报告地点：行健楼学术活动室526
邀请人：杜佳丽副教授
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-semi- regular 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 $\Omega\times\Omega$.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, Maru\v{s}i\v{c} 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, Malni\v{c}, Mart\'{a}nez and Maru\v{s}i\v{c} proposed the quasi-semir- egular 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 on prime-valent symmetric graphs with a quasi-semiregular automorphism, published in [Journal of Combinatorial Theory B 159 (2023)101-125].
报告人简介：冯衍全，北京交通大学数学与统计学院教授，博士生导师。自1997年获北京大学理学博士学位以来，一直从事代数与组合，群与图以及互连网络方面的研究，2001年晋升教授，2002年被评为博士生导师。北京市数学会常务理事，中国数学会组合与图论专业委员会理事，SCI杂志《Ars Mathe- matica Contemporanea》编委。2010年主持《图的对称性》获教育部高等学校科学研究优秀成果奖自然科学二等奖，2011年获国务院政府特殊津贴，主持国家自然基金项目多项，其中重点项目两项。
