Visualizing Irreducible \(T\)-modules for the 13 Distance-Regular Graphs with Valency Three

This website is supplementary material for "The Terwilliger algebra for the distance-regular graphs with valency three," a paper developed as a part of a project for the Madison Experimental Mathematics Laboratory (MXM) at the University of Wisconsin–Madison during the academic year 2025/26. The project was headed by Professor Paul Terwilliger, with undergraduate contributors Kevin Kauflin, Barnabás Valkó, and Hanyi Wu. The graduate mentor for the project was Jimmy Vineyard.

The goal of the project follows. It is known by a classical theorem that there are exactly 13 distance-regular graphs of valency three. The paper gives, for each, an orthogonal direct-sum decomposition of the standard module \(V = \mathbb{C}^X\) into irreducible \(T\)-modules, with an explicit pure basis for each module. With these decompositions, we can calculate \(\dim T\) for each graph.

This visualization is intended to make those decompositions visually clear; for any selected graph and module, one can interactively verify closure under the adjacency matrix.

Let \(\Gamma = (X, \mathcal{E})\) be a finite undirected distance-regular graph. For a fixed base vertex \(x \in X\), the vertices are partitioned into subconstituents \(\Gamma_0(x), \Gamma_1(x), \dots, \Gamma_D(x)\) by shortest-path distance from \(x\), where \(D\) is the diameter of \(\Gamma\).

The Terwilliger algebra \(T = T(x)\) is the algebra of matrices generated by the adjacency matrix \(A\) and the diagonal dual primitive idempotents \(\{E_i^*\}_{i=0}^D\). Note that the action of \(E_i^*\) retains only the coordinates of vertices at distance \(i\) from \(x\) and zeroes out the rest.

A \(T\)-module is a subspace of \(\mathbb{C}^X\) closed under the action of every element of \(T\). It is called irreducible if it contains no smaller nonzero \(T\)-module. The standard module \(V = \mathbb{C}^X\) decomposes as an orthogonal direct sum of irreducible \(T\)-modules, and it is this decomposition that the paper derives for each of the 13 graphs.

All 13 graphs, save for Tutte's 12-cage, are distance-transitive, so the choice of \(x\) is unimportant. For the Tutte's 12-cage, derivations for both choices of \(x\) up to isomorphism are shown (these are the cases \(x \in X^+\) and \(x \in X^-\), where \(X^+,X^-\) are the two bipartite halves that partition vertex set \(X\)).

Select a graph and a \(T\)-module, then choose an initial basis vector \(\mathbf{v}_i\) to load its values onto the graph's vertices. Clicking "Apply Adjacency Matrix \(A\)" updates each vertex's value to the sum of its neighbors' current values, exactly the action of the adjacency matrix. The visualization shows that the result stays within the span of the module's basis, demonstrating closure under \(A\). Closure under each \(E_i^*\) holds automatically, since every basis vector is supported on a single subconstituent.

Because basis vectors contain many trivial zeros outside their subconstituent, hovering over a vector in Configuration displays its essential part: the restriction to its subconstituent shown as a row vector, with the overset number indicating which subconstituent.

Irreducibility of each non-primary module follows from the fact that it is generated by a single seed vector \(\nu \in E_r^* W\), with \(\dim(E_r^* W) = 1\), where \(r\) is the endpoint of the module (Theorem 6 in the paper). The unique primary module is the span of the main diagonals of the nonzero \(E_i^*\) matrices (Theorem 2).

Color Mapping. Each basis vector is assigned a distinct color, used consistently across the graph, the current linear combination display, and the matrix. Vertex \(y\) is colored the color of a basis vector \(\mathbf{v}_i\) if the \(y\)-th entry of the current linear combination of vectors includes a nonzero contribution from \(\mathbf{v}_i\). Vertices with contributions from multiple basis vectors split into multiple colors. Note that this coloring is unrelated to the subconstituent coloring used in the paper.

Graph Tooltips. Hover over any vertex to see its current total value, a history of incoming contributions, and a breakdown by basis vector.

Basis Vector Overlays. There is a checkbox in Configuration that says "Allow basis vector overlay upon hover." When the box is checked, hover over any basis vector \(\mathbf{v}_i\) in Configuration to overlay that vector onto the graph. This feature is intended to aid comparisons of basis vectors to the current values shown.

Subconstituent Highlights. Click or hover a row in the \(\Gamma_i\) table to highlight that subconstituent on the graph. Clear with "Reset Highlights."

Matrix representing \(A\). The matrix of \(A\) in the selected basis is shown bottom right; the column for \(A\mathbf{v}_i\) is highlighted when that vector is acted on.

Animation Controls. Press "Animation Settings" to adjust or disable transition animations.

Data Downloads: Use the buttons in the top left of the graph viewport to download the adjacency matrix \(A\) of the current graph as a text file, or the dual primitive idempotents \(\{E_i^*\}_{i=0}^D\) as a ZIP. Click the green download button in the top right to download all matrices for all 13 graphs at once (~893KB). Click the "Download Current Vector" button in the bottom right to download the current vector displayed on the graph as a .txt file.