Constructing Clifford Graph Algebras from Classical Clifford Algebras

Abstract

There is a special case of a generalized Clifford algebra, known as a Clifford graph algebra, which is useful for studying a simple graph $G_n$, with $n$ vertices. We will discuss how this algebra $GA(G_n)$ can represent $G_n$, and prove that it exists in general by defining it as an appropriate sub-algebra of a classical Clifford algebra. We will then refine this process of “construction by inclusion” for the path graph $P_n$, and the complete star graph $K_{1,n}$, by choosing from a parent classical Clifford algebra as many bi-vectors as possible for the generators which define $GA(P_n)$ and $GA(K_{1,n})$.