Tag Archives: math

Satake homomorphisms for the 2-fold cover of p-adic SL(2), revisited

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In very vague, rough terms, the Langlands program gives a way of understanding certain types of maps between two specific types of groups that are related in a particular way. One way I like to think about it is that it gives mathematicians a template for conjectures of such “correspondences” which are likely to be true. Around 20-30 years ago, I was very interested specifically in the metaplectic covers of SL(2,F), where F is a p-adic field. I seems to have roped a number of victims into thinking and talking about such things with me. My colleagues Courtney Moen (who has retired from mathematics) and Jason Levy (who seems to have also left academia) are two examples.

Around 2004, with Jason, I wrote down a form of the correspondence that I thought the Langlands program template gave me as a conjectural “correspondence” between SL(2) over a p-adic field and the 2-fold cover. It’s very technical to write up, yes, but technically complicated and original are completely different things. I want to be clear, the notes/conversations had nothing original – we were writing down this template due to Robert Langlands in the case we wanted to investigate. I vaguely remember knowing that the conjectural guess was false, but it was just a template and I knew I’d have to sculpt it, like an artist, into a correct one. Sadly, my brain did not have the sharpness needed for the job, and I gave up.

That would have ended it forever if not for the facts that (a) I typed up notes of the template conjecture (or my version of how I remembered them) and, (b) now in 2026 20-something years later, AI is being used to prove the quasi-Riemann hypothesis. So, can it fix the false template guess I made long ago?

Thanks to both Opus-Claude and Sol-ChatGPT, the answer is yes. The AI-written paper The Satake Isomorphism, the Fundamental Lemma, and Transfer of Orbital Integrals for the Metaplectic Double Cover of SL(2, F) is the result. Basically, I tag-teamed one AI agent against another, one acting as the researcher and the other as the referee, and I would be adding or subtracting material at each stage to increase readability, preserve topicality, and maintain correctness. This went through 7 rounds of revision and the pdf above is the result.

The real work was done by AI, not Jason, not me. I think the reason it was so much fun working with that AI team in this way was solely because it taught me something new and interesting. I don’t like to give up on a problem. No one does. This helped me finish the job I gave up on in 2004.

Odd king tours on even chessboards, revisited

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Almost 30 years ago this year, a former student Michael Fourte and I wrote a combinatorics paper (original paper, 1997) with the title above, eventually published in the now defunct Journal of Recreational Mathematics 31(2003)173–177.That paper was a pleasure to write, but was it correct?

With AI tackling open math problems left and right, I asked AI to review our paper and correct it, if any errors were found. In fact, I expected no errors but it turns out the main theorem was not as carefully stated as it should be, and some conjectural statements were not formulated in the best way (though our proofs were correct). A corrected version, made with the help of AI, can be found here: 2026 corrected version. An AI-written report on the paper with some possible directions for further research is here: 2026 AI-written report/review.

The shell algorithm and a new book

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A Peek Behind the Curtain

One feature of my forthcoming book with Caroline Melles, Exploring Graphs via Harmonic Morphisms, is its computational emphasis. Many of the examples and insights emerged not from pure contemplation, but from extensive experimentation using SageMath. This post gives a glimpse into the algorithmic machinery that made this exploration possible.

The fundamental problem I faced in generating examples was this: given two graphs Γ₁ and Γ₂, find all harmonic morphisms φ: Γ₂ → Γ₁. This approach is direct—a brute force algorithm that constructs all possible “color lists” (vertex labelings that encode potential morphisms) and tests each one for the harmonic property. Straightforward? Yes. Fast? Definitely not.

However, when the target graph Γ₁ is a path graph, one can do better. The “shell algorithm” exploits the geometry of distance in the source graph. Starting from a chosen vertex v₀, one builds outward through successive neighborhoods—vertices at distance 1, then 2, and so on—assigning colors constrained by distance from the starting point. At each stage, one checks necessary conditions before proceeding, pruning impossible branches early. The automorphism groups of both graphs then generate equivalent colorings, and after removing duplicates, one has a complete enumeration of the harmonic morphisms.

That’s what was done for various choices of small graphs, printing computational data to a file whose name used notation or terminology for the graphs \Gamma_1 and \Gamma_2. Months of computer searches resulted in a number of conjectures that were used to shape the material in the book.

The Elevator Problem

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You are on the bottom floor (floor 0, lets call it) of an apartment building with no basement. There are n elevators, which we index 1,2,…,n. Assume the elevators are on floors f1, f2, … , where fk > 0 is the floor elevator k is currently on, 1 <= k <= n. Assume you only like one of the elevators, elevator e.


The way the elevator logic works is this: When you press the elevator button, one closest to you ( = one on floor number min(fk, k>0)) is told to go to 0. If there is a tie then, of those on the same lowest floor, the elevator with the smallest index is told to go to 0.
Move: If you press the button and some other elevator than elevator e arrives, you can tell it to go to any floor you wish.
Taboo: You can press the elevator button if and only if no elevator is moving down.
Goal: You want to use elevator e (for some fixed e=1, 2, …, n).

Problem: Is there a finite sequence of moves that allows you to ride in elevator e?

My plan is to post the answer sometime later, but have fun with it!

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

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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.

  1. 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.
  2. 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.
  3. 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.
  4. 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,
  5. introduces Laplacian matrices as differences of degree and adjacency matrices, connecting to the incidence matrix framework.
  6. Introduced graph blowup morphisms via a blowup construction where vertices are replaced by independent sets, creating natural homomorphisms with specific structural properties.
  7. Some functorial properties of graph morphisms are established, such as how morphisms behave under graph constructions like subdivisions, smoothing, deletions, and substitutions.
  8. The chapter ends with exercises and a chapter summary.

Chapter 2 on Harmonic Morphisms

    This chapter is devoted to the basics of harmonic morphisms.
  1. Introduces the core definition: a graph morphism is harmonic if local horizontal multiplicities are constant across edges incident to each vertex’s image.
  2. 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.
  3. 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.
  4. The Riemann-Hurwitz formula – Presents the graph-theoretic analogue relating genera of graphs via harmonic morphisms, with matrix proof and applications to regular graphs.
  5. 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.
  6. The chapter ends with exercises and a chapter summary.

All harmonic morphisms from this graph to C4 are covers.

  1. Fundamental Problem: Given a graph Gamma1, for which graphs Gamma2 is there a non-trivial harmonic morphism phi from Gamma2 to Gamma1?
  1. 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.

  1. Regarding harmonic morphisms between path graphs, we show how to construct and count the harmonic morphisms from longer path graphs to shorter ones.
  2. 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.
  3. 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.
  4. Harmonic morphisms to P2 (arising from the Baker-Norin Theorem) can be counted.
  5. 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.
  6. 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.
  1. Quotient graphs from group actions. Harmonic actions and transitive actions are studied separately.
  2. Quotient graphs from paritions. Orbit partitions and equitable partitions are studied.
  3. As a nice application of harmonic morphisms with particularly nice structural properties, we consider multicovers and blowup graphs.
  4. 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.
  5. The chapter ends with exercises and a chapter summary.

Chapter 5 on Graph Morphisms and Graph Products

    This chapter studies graph morphisms associated to tensor products of graphs and lexicographical products of graphs.

    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?

  1. For example, we show that projection morphisms from tensor products are always harmonic with explicit horizontal multiplicity formulas.
  2. 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.
  3. 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.
  4. 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

  1. This chapter studies graph morphisms associated to Cartesian/strong/disjunctive products of graphs as well as joins and NEPS graphs.
  2. For example, we show that projection morphisms from Cartesian products or from strong products are always harmonic with explicit horizontal multiplicity formulas.
  3. 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).
  4. 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.
  5. Can one classify the graphs for which the disjunctive product projections pr1 or pr2 are graph morphisms?
  6. 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.
  7. 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.
  8. 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.
  9. 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

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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.

Another mathematician visits the ballpark – WHIP

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This is the second in the series of blog posts inspired by the 2004 Ken Ross book entitled A Mathematician at the Ballpark. The first one is here. In this post, again, we illustrate all these notions using the Baltimore Orioles’ 2022 season.

For an experienced baseball fan, baseball is a game of patterns. We “know” what a well-executed pitch looks like, how a double play is to be executed, how a pop-up is to be fielded, and so on. Because of these expected patterns, we know the plays which should emerge and so the desire to track their occurrences should come as no surprise. It’s been done since professional baseball started in the 1870s.
This week we discuss a pitching statistic you see on televised games, WHIP. WHIP is short for “walks plus hits allowed per innings pitched”.

Earned run average

We start off with the most basic pitching statistic, the Earned Run Average or ERA. This is the number of earned runs per 9 innings pitched:

ERA = 9·ER/IP,

where 

  • IP, the number of Innings Pitched,
  • ER, the number of Earned Runs allowed by the pitcher. That is, it counts the number of runs enabled by the offensive team’s production in the face of competent play from the defensive team.

It is possible to have ERA = ∞, since innings are measured by the number of outs achieved (so if the pitcher doesn’t get any batters out, his IP=0). The lower the ERA the better the pitcher. In the 2022 season, right-handed Félix Bautista, who entered late innings as either a closer or a reliever, had an ERA of 2.19. Left-handed closer Cionel Pérez had an ERA of 1.40.

WHIP

We define walks plus hits allowed per innings pitched by:

WHIP = (BB+H)/IP,

where (as in the previous post) BB is the number of walks and H stands for the number of Hits allowed by the pitcher (so, for example, reaching base due to a fielding error doesn’t count). WHIP reflects a pitcher’s propensity for allowing batters to reach base, therefore a lower WHIP indicates better performance.

When we plot the ERA vs the WHIP for the top 20 Orioles pitchers in 2022, we get 

Again, the line shown is the line that best fits the data. As the line of best fit doesn’t fit the date too well, this tells us that these two statistical measurements aren’t too well-correlated. In other words, low ERA indicates a good pitcher and low WHIP indicates a good pitcher, but the values for “average” pitchers seem less related to each other.

Another mathematician visits the ballpark – OPS

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Yes, I more-or-less stole the above title from the 2004 Ken Ross book entitled A Mathematician at the Ballpark. Like that book, anyone familiar with middle-school (or junior high school) math, should have no problem with most of what we do here. However, I will try to go into baseball in more detail than the book did.

Paraphrasing slightly, I read somewhere the following facetious remark:

From a survey of 1000 random baseball fans 

across the nation,  183% of them hate math. 

If you are one of these 183%, then this series could be for you. Hopefully, even if you aren’t a baseball expert, but you would like to learn some baseball statistics, (now often called “sabermetrics”), these posts will help. I’m no expert myself, so we’ll learn together.

In this series of blog posts, each post will introduce a particular metric in baseball statistics as well as some of the math and baseball behind it. We illustrate all these notions using the Baltimore Orioles’ 2022 season.

This week we look at one of the most popular statistics you see on televised games: OPS or “On-base Plus Slugging,” which is short for on-base percentage plus slugging percentage. Don’t worry, we’ll explain all these terms as we go. 

On-base percentage

First, On-Base Percentage or OBP is a more recent version of On-Base Average or OBA (the same as OBP but the SF term is omitted). We define 

OBP = (H + BB + HBP)/(AB + BB + HBP + SF), 

where 

  • H is the number of Hits (the times the batter reaches base because of a batted, fair ball without error by the defense), 
  • BB is the number of Base-on-Balls (or walks), where a batter receives four pitches that the umpire calls balls, and is in turn awarded first base,
  • HBP, or Hit By Pitch, counts the times this hitter is touched by a pitch and awarded first base as a result, and 
  • SF is the number of Sacrifice Flies and AB the number of At-Bats, which are more complicated to carefully define.

The official scorer keeps track of all these numbers, and more, as the baseball game is played. We still have to define the expressions AB and SF.

First, SF, or Sacrifice Flies, counts the number of fly balls hit to the outfield for which both of the following are true:

  • this fly is caught for an out, and a baserunner scores after the catch (so there must be at most one hit at the time),
  • the fly is dropped, and a runner scores, if in the scorer’s judgment the runner could have scored after the catch had the fly ball been caught.

A sacrifice fly is only credited if a runner scores on the play. (By the way, this is a “recent” statistic, as they weren’t tabulated before 1954. Between 1876, when the major league baseball national league was born, and 1954 baseball analysts used the OBA instead.)

Second, AB, or At-Bats, counts those plate appearances that are not one of the following:

  • A walk,
  • being hit by a pitch,
  • a bunt (or Sacrifice Hit, SH),
  • a sacrifice fly,
  • interference (the catcher hitting the bat with his glove, for example), or
  • an obstruction (by the first baseman blocking the base path, for example).

Incidentally, the self-explanatory number Plate Appearances, or PA, can differ from AB by as much as 10%, mostly due to the number of walks that a batter can draw.

The main terms in the OBP expression are H and AB. So we naturally expect OBP to be approximately equal to the Batting Average, defined by

BA = H/AB,

For example, if we take the top 18 Orioles players and plot the BA vs the OBP, we get the following graph:

The line shown above is simply the line of best fit to visually indicate the correlation.

Example: As an example, let’s look at the Orioles’ All-Star center fielder,  Curtis Mullins, who had 672 plate appearances and 608 at bats, for a difference of 672 − 608 = 64. He had 1 bunt, 5 sacrifice flies, he was hit by a pitch 9 times, and walked 47 times. These add up to 62, so (using the above definition of AB) the number of times he was awarded 1st base due to interference or obstruction was 64 − 62 = 2.

Mullins’ H = 157 hits break down into 105 singles, 32 doubles, 4 triples, and 16 home runs.

Second, let’s add to these his 126 strikeouts, for a total of 157+126+64 = 347.

The remaining 608 − 347 = 261 plate appearances were pitches hit by Mullins, but either caught on the fly but a fielder or the ball landed fair and Mullins was thrown out at a base.

These account for all of Mullins’ plate appearances. Mullins has a batting average of BA = 157/608 = 0.258 and an on-base percentage of OBP = 0.318.

Slugging percentage

The slugging percentage, SLG, (SLuGging) is the total bases achieved on hits divided by at-bats:

SLG = TB/AB.

Here, TB or Total Bases, is the weighted sum

TB = 1B + 2*2B + 3*3B + 4*HR,

where

  • 1B is the number of “singles” (hits where the batter makes it to 1st Base),
  • 2B is the number of doubles,
  • 3B is the number of triples, and
  • HR denotes the number of Home Runs.

On-base Plus Slugging

With all these definitions under own belt, finally we are ready to compute “on-base plus slugging”, that is the on-base percentage plus slugging percentage:

OPS = OBP + SLG.

Example: Again, let’s consider Curtis Mullins. He had 1B = 105 singles, 2B = 32 doubles, 3B = 4 triples, and HR = 16 home runs, so his TB = 105+64+12+64 = 245. Therefore, his SLG = 245/608 = 0.403, so his on-base plus slugging is OPS = OBP + SLG = 0.318 + 0.403 = 0.721.

This finishes our discussion of OPS. I hope this helps explain it better. For more, see the OPS page at the MLB site or the wikipedia page for OPS. 

A table of small quartic graphs

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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)\}.