On the split structure of lifted groups

Let ▫$wp colon tilde{X} to X$▫ be a regular covering projection of connected graphs with the group of covering transformations ▫$rm{CT}_wp$▫ being abelian. Assuming that a group of automorphisms ▫$G le rm{Aut} X$▫ lifts along $wp$ to a group ▫$tilde{G} le rm{Aut} tilde{X}$▫, the problem whether the corresponding exact sequence ▫$rm{id} to rm{CT}_wp to tilde{G} to G to rm{id}$▫ splits is analyzed in detail in terms of a Cayley voltage assignment that reconstructs the projection up to equivalence. In the above combinatorial setting the extension is given only implicitly: neither ▫$tilde{G}$▫ nor the action ▫$Gto rm{Aut} rm{CT}_wp$▫ nor a 2-cocycle ▫$G times G to rm{CT}_wp$▫, are given. Explicitly constructing the cover ▫$tilde{X}$▫ together with ▫$rm{CT}_wp$▫ and ▫$tilde{G}$▫ as permutation groups on ▫$tilde{X}$▫ is time and space consuming whenever ▫$rm{CT}_wp$▫ is largethus, using the implemented algorithms (for instance, HasComplement in Magma) is far from optimal. Instead, we show that the minimal required information about the action and the 2-cocycle can be effectively decoded directly from voltages (without explicitly constructing the cover and the lifted group)one could then use the standard method by reducing the problem to solving a linear system of equations over the integers. However, along these lines we here take a slightly different approach which even does not require any knowledge of cohomology. Time and space complexity are formally analyzed whenever ▫$rm{CT}_wp$▫ is elementary abelian.Naj bo ▫$wp colon tilde{X} to X$▫ regularna krovna projekcija povezanih grafov, grupa krovnih transformacij ▫$rm{CT}_wp$▫ pa naj bo abelova. Ob predpostavki, da se grupa avtomorfizmov ▫$G le rm{Aut} X$▫ dvigne vzdolž ▫$wp$▫ do grupe ▫$tilde{G} le rm{Aut} tilde{X}$▫, podrobno analiziramo problem, ali se ustrezno eksaktno zaporedje ▫$rm{id} to rm{CT}_wp to tilde{G} to G to rm{id}$▫ razcepi glede na Cayleyevo dodelitev napetosti, ki rekonstruira projekcijo do ekvivalence natančno. V gornjem kombinatoričnem sestavu je razširitev podana samo implicitno: podani niso ne ▫$tilde{G}$▫ ne delovanje ▫$Gto rm{Aut} rm{CT}_wp$▫ ne 2-kocikel ▫$G times G to rm{CT}_wp$▫. Eksplicitno konstruiranje krova ▫$tilde{X}$▫ ter ▫$rm{CT}_wp$▫ in ▫$tilde{G}$▫ kot permutacijskih grup na ▫$tilde{X}$▫ je časovno in prostorsko zahtevno vselej, kadar je ▫$rm{CT}_wp$▫ veliktako je uporaba implementiranih algoritmov (na primer, HasComplement v Magmi) vse prej kot optimalna. Namesto tega pokažemo, da lahko najnujnejšo informacijo o delovanju in 2-kociklu učinkovito izluščimo neposredno iz napetosti (ne da bi eksplicitno konstruirali krov in dvignjeno grupo)zdaj bi bilo mogoče uporabiti standardno metodo reduciranja problema na reševanje sistema linearnih enačb nad celimi števili. Vendar tukaj uberemo malce drugačen pristop, ki sploh ne zahteva nobenega poznavanja kohomologije. Časovno in prostorsko zahtevnost formalno analiziramo za vse primere, ko je ▫$rm{CT}_wp$▫ elementarna abelova

Regular maps with nilpotent automorphism groups

AbstractWe study regular maps with nilpotent automorphism groups in detail. We prove that every nilpotent regular map decomposes into a direct product of maps H×K, where Aut(H) is a 2-group and K is a map with a single vertex and an odd number of semiedges. Many important properties of nilpotent maps follow from this canonical decomposition, including restrictions on the valency, covalency, and the number of edges. We also show that, apart from two well-defined classes of maps on at most two vertices and their duals, every nilpotent regular map has both its valency and covalency divisible by 4. Finally, we give a complete classification of nilpotent regular maps of nilpotency class 2

The strongly distance-balanced property of the generalized Petersen graphs

A graph ▫$X$▫ is said to be strongly distance-balanced whenever for any edge ▫$uv$▫ of ▫$X$▫ and any positive integer ▫$i$▫, the number of vertices at distance ▫$i$▫ from ▫$u$▫ and at distance ▫$i + 1$▫ from ▫$v$▫ is equal to the number of vertices at distance ▫$i + 1$▫ from ▫$u$▫ and at distance ▫$i$▫ from ▫$v$▫. It is proven that for any integers ▫$k ge 2$▫ and ▫$n ge k^2 + 4k + 1$▫, the generalized Petersen graph GP▫$(n, k)$▫ is not strongly distance-balanced

Semiregular automorphisms of vertex-transitive graphs of certain valencies

AbstractIt is shown that a vertex-transitive graph of valency p+1, p a prime, admitting a transitive action of a {2,p}-group, has a non-identity semiregular automorphism. As a consequence, it is proved that a quartic vertex-transitive graph has a non-identity semiregular automorphism, thus giving a partial affirmative answer to the conjecture that all vertex-transitive graphs have such an automorphism and, more generally, that all 2-closed transitive permutation groups contain such an element (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69–81; P.J. Cameron (Ed.), Problems from the Fifteenth British Combinatorial Conference, Discrete Math. 167/168 (1997) 605–615])

Invariant subspaces, duality, and covers of the Petersen graph

AbstractA general method for finding elementary abelian regular covering projections of finite connected graphs is applied to the Petersen graph. As a result, a complete list of pairwise non-isomorphic elementary abelian covers admitting a lift of a vertex-transitive group of automorphisms is given. The resulting graphs are explicitly described in terms of voltage assignments