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

TL;DR

This paper explores algebraic structures associated with finite graphs, including Terwilliger algebras and related algebras defined by various partitions, providing methods for their computation and illustrating with examples.

Contribution

It introduces new algebraic constructions based on different graph partitions and offers computational techniques for these algebras, expanding understanding of graph symmetries.

Findings

Methods to compute these algebras are developed.

Examples demonstrate the algebraic structures for various graphs.

Connections between different algebraic constructions are established.

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.