報(bào) 告 人：馮衍全 教授
報(bào)告題目：Semiregular and quasi-semiregular automorphisms of digraphs
報(bào)告時(shí)間：2024年12月15日（周日）下午3：00
報(bào)告地點(diǎn)：靜遠(yuǎn)樓1508會(huì)議室
主辦單位：數(shù)學(xué)與統(tǒng)計(jì)學(xué)院、數(shù)學(xué)研究院、科學(xué)技術(shù)研究院
報(bào)告人簡(jiǎn)介：
馮衍全，北京交通大學(xué)二級(jí)教授，自1997年獲北京大學(xué)理學(xué)博士學(xué)位以來，一直從事代數(shù)與組合，群與圖以及互連網(wǎng)絡(luò)方面研究。現(xiàn)任中國(guó)工業(yè)與應(yīng)用數(shù)學(xué)學(xué)會(huì)理事、中國(guó)數(shù)學(xué)會(huì)理事等，代數(shù)組合JACO等雜志編委。2010年主持《圖的對(duì)稱性》獲教育部?jī)?yōu)秀成果二等獎(jiǎng)，2011年獲政府特殊津貼。共發(fā)表SCI科研論文150余篇，主持完成國(guó)家自然科學(xué)基金10余項(xiàng)，包括重點(diǎn)項(xiàng)目1項(xiàng)。正在承擔(dān)國(guó)家自然科學(xué)基金重點(diǎn)項(xiàng)目1項(xiàng)、面上項(xiàng)目1項(xiàng)、國(guó)際合作研究項(xiàng)目1項(xiàng)。
報(bào)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.