Spectra of graphs and closed distance magic labelings
Anholcer, Marcin ; Cichacz, Sylwia ; Peterin, Iztok
Naj bo ▫$G=(V,E)$▫ graf z ▫$n$▫ vozlišči. Zaprta razdaljno čarobna označitev grafa ▫$G$▫ je bijekcija ▫$\ell \colon V(G)\rightarrow \{1,\ldots ,n\}$▫, za katero obstaja naravno število ▫$k$▫, da ...velja ▫$\sum_{x\in N[v]}\ell (x)=k$▫ za vsak ▫$v\in V $▫, kjer je ▫$N[v]$▫ zaprta okolica vozlišča ▫$v$▫. V tem delu obravnavamo zaprte razdaljno čarobne grafe iz algebrajskega stališča. Posebej analiziramo relacije med zaprtimi razdaljno čarobnimi označitvami in spektrom grafa ▫$G$▫. Te rezultate nato uporabimo na krepkem produktu, kjer je en faktor poln graf ali cikel in na cirkularnih grafih. Končamo z problemom iz teorije števil, katerega rešitev nam poraja novo družino zaprtih razdaljno čarobnih grafov, ki so sorodni s krepkimi produkti.
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