Asymptotic properties of Fibonacci cubes and Lucas cubes
Klavžar, Sandi ; Mollard, Michel
Dokazano je, da imajo Fibonaccijeve kocke in Lucasove kocke asimptotično povprečno ekscentričnost enako ▫$(5+\sqrt 5)/10$▫ in asimptotično povprečno stopnjo enako ▫$(5-\sqrt 5)/5$▫. Vpeljano je novo ...označevanje Fibonaccijevih dreves in dokazano je, da je ekscentričnost vozlišča dane Fibonaccijeve kocke enaka globini prirejenega lista v pripadajočem Fibonaccijevem drevesu. Vpeljana in študirana je hiperkockina gostota. Za Fibonaccijeve kocke in Lucasove kocke je dokazano, da je ta gostota enaka ▫$(1-1/\sqrt 5)/\log_2\varphi$▫, kjer je ▫$\varphi$▫ razmerje zlatega reza. S pomočjo kartezičnega produkta grafov so konstruirane družine grafov s fiksno, neničelno gostoto. Dokazano je tudi, da povprečno razmerje števila Fibonaccijevih nizov, ki imajo 0 oz. 1 na danem mestu, in kjer je povprečje narejeno preko vseh mest, konvergira k ▫$\varphi^2$▫. Podobno je narejeno tudi za Lucasove kocke.
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