A New Adjacency Matrix for Finite Graphs

Abstract

This paper expands on the graph-theoretic content of my contributed talk at the Seventh International Conference on Clifford Algebras and Their Applications. A well-known result in graph theory states that when A is the adjacency matrix of a finite graph G, the entries of Ak represent numbers of k-step walks existing in G. However, the adjacency matrix fails to distinguish between walks and “self-avoiding” walks (i.e., walks without repeated vertices). Utilizing elements of abelian, nilpotent-generated algebras, a “new” adjacency matrix is associated with a finite graph on n vertices. By considering entries of \({\mathcal{A}}^k\), where \({\mathcal{A}}\) is an appropriate nilpotent adjacency matrix, one is able to recover the self-avoiding k-walks in any finite graph. In particular, a graph’s Hamiltonian cycles are enumerated by the top-form coefficient in the trace of \({\mathcal{A}}^n\) when n is the number of vertices in the graph. By considering the lth power of the trace of \({\mathcal{A}}^k\), l-tuples of pairwise-disjoint k-cycles are recovered. By defining a nilpotent transition matrix associated with a time-homogeneous Markov chain, a method of computing probabilities of self-avoiding random walks on finite graphs is developed. Expected hitting times of specific states in Markov chains and expected times of first self-intersection of random walks are also recovered using these methods. The algebra used to define the nilpotent adjacency matrix of a graph on n vertices is not itself a Clifford algebra, but it can be constructed within the 2n-particle fermion algebra \({\mathcal{C}}\ell_{2n,2n}\), indicating potential connections to quantum computing.