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  • A local approach to 1-homogeneous graphs
    Jurišić, Aleksandar ; Koolen, Jack
    Let ▫$\Gamma$▫ be a distance-regular graph with diameter ▫$d$▫. For vertices ▫$x$▫ and ▫$y$▫ of ▫$\Gamma$▫ at distance ▫$i$▫, ▫$1 \le i \le d$▫, we define the sets ▫$C_i = \Gamma_{i-1}(x) \cap ... \Gamma(y)$▫, ▫$A_i(x,y) = \Gamma_i(x) \cap \Gamma(y)$▫ and ▫$B_i(x,y) = \Gamam_{i+1}(x) \cap \Gamma(y)$▫. Then we say ▫$\Gamma$▫ has CAB▫$_j$▫ property, if the partition ▫$CAB_i(x,y) = \{C_I(x,y), A_i(x,y), B_i(x,y)\}$▫ of the local graph of ▫$y$▫ is equitable for each pair of vertices ▫$x$▫ and ▫$y$▫ of ▫$\Gamma$▫ at distance ▫$i<j$▫. We show that if ▫$\Gamam$▫ has a CAB▫$_j$▫ property then the parameteres of the equitable partitions ▫$CAB_i(x,y)$▫ do not depend on the choice of vertices ▫$x$▫ and ▫$y$▫ at distance ▫$i$▫ for all ▫$i>j$▫. The graph ▫$\Gamma$▫ has the CAB property if it has the CAB▫$_d$▫ property. We show the equivalence of CAB property and 1-homogeneous property in a distance-regular graph with ▫$a_1 \ne 0$▫. Finally, we classify the 1-homogeneous Terwilliger graphs with ▫$c_2 \ge 2$▫.
    Vir: Preprint series. - ISSN 1318-4865 (Vol. 37, [št.] 663, 1999, str. 1-20)
    Vrsta gradiva - članek, sestavni del
    Leto - 1999
    Jezik - angleški
    COBISS.SI-ID - 8982617

vir: Preprint series. - ISSN 1318-4865 (Vol. 37, [št.] 663, 1999, str. 1-20)

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