279 KB25 KB200820252016Mednykh, AlexanderMednykh, Ilyaštevilka:1letnik:10str. 183-192COBISSID:17735513ISSN:1855-3966URN:URN:NBN:SI:doc-8HJJF94NenUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneagrafgraphhipereliptičen grafhomology grouphomološka grupahyperelliptic graphRiemann--Hurwitz formulaRiemann-Hurwitzeva formulaSchreier formulaSchreierjeva formulaOn gamma-hyperellipticity of graphs|The basic objects of research in this paper are graphs and their branched coverings. By a graph, we mean a finite connected multigraph. The genus of a graph is defined as the rank of the first homology group. A graph is said to be ?$\gamma$?-hyperelliptic if it is a two fold branched covering of a genus ?$\gamma$ graph?. The corresponding covering involution is called ?$\gamma$?-hyperelliptic. The aim of the paper is to provide a few criteria for the involution ?$\tau$? acting on a graph ?$X$? of genus ?$g$? to be ?$\gamma$?-hyperelliptic. If $\tau$ has at least one fixed point then the first criterium states that there is a basis in the homology group ?$H_1(X)$? whose elements are either invertible or split into ?$\gamma$? interchangeable pairs under the action of ?$\tau_\ast$?. The second criterium is given by the formula ?$\text{tr}_{\rm{H_1}(X)} (\tau_\ast) = 2 \gamma - g$?. Similar results are also obtained in the case when ?$\tau$? acts fixed point freeOsnovni objekti raziskave v tem članku so grafi in njihovi razvejani krovi. Graf nam tukaj pomeni okrajšavo za: končen povezan multigraf. Rod grafa je definiran kot red prve homološke grupe. Graf se imenuje ?$\gamma$?-hipereliptičen, če ga lahko dobimo kot dvolistni razvejani krov nekega grafa roda ?$\gamma$?. Ustrezni krovni involuciji pravimo ?$\gamma$?-hipereliptična involucija. Namen članka je najti nekaj kriterijev, ki povedo, kdaj je involucija ?$\tau$?, delujoča na grafu ?$X$? roda ?$g$?, ?$\gamma$?-hipereliptična. Če ima ?$\tau$? vsaj eno fiksno točko, potem prvi kriterij pove, da obstaja baza v homološki grupi ?$H_1(X)$)?, katere elementi so bodisi obrnljivi bodisi se razcepijo v ?$\gamma$?-izmenljive pare pod delovanjem ?$\tau_\ast$?. Drugi kriterij je dan s formulo ?$\text{tr}_{\rm{H_1}(X)} (\tau_\ast) = 2 \gamma - g$?. Podobne rezultate dobimo tudi v primeru, ko ?$\tau$? deluje brez fiksnih točkTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije