# 群论讨论班 Group Theory Seminars

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# Covers of locally-$$s$$-distance-transitive graphs

## 周慧 Hui Zhou

Beijing University of Chemical Technology

Time: 16:00-18:00 (GMT+8), Monday April 17th, 2023
Location: Zoom

Abstract: We first introduce some basic concepts in algebraic graph theory, such as (locally-)$$s$$-arc-transitive graphs, distance-transitive graphs, (locally-)$$s$$-distance-transitive graphs, and the study on covers of these graphs. Taking normal quotients is a powerful tool on the research of graphs with some symmetry, and this method was first used by Cheryl Praeger in 1990's. By Cheryl' method, researchers around the world made a lot of contributions on studying symmetries of combinatorial structures. Usually, a graph is a normal multi-cover of its normal quotients, when the graph has higher symmetry the multi-cover is actually a cover. By Conway's voltage assignment, each (locally-)$$s$$-arc-transitive graph has a nontrivial (locally-)$$s$$-arc-transitive normal cover, however, this is not true for locally-$$s$$-distance-transitive graphs. Locally-$$s$$-distance-transitive graphs were initiated by Alice Devillers, Michael Giudici, Cai Heng Li and Cheryl Praeger. They gave complete multipartite graphs $$K_{m[b]}$$ as an example of locally-$$s$$-distance-transitive graphs with property having no nontrivial normal covers. By analysing the structure of complete multipartite graphs $$K_{m[b]}$$, we give a generalization of graphs with such property, which are called $$E$$-graphs. Some examples and characterisations of $$E$$-graphs are given.

Host: 丁兆宸 Zhaocheng Ding