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1-homogeneous graphs with Coctail Party ▫$\mu$▫-graphs
Jurišić, Aleksandar ; Koolen, Jack
Na bo ▫$\Gamma$▫ 1-homogen graf v smislu Nomure z diametrom ▫$d \ge 2$▫, tj. za vsako povezavo ▫$xy$▫ grafa ▫$\Gamma$▫ je razdaljna particija ▫$$\{\{z \in V(\Gamma)| {\rm dist}(x,y)=i, {\rm ...dist}(x,z)=j\} | 0 \le i,j \le {\rm diam}(\Gamma)\}$$▫ ekvitabilna in njeni parametri niso odvisni od povezave ▫$xy$▫. Potem je ▫$\Gamma$▫ razdaljno regularen graf in tudi lokalno krepko regularen. Lokalni grafi imajo enake parametre. Označimo jih z ▫$(v',k', \lambda',\mu')$▫. Očitno velja ▫$v'=k$▫, ▫$k'=a_1$▫, ▫$(v'-k'-1)\mu' = k'(k'-1-\lambda')$▫ in ▫$c_2 \ge \mu'+1$▫, saj je vsak ▫$\mu$▫-graf regularen graf z valenco ▫$\mu'$▫. Poznamo številne zanimive primere 1-homogenih grafov, katerih ▫$\mu$▫-grafi so izomorfni nekaj kopijam polnih večpartitnih grafov. Na primer, če velja enakost v zgornji oceni in je ▫$c_2 \ne 1$▫, potem je ▫$\Gamma$▫ Terwilligerjev graf (tj. vsi njegovi ▫$\mu$▫-grafi so polni) in te grafe smo že klasificirali. V članku študiramo primer ▫$c_2 = \mu'+2 \ge 3$▫, to je primer, ko so ▫$\mu$▫-grafi grafa ▫$\Gamma$▫ Coktail party grafi in izpeljemo, da velja bodisi ▫$\lambda'=0$▫, ▫$\mu'=2$▫ ali pa je graf ▫$\Gamma$▫ eden od naslednjih grafov: (i) Johnsonov graf ▫$J(2m,m)$▫ ▫$m \ge 2$▫, (ii) Prepognjen Johnsonov graf ▫$\overline{J}(4m,2m}$▫ ▫$m \ge 3$▫, (iii) polovična ▫$m$▫-kocka, ▫$m \ge 4$▫, (iv) prepognjena polovična ▫$(2m)$▫-kocka ▫$m \ge 5$▫, (v) Cocktail Party graf ▫$K_{m \times 2}$▫, ▫$m \ge 3$▫, (vi) Schläflijev graf , (vii) Gossetov graf.
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