Problem, ki ga obravnavamo, sprašuje, kateri razdaljno-regularni grafi z majhno valenco so Cayleyjevi grafi. Poišcemo Cayleyjeve grafe med razdaljno-regularnimi grafi z valenco kvecjemu 4, med ...razdaljno-regularnimi grafi za znane domnevne presečne tabele z valenco 5 in med vsemi razdaljno-regularnimi grafi z ožino 3 in valenco 6 ali 7. Ugotovimo, da so incidenčni grafi Desarguesovih afinih ravnin brez enega razreda vzporednih premic Cayleyjevi grafi. Pokažemo, da incidenčni grafi znanih posplošenih šestkotnikov niso Cayleyjevi grafi; prav tako niso Cayleyjevi nekateri drugi razdaljno-regularni grafi, ki izhajajo iz majhnih posplošenih štirikotnikov ali šestkotnikov. Med ''izjemnimi'' razdaljno-regularnimi grafi z majhno valenco smo našli dva, ki sta Cayleyjeva, in sicer sta to Armanios-Wellsov graf in Kleinov graf.
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