Distance-regular graphs with complete multipartite ▫$\mu$▫-graphs and AT4 family
Jurišić, Aleksandar ; Koolen, Jack
Naj bo ▫$\Gamma$▫ antipoden razdaljno-regularen graf premera 4 z lastnimi vrednostmi ▫$\theta_1 > \theta_2 > \theta_3 > \theta_4$▫. Potem je njegov Kreinov parameter ▫$q_{11}^4$▫ enak 0 natanko ...tedaj, ko je graf ▫$\Gamma$▫ tesen v smislu Jurišića, Koolna in Terwilligerja, in nadalje natanko tedaj, ko je ▫$\Gamma$▫ lokalno krepko regularen z netrivialnimi lastnimi vrednostmi ▫$p:=\theta_2$▫ in ▫$-q:=\theta_3$▫. V tem primeru lahko parametriziramo presečno zaporedje grafa ▫$\Gamma$▫ s ▫$p$▫, ▫$q$▫ in velikostjo ▫$r$▫ antipodnega razreda grafa ▫$\Gamma$▫. Naj bo ▫$\Gamma$▫ antipodni tesni graf premera 4, oznaka AT4▫$(p,q,r)$▫, in naj bo ▫$\mu$▫-graf tisti graf, ki ga inducirajo skupni sosedi dveh vozlišč na razdalji 2. Potem pokažemo, da so vsi ▫$\mu$▫-grafi v ▫$\Gamma$▫ polni večdelni, če in samo če je ▫$\Gamma$▫ enak AT4▫$(sq,q,q)$▫ za neko naravno število ▫$s$▫. Kot posledico izpeljemo nove neeksistenčne pogoje za grafe iz družine AT4, katerih ▫$\mu$▫-grafi niso polni večdelni. Druga zanimiva uporaba naših rezultatov pa je dokaz, da so ▫$\mu$▫-grafi razdaljno-regularnega grafa z enakim presečnim zaporedjem, kot ga ima Pattersonov graf, polni dvodelni grafi ▫$K_{4,4}$▫.
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