Cellular Algebras and Graph Invariants Based on Quantum Walks

TL;DR

This paper explores how cellular algebras derived from quantum walk-based graph invariants can fail to distinguish certain non-isomorphic graphs, highlighting limitations in these quantum-inspired methods.

Contribution

It introduces a connection between cellular algebras and quantum walk invariants, showing that similar algebraic structures imply indistinguishability by these invariants.

Findings

Cellular algebras can be associated with graphs based on quantum walk invariants.

Similar cellular algebra structures mean the invariants cannot distinguish the graphs.

The invariants may have limitations in graph isomorphism testing.

Abstract

We consider two graph invariants inspired by quantum walks- one in continuous time and one in discrete time. We will associate a matrix algebra called a cellular algebra with every graph. We show that, if the cellular algebras of two graphs have a similar structure, then they are not distinguished by either of the proposed invariants.