講座主題：Symmetric cubic graphs as Cayley graphs
專家姓名：Marston Conder
工作單位：新西蘭奧克蘭大學
講座時間：2017年11月6日15:00-16:00
講座地點：數學院大會議室
主辦單位：煙臺大學數學與信息科學學院
內容摘要：
A graph is symmetric if its automorphism group acts transitively on the arcs of , and -arc-transitive if its automorphism group acts transitively on the set of -arcs of . Furthermore, if the latter action is sharply-transitive on -arcs, then is -arc-regular. It was shown by Tutte (1947, 1959) that every finite symmetric cubic graph is -arc-regular for some . Djokovic and Miller (1980) took this further by showing that there are seven types of arc-transitive group action on finite cubic graphs, characterised by the stabilisers of a vertex and an edge. The latter classification was refined by Conder and Nedela (2009), in terms of what types of arc-transitive subgroup can occur in the automorphism group of $X$. In this talk we consider the question of when a finite symmetric cubic graph can be a Cayley graph. We show that in five of the 17 Conder-Nedela classes, there is no Cayley graph, while in two others, every graph is a Cayley graph. In eight of the remaining ten classes, we give necessary conditions on the order of the graph for it to be Cayley; there is no such condition in the other two. Also we use covers (and the `Macbeath trick') to show that in each of those last ten classes, there are infinitely many Cayley graphs, and infinitely many non-Cayley graphs. This research grew out of some discussions with Klavdija Kutnar and Dragan Marusic (in Slovenia).
主講人介紹：
Marston is a Distinguished Professor of Mathematics in Aucland University (and former Co-Director of the New Zealand Institute of Mathematics and its Applications (the NZIMA)). His main areas of interest are group theory and graph theory (sections 20 and 05 in Math Reviews). He is especially interested in the methods and applications of combinatorial group theory, including computational techniques for handling finitely-presented groups and their images. Professor Conder has published 169 distinguished papers from 1980. He has contributed to the graph and group theory as much as you can imagine.
