In 1936, Kőnig conjectured that every finite group can be represented as the automorphism group of a finite graph. This was answered in the affirmative by Frucht in 1939. Since then, this problem has been explored under various constraints on the graph. A particularly interesting case arises when the graph is vertex-transitive and has the same order as the group; such a graph is known as a Graphical Regular Representation (GRR). In the 1970s, a series of papers established which groups admit GRRs. In this talk I will first provide an overview of these classic results and then discuss some recent developments, including asymptotic enumeration, GRRs with specific valencies, and new variations of GRRs.

夏彬绉，2009年本科毕业于浙江大学；2014年获得北京大学博士学位；2014-2016年于北京国际数学研究中心从事博士后研究；2016-2017年西澳大利亚大学Research associate；2017年起任职于墨尔本大学。国际组合数学与应用学会(ICA)2017年Kirkman奖获得者。主要研究方向为代数图论、组合学与置换群论。在诸多国际著名期刊（如Mem. Amer. Math. Soc., Proc. Lond. Math. Soc., J. Lond. Math. Soc., J. Combin. Theory Ser. B等)发表论文50余篇，近三年来的研究工作集中在具有特定对称性的凯莱图的分类、渐近计数和特征值等方向。