Constructing Clifford Graph Algebras from Classical Clifford Algebras

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 \( GA(K_{1,n}) \).