Terwilliger algebras and some related algebras defined by finite connected simple graphs

TL;DR

The article systematizes five interrelated algebras $\mathcal{T}_\ell(\Gamma,x_0)$ (for $\ell=0,1,2,3,4$) attached to a finite connected simple graph $\Gamma$ and a base vertex $x_0$, including the Terwilliger algebra and a centralizer variant, and provides practical methods to compute them across graph families. It derives structural results and explicit decompositions for key classes: distance-regular and strongly regular graphs, as well as concrete graphs such as path, star, and cycle graphs, plus Paley graphs, revealing precise conditions under which $\mathcal{T}_2$ equals or differs from $\mathcal{T}_3$ and $\mathcal{T}_4$. Notably, it shows $E_{X_1}\mathcal{T}_2E_{X_1}$ is commutative in SRGs and gives dimension bounds, identifies infinite families with $\mathcal{T}_2\neq\mathcal{T}_3$, and provides explicit counterexamples where $\mathcal{T}_3\neq\mathcal{T}_4$ (e.g., Paley graphs) as well as minimal graphs with $\mathcal{T}_2\neq\mathcal{T}_3$ (Delta$_5$). Together, these results clarify when these natural algebras coincide with full matrix algebras and supply detailed, computable descriptions for a broad spectrum of graphs.

Abstract

For a finite connected simple graph, the Terwilliger algebra is a matrix algebra generated by the adjacency matrix and idempotents corresponding to the distance partition with respect to a fixed vertex. We will consider algebras defined by two other partitions and the centralizer algebra of the stabilizer of the fixed vertex in the automorphism group of the graph. We will give some methods to compute such algebras and examples for various graphs.

Theorems & Definitions (67)

• Conjecture : J. Koolen
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 3.4