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Triangle-free distance-regular graphs with an eigenvalue multiplicity equal to their valency and diameter 3
Jurišić, Aleksandar ; Koolen, Jack ; Žitnik, Arjana
Študiramo razdaljno-regularne grafe premera 3, ki so brez trikotnikov in imajo lastno vrednost ▫$\theta$▫, katere večkratnost je enaka valenci grafa. Naj bo ▫$\Gamma$▫ tak graf. Najprej pokažemo da ...je ▫$\theta = -1$▫ natanko tedaj, ko je graf ▫$\Gamma$▫ antipoden. Nato privzamemo, da je graf ▫$\Gamma$▫ primitiven. Dokažemo, da je graf ▫$\Gamma$▫ formalno sebi-dualen (ter zato tudi ▫$Q$▫-polinomski in 1-homogen),vse njegove lastne vrednosti so celoštevilske, lastna vrednost, katere večkratnost je enaka valenci, pa je bodisi druga največja ali pa najmanjša. Naj bosta ▫$x,y \in V\Gamma$▫ sosednjiin ▫$z\in \Gamma_2(x)\cap Gamma_2(y)$▫. Potem je presečno število ▫$\tau_2:=|\Gamma(z)\cap \Gamma_3(x)\cap \Gamma_3(y)|$▫ neodvisno od izbire vozlišč ▫$x$▫, ▫$y$▫ in ▫$z$▫. V primeru grafa odsekov dvakrat skrajšane binarne Golayeve kode velja ▫$b_2=\tau_2$▫. Klasificiramo vse grafe z ▫$b_2=\tau_2$▫ in ugotovimo, da je pravkar omenjeni graf edini tak primer. S tem pokažemo, da ne obstaja graf katerega presečno zaporedje bi bilo element neke neskončne družine (dotlej) dopustnih presečnih zaporedij.
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