報 告 人：馮衍全 教授

報告題目：Semiregular and quasi-semiregular automorphisms of digraphs

報告時間：2024年12月15日（周日）下午3：00

報告地點：靜遠樓1508會議室

主辦單位：數學與統計學院、數學研究院、科學技術研究院

報告人簡介：

       馮衍全，北京交通大學二級教授，自1997年獲北京大學理學博士學位以來，一直從事代數與組合，群與圖以及互連網絡方面研究。現任中國工業與應用數學學會理事、中國數學會理事等，代數組合JACO等雜志編委。2010年主持《圖的對稱性》獲教育部優秀成果二等獎，2011年獲政府特殊津貼。共發表SCI科研論文150余篇，主持完成國家自然科學基金10余項，包括重點項目1項。正在承擔國家自然科學基金重點項目1項、面上項目1項、國際合作研究項目1項。

報告摘要：

       Let G be a permutation group on a finite set Omega . An non-identity element g in G is said to be semiregular if every cycle in the unique cycle decomposition of g has the same length, and quasi-semiregular if g has an unique 1-cycle in the cycle decomposition of g and every other cycle has the same length. An automorphism of a digraph is called semiregular or quasi-semiregular if it is a semiregular or quasi-semiregular permutation on the vertex set of the digraph. The permutation group G is called 2-closed if G is the largest subgroup of the symmetric group S_Omega on Omega with the same orbits as G on Omega× Omega.

       In 1981 Fein, Kantor and Schacher proved that a transitive permutation group on a finite set with degree at least 2 has an element of prime-power order with no fixed point, but may not have a semiregular element. In the same year, Marusic conjectured that every finite vertex-transitive digraph has a semiregular automorphism, and in 1995, Klin proposed the well-known Polycirculant Conjecture: Every 2-closed transitive permutation group has a semiregular element. Note that the automorphism group of any digraph is 2-closed. In 2013, Kutnar, Malnic, Martanez and Marusic proposed the quasi-semiregular automorphism of a digraph and investigated strongly regular graphs with such an automorphism.

        A lot of work relative to semiregular or quasisemiregular automorphisms of digraphs has been done and in this talk, we review some progress on this line. Furthermore, we talk about a recent work by Yin, Feng, Zhou and Jia [Journal of Combinatorial Theory B 159 (2023) 101-125] on prime-valent symmetric graphs with a quasi-semiregular automorphism.