On the least eigenvalue of genuine strongly 3-walk-regular graphs

As a generalization of strongly regular graphs, van Dam and Omidi [8] introduced the concept of strongly walk-regular graphs. A graph is called strongly t degrees-walk-regular if the number of walks of length t degrees from a vertex to another vertex depends only on whether the two vertices are adjacent, not adjacent, or identical. They proved that this class of graphs falls into several subclasses including connected regular graphs with four eigenvalues, which are called genuine strongly t degrees-walk-regular. In this paper, we prove that the least eigenvalue of a connected genuine strongly 3-walk-regular graph is no more than -2 and characterize all graphs reaching this upper bound.