4, 0, 4)), vertex transitive, edge transitive graph.
graph4reg8c
We have

diameter = 2
girth = 4
If G denotes the automorphism group then G has cardinality $1152=2^7\cdot 3^2$ and is generated by (5,6), (4,7), (3,4), (2,5), (1,2), (0,1)(2,3)(4,5)(6,7).

4reg8d: The 4th such 4-regular graph is the graph $\Gamma$ having edge set: $\{(0, 1), (0, 2), (0, 4), (0, 6), (1, 3), (1, 5), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 6), (5, 7), (6, 7)\}$ . This graph is not vertex transitive, nor edge transitive.
graph4reg8d
We have

diameter = 2
girth = 3
If G denotes the automorphism group then G has cardinality 16 and is generated by (3,5), (1,4), (0,2)(1,3)(4,5)(6,7), (0,6)(2,7).

4reg8e: The 5th such 4-regular graph is the graph $\Gamma$ having edge set: $\{(0, 1), (0, 2), (0, 6), (0, 7), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 7), (3, 5), (3, 7), (4, 5), (4, 6), (5, 6), (6, 7)\}$ . This graph is not vertex transitive, nor edge transitive.
graph4reg8e
We have

diameter = 2
girth = 3
If G denotes the automorphism group then G has cardinality 4 and is generated by (0,1)(2,4)(3,6)(5,7), (0,2)(1,4)(3,6).

4reg8f: The 6th (and last) such 4-regular graph is the bipartite graph $\Gamma=K_{4,4}$ having edge set: $\{(0, 1), (0, 2), (0, 6), (0, 7), (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 5), (3, 7), (4, 5), (4, 6), (5, 6), (5, 7), (6, 7)\}$ . This graph is not vertex transitive, nor edge transitive.
graph4reg8f
We have

diameter = 2
girth = 3
If G denotes the automorphism group then G has cardinality 12 and is generated by (3,4)(6,7), (1,2), (0,3)(5,6).

9 vertices: Let $V=\{0,1,\dots, 8\}$ denote the vertex set. There are (up to isomorphism) exactly 16 4-regular connected graphs on 9 vertices. Perhaps the most interesting of these is the strongly regular graph with parameters (9, 4, 1, 2) (also distance regular, as well as vertex- and edge-transitive). It has an automorphism group of cardinality 72, and is referred to as d4reg9-14 below.

Without going into details, it is possible to theoretically prove that there are no harmonic morphisms from any of these graphs to either the cycle graph $C_4$ or the complete graph $K_4$ . However, both d4reg9-3 and d4reg9-14 not only have harmonic morphisms to $C_3$ , they each may be regarded as a multicover of $C_3$ .

d4reg9-1
Gamma edges: E1 = [(0, 1), (0, 2), (0, 7), (0, 8), (1, 2), (1, 3), (1, 7), (2, 3), (2, 8), (3, 4), (3, 5), (4, 5), (4, 6), (4, 8), (5, 6), (5, 7), (6, 7), (6, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  12 
aut gp gens:  [(1,2)(4,5)(7,8), (0,1)(3,8)(5,6), (0,4)(1,5)(2,6)(3,7)]

d4reg9_1

d4reg9-2 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 3), (1, 7), (2, 3), (2, 5), (2, 8), (3, 4), (4, 5), (4, 6), (4, 8), (5, 6), (5, 7), (6, 7), (6, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  2 
aut gp gens:  [(0,5)(1,6)(2,8)(3,4)]

d4reg9_2

d4reg9-3 
Gamma edges: E1 = [(0, 1), (0, 2), (0, 7), (0, 8), (1, 2), (1, 3), (1, 4), (2, 3), (2, 8), (3, 4), (3, 5), (4, 5), (4, 6), (5, 6), (5, 7), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  18 
aut gp gens:  [(1,7)(2,8)(3,6)(4,5), (0,1,4,6,8,2,3,5,7)]

d4reg9_3

d4reg9-4 
Gamma edges: E1 = [(0, 1), (0, 5), (0, 7), (0, 8), (1, 2), (1, 4), (1, 7), (2, 3), (2, 4), (2, 5), (3, 4), (3, 6), (3, 8), (4, 5), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  4 
aut gp gens:  [(2,4), (0,6)(1,3)(7,8)]

d4reg9_4

d4reg9-5 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 5), (0, 8), (1, 2), (1, 4), (1, 7), (2, 3), (2, 5), (2, 7), (3, 4), (3, 8), (4, 5), (4, 6), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  12 
aut gp gens:  [(1,5)(2,4)(6,7), (0,1)(2,3)(4,5)(7,8)]

d4reg9_5

d4reg9-6 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 5), (1, 6), (2, 3), (2, 5), (2, 6), (3, 4), (3, 8), (4, 5), (4, 7), (4, 8), (5, 6), (6, 7), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  8 
aut gp gens:  [(2,6)(3,7), (0,3)(1,2)(4,7)(5,6)]

d4reg9_6

d4reg9-7 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 4), (0, 8), (1, 2), (1, 3), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (4, 5), (4, 8), (5, 6), (5, 7), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  2 
aut gp gens:  [(0,3)(1,4)(2,8)(5,6)]

d4reg9_7

d4reg9-8 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 3), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (4, 5), (4, 6), (4, 8), (5, 6), (5, 8), (6, 7), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  2 
aut gp gens:  [(0,8)(1,5)(2,6)(3,4)]

d4reg9_8

d4reg9-9 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 6), (0, 8), (1, 2), (1, 3), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (4, 5), (4, 7), (4, 8), (5, 6), (5, 8), (6, 7), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  4 
aut gp gens:  [(5,7), (0,3)(2,6)(4,8)]

d4reg9_9

d4reg9-10 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 5), (0, 8), (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 8), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  16 
aut gp gens:  [(2,6)(3,8), (1,5), (0,1)(2,3)(4,5)(6,8)]

d4reg9_10

d4reg9-11 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 5), (4, 5), (4, 8), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  8 
aut gp gens:  [(2,4)(7,8), (0,2)(3,7)(4,6)(5,8)]

d4reg9_11

d4reg9-12 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 6), (0, 8), (1, 2), (1, 4), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 8), (5, 6), (5, 8), (6, 7), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  18 
aut gp gens:  [(1,6)(2,5)(3,8)(4,7), (0,1,6)(2,7,3)(4,5,8), (0,2)(1,3)(5,8(6,7)]

d4reg9_12

d4reg9-13 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 4), (0, 8), (1, 2), (1, 5), (1, 6), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 8), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  8 
aut gp gens:  [(2,6)(3,8), (0,1)(2,3)(4,5)(6,8), (0,4)(1,5)]

d4reg9_13

d4reg9-14 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 4), (0, 8), (1, 2), (1, 5), (1, 8), (2, 3), (2, 5), (2, 7), (3, 4), (3, 7), (4, 5), (4, 6), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  72 
aut gp gens:  [(2,5)(3,4)(6,7), (1,3)(4,8)(5,7), (0,1)(2,3)(4,5)]

d4reg9_14

d4reg9-15 
Gamma edges: E1 = [(0, 1), (0, 4), (0, 6), (0, 8), (1, 2), (1, 3), (1, 5), (2, 3), (2, 4), (2, 7), (3, 4), (3, 7), (4, 5), (5, 6), (5, 8), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  32 
aut gp gens:  [(6,8), (2,3), (1,4), (0,1)(2,6)(3,8)(4,5)]

d4reg9_15

d4reg9-16 
Gamma edges: E1 = [(0, 1), (0, 3), (0, 7), (0, 8), (1, 2), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5), (3, 7), (3, 8), (4, 5), (4, 6), (5, 6), (6, 7), (6, 8), (7, 8)] 
diameter:  2 
girth:  3 
is connected:  True 
aut gp size:  16 
aut gp gens:  [(7,8), (4,5), (0,1)(2,3)(4,7)(5,8), (0,2)(1,3)(4,7)(5,8)]

d4reg9_16

10 vertices: Let $V=\{0,1,\dots, 9\}$ denote the vertex set. There are (up to isomorphism) exactly 59 4-regular connected graphs on 10 vertices. One of these actually has an automorphism group of cardinality 1. According to SageMath: Only three of these are vertex transitive, two (of those 3) are symmetric (i.e., arc transitive), and only one (of those 2) is distance regular.

Example 1: The quartic, symmetric graph on 10 vertices that is not distance regular is depicted below. It has diameter 2, girth 4, chromatic number 3, and has an automorphism group of order 320 generated by $\{(7,8), (4,6), (1,2), (1,7)(2,8)(3,4)(5,6), (0,1,3,4,7)(2,5,6,8,9)\}$ .

d4reg10-46a

Example 2: The quartic, distance regular, symmetric graph on 10 vertices is depicted below. It has diameter 3, girth 4, chromatic number 2, and has an automorphism group of order 240 generated by $\{(2,5)(4,7), (2,8)(3,4), (1,5)(7,9), (0,1,3,2,7,6,9,8,4,5)\}$ .

d4reg10-51a

11 vertices: There are (up to isomorphism) exactly 265 4-regular connected graphs on 11 vertices. Only two of these are vertex transitive. None are distance regular or edge transitive.

Example 1: One of the vertex transitive graphs is depicted below. It has diameter 2, girth 4, chromatic number 3, and has an automorphism group of order 22 generated by $\{(1,10)(2,9)(3,4)(5,6)(7,8), (0,1,5,4,2,7,8,9,3,6,10)\}$ .

Example 2:The second vertex transitive graph is depicted below. It has diameter 3, girth 3, chromatic number 4, and has an automorphism group of order 22 generated by $\{(1,5)(2,7)(3,6)(4,8)(9,10), (0,1,3,2,4,10,9,8,7,6,5)\}$ .

Harmonic morphisms from cubic graphs of order 8 to a graph of order 4

Posted on 2020/06/08 by wdjoyner

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 $\Gamma_1$ be a simple graph on n1 vertices, $\Gamma_2$ a simple graph on n2 vertices, and assume there is a harmonic morphism $\phi:\Gamma_1 \to \Gamma_2$ . Call an n1-tuple of “colors” $\{0,1,2,..., n2-1\}$ a harmonic color list (HCL) if it’s attached to a harmonic morphism in the usual way (the ith coordinate is j if $\phi$ sends vertex i of $\Gamma_1$ to vertex j of $\Gamma_2$ ). Let S be the set of all such HCLs. The automorphism group $G_1$ of $\Gamma_1$ acts on S (by permuting coordinates associated to the vertices of $\Gamma_1$ , as does the automorphism group $G_2$ of $\Gamma_2$ (by permuting the “colors” associated to the vertices of $\Gamma_2$ ). These actions commute. Clearly S decomposes as a disjoint union of distinct $G_1\times G_2$ orbits. The conjecture is that there is only one such orbit.

Note: Caroline Melles has disproven this conjecture. Still, the question of the number of orbits is an interesting one, IMHO.

Onto the topic of the post! The 6 connected graphs of order 4 are called P4 (the path graph), D3 (the star graph, also $K_{3,1}$ ), 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 $\Gamma_1$ 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 $\Gamma_1$ 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), $|G|=16$ . This graph is not vertex transitive. Its characteristic polynomial is $x^8 - 12x^6 - 8x^5 + 38x^4 + 48x^3 - 12x^2 - 40x - 15$ . 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 $\Gamma_2=K4$ . 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 $\Gamma_2={\rm Diamond}$ . 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 $\Gamma_1$ acts by permuting some of the coordinates, for example, it can swap the 5th and 6th coordinates.Next, we take for $\Gamma_1$ 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 $\Gamma_1$ 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), $|G|=4$ (obviously too small to act transitively on the vertices). Its characteristic polynomial is $x^8 - 12x^6 - 4x^5 + 38x^4 + 16x^3 - 36x^2 - 12x + 9$ , 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 $\Gamma_1$ 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 $\Gamma_1$ has diameter 2, girth 3, and its automorphism group G is generated by (4,6), (1,2)(3,5), (0,1)(5,7), $|G|=12$ . It does not act transitively on the vertices. Its characteristic polynomial is $x^8 - 12x^6 - 2x^5 + 36x^4 - 31x^2 + 12x$ 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 $\Gamma_1$ 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” $Q_3$ , this graph $\Gamma_1$ 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), $|G|=48$ . It is vertex transitive. Its characteristic polynomial is $x^8 - 12x^6 + 30x^4 - 28x^2 + 9$ 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.

Duursma zeta function of a graph

Posted on 2020/05/28 by wdjoyner

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.

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Category Archives: graph theory