428 KB46 KB129 KB3 KB200820252023Mirafzal, Seyed Mortezaštevilka:2letnik:23P2.06 (16 str.)DOI:10.26493/1855-3974.2621.26fCOBISSID_HOST:151635459ISSN:1855-3966URN:URN:NBN:SI:doc-2IC0BDX5enUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaautomorphic graphautomorphism groupavtomorfni grafdistance-transitive graphgrupa avtomorfizmovhiperkockahypercubeJohnson graphJohnsonov grafkvadrat grafarazdaljno tranzitiven grafsquare of a graphSome remarks on the square graph of the hypercube|Let ?$\Gamma = (V, E)$? be a graph. The square graph ?$\Gamma^2$? of the graph ?$\Gamma$? is the graph with the vertex set ?$V (\Gamma^2) = V$? in which two vertices are adjacent if and only if their distance in ?$\Gamma$? is at most two. An interesting property of the square graph of the hypercube ?$Q_n$? is that it is highly symmetric and panconnected. In this article, the author investigates some algebraic properties of the graph ?$Q^2_n$?. He shows that the graph ?$Q^2_n$? is distance-transitive and that the graph ?$Q^2_n$? is an imprimitive distance-transitive graph if and only if ?$n$? is an odd integer and determines the spectrum of the graph ?$Q^2_n$?. Finally, the author shows that when ?$n > 2$? is an even integer, then ?$Q^2_n$? is an automorphic graph, that is, ?$Q^2_n$? is a distance-transitive primitive graph which is not a complete or a line graphNaj bo ?$\Gamma = (V, E)$? graf. Kvadratni graf ?$\Gamma^2$? grafa ?$\Gamma$? je graf z množico vozlišč ?$V (\Gamma^2) = V$?, v katerem sta dve vozlišči sosednji, če je njuna razdalja v grafu ?$\Gamma$? največ dve. Kvadratni graf hiperkocke ?$Q_n$? ima določene zanimive lastnosti. Tako je npr. visoko simetričen in vsepovezan. V tem članku raziskujemo nekatere algebraične lastnosti grafa ?$Q^2_n$?. V prvi vrsti pokažemo, da je graf ?$Q^2_n$? razdaljno tranzitiven. Dokažemo tudi, da je graf ?$Q^2_n$? neprimitiven razdaljno tranzitiven graf natanko takrat, ko je ?$n$? sodo število. Določimo tudi spekter grafa ?$Q^2_n$?. Nazadnje dokažemo: če je ?$n > 2$? sodo število, potem je ?$Q^2_n$? avtomorfen graf, kar pomeni, da je ?$Q^2_n$? razdaljno tranzitiven primitiven graf, ki ni ne polni ne povezavni grafTEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije