Taut distance-regular graphs of odd diameter

MacLean, Mark S.

Journal of Algebraic Combinatorics, Tome 17 (2003) no. 2, pp. 125-147.

Résumé

Summary: Let _boxclose _i - _i - 1 = , = begingathered frac$\sigma $_i + 1 - $\alpha \sigma }}{{\sigma \sigma $_i - $\sigma $_i - 1 = frac$\beta \rho $_i - $\rho $_i - 1 $\rho \rho $_i - $\rho $_i - 1 , $\hfill $ frac$\rho $_i + 1 - $\beta \rho $_i $\rho \rho $_i - $\rho $_i - 1 = frac$\alpha \sigma $_i - $\sigma $_i - 1 $\sigma \sigma $_i - $\sigma $_i - 1 $\hfill $ endgathered for 1 $leileD - 1$, where $sgr = sgr _{1}, rgr = rgr _{1}$. Using these equations, we recursively obtain $sgr _{0}, sgr _{1}, \dots , sgr _{D}$ and $rgr _{0}, rgr _{1}, \dots , rgr_{ D }$ in terms of the four real scalars $sgr, rgr, agr, beta$. From this we obtain all intersection numbers of $Gamma$ in terms of $sgr, rgr, agr, beta$. We showed in an earlier paper that the pair $E _{1}, E _{d}$ is taut, where $d = ( D - 1)/2$. Applying our results to this pair, we obtain the intersection numbers of $Gamma$ in terms of $k, mgr, theta _{1}, theta _{d}$, where $mgr$ denotes the intersection number $c _{2}$. We show that if $Gamma$ is taut and $D$ is odd, then $Gamma$ is an antipodal 2-cover.

Détail

Classification : homogeneous, bipartite, distance-regular, graphs, in, [14]
Keywords: distance-regular graph, association scheme, bipartite graph, tight graph, taut graph