A new algebraic approach to finding all simple paths and cycles in undirected graphs

Finding all simple paths and cycles in undirected graphs is a generic problem that covers a wide range of research areas. The problem has been discussed only to a minor extent in recent years. This paper investigates the problem and provides a necessary and sufficient condition for simple paths and cycles. With the theory of the semi-tensor product, a necessary condition for simple paths and cycles is then proposed to reduce the search space. Based on the above-mentioned results , a new algebraic algorithm to find all simple paths and cycles in undirected graphs is presented. The algorithm involves only algebraic operations, reduces the search space, and decreases the computational complexity. An illustrative example is finally given to demonstrate the effectiveness of the presented algorithm.