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2. On 3-isoregularity of multicirculantsKlavdija Kutnar, Dragan Marušič, Štefko Miklavič, 2025, original scientific article Abstract: A graph is said to be k-isoregular if any two vertex subsets of cardinality at most k, that induce subgraphs of the same isomorphism type, have the same number of neighbors. It is shown that no 3-isoregular bicirculant (and more generally, no locally 3-isoregular bicirculant) of order twice an odd number exists. Further, partial results for bicirculants of order twice an even number as well as tricirculants of specific orders, are also obtained. Since 3-isoregular graphs are necessarily strongly regular, a motivation for the above result about bicirculants is that it brings us a step closer to obtaining a direct proof of a classical consequence of the Classification of Finite Simple Groups, that no simply primitive group of degree twice a prime exists for primes greater than 5.
Keywords: 3-isoregularity, strongly regular graph, bicirculant, tricirculant
Published in RUP: 06.08.2025; Views: 403; Downloads: 3
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3. Semiregular automorphisms in vertex-transitive graphs with a solvable group of automorphismsDragan Marušič, 2017, original scientific article Abstract: It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally 2-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. The known affirmative answers for graphs with primitive and quasiprimitive groups of automorphisms suggest that solvable groups need to be considered if one is to hope for a complete solution of this conjecture. It is the purpose of this paper to present an overview of known results and suggest possible further lines of research towards a complete solution of the problem.
Keywords: solvable group, semiregular automorphism, fixed-point-free automorphism, polycirculant conjecture
Published in RUP: 03.01.2022; Views: 2111; Downloads: 26
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6. Hamilton cycles in primitive vertex-transitive graphs of order a product of two primes - the case PSL(2, q[sup]2) acting on cosets of PGL(2, q)Shao Fei Du, Klavdija Kutnar, Dragan Marušič, 2020, original scientific article Abstract: A step forward is made in a long standing Lovász problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes arising from the group action of the projective special linear group PSL▫$(2, q^2)$▫ on cosets of its subgroup isomorphic to the projective general linear group PGL$(2, q)$ contains a Hamilton cycle.
Keywords: vertex-transitive graph, Hamilton cycle, automorphism group, orbital graph
Published in RUP: 20.07.2020; Views: 3120; Downloads: 56
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