Hypergroups derived from random walks on some infinite graphs (Q2416345)

The classification of finite groups of small order is one of the classical topics of finite group theory. On the other hand, it is very hard to classify finite hypergroups, although structures of finite hypergroups of order two and three are completely determined. \textit{N. J. Wildberger} [in: Applications of hypergroups and related measure algebras. A joint summer research conference on applications of hypergroups and related measure algebras. Providence: American Mathematical Society. 413--434 (1995; Zbl 0826.43003)] gave a systematic way to construct a finite hypergroup (commutative) from a random walk on certain finite graphs. Further, Wildberger, in his paper, has also mentioned that a random walk on any strong regular graph and any distance transitive graph produces a hypergroup of order three. He has also suggested that a random walk on any distance-regular graph produces a finite hypergroup. \par In this paper under review, the authors extend Wildberger's contruction on some infinite graphs. In fact, a random walk on some infinite graph is used to produce a discrete countable hypergroup. Section 3 deals with infinite distance-regular graphs. Here they show that these graphs produce a hermitian discrete hypergroup. Section 4 deals with non-distance-regular graphs. Some examples of this kind of graphs, which gives rise to discrete hypergroups, are discussed.