Catherine H.F. Yan

In 1997, Spencer developed a new way to enumerate connected graphs with a fixed number of edges, which led to a new characterization of Wright constant, and new insight to the asymptotics. The main idea is to run the breadth-first search algorithm on random trees as a queue process. Information of such queues can be expressed as parking functions and its statistics, a subject that appears in many combinatorial structures, including hashing functions, labeled trees, hyperplane arrangement, noncrossing partitions, polytope triangulation, and analysis of diagonal harmonics. Recently there were notions of $G$-parking functions, arisen from the study of certain quotient of the polynomial ring. In this talk we show how to extend above results to an arbitrary graph $G$. We describe an algorithmic bijection between $G$-parking functions and spanning trees of $G$, and connect it to graph enumeration, Tutte polynomial, and chip-firing games.