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GI-graphs: a new class of graphs with many symmetries
Conder, Marston D. E. ; Pisanski, Tomaž ; Žitnik, Arjana
Vpeljemo novo družino ▫$GI$▫-grafov, ki posplošijo družino posplošenih Petersenovih grafov. Za vsako naravno število ▫$n$▫ in poljubno zaporedje ▫$j_0,j_1, \dots ,j_{t-1}$▫ naravnih števil po modulu ...▫$n$▫ je ▫$GI$▫-graf ▫$GI(n;j_0,j_1, \dots ,j_{t-1})$▫ ▫$(t+1)$▫-valenten graph na množici vozlišč ▫${\mathbb{Z}}_t \times {\mathbb{Z}}_n$▫, s povezavami dveh tipov: (1) povezave od ▫$(s,v)$▫ do ▫$(s',v)$▫, za vse različne ▫$s,s' \in {\mathbb{Z}}_{t}$▫ in vse ▫$v \in {\mathbb{Z}}_n$▫, (2) povezave od ▫$(s,v)$▫ do ▫$(s,v + j_s)$▫ in ▫$(s,v - j_s)$▫, za vse ▫$s \in {\mathbb{Z}}_{t}$▫ in ▫$v \in {\mathbb{Z}}_n$▫. Klasificiramo avtomorfizme ▫$GI$▫-grafov, kar nam omogoči, da opišemo grupo avtomomorfizmov za vsakega od ▫$GI$▫-grafov in določimo, kateri ▫$GI$▫-grafi so točkovno tranzitivni ali Cayleyevi. ▫$GI$▫-graf je lahko povezavno tranzitiven le za ▫$t \leq 3$▫. Najdemo tudi vložitev grafa ▫$GI(7;1,2,3)$▫ v ravnino, pri kateri imajo vse povezave isto dolžino.
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