Jurišić, Aleksandar; Terwilliger, Paul

Pseudo 1-homogeneous distance-regular graphs

J. Algebr. Comb. 28(4), 509-529 (2008)
DOI: 10.1007/s10801-007-0115-y


Summary: Let $\Gamma $be a distance-regular graph of diameter $d\geq 2$ and $a _{1}\neq 0$. Let $\theta $ be a real number. A pseudo cosine sequence for $\theta $ is a sequence of real numbers $\sigma _{0},\cdots , \sigma _{ d }$ such that $\sigma _{0}=1$ and $c _{ i } \sigma _{ i - 1}+ a _{ i } \sigma _{ i }+ b _{ i } \sigma _{ i+1}= \theta \sigma _{ i }$ for all $i\in {0,\cdots , d - 1}$. Furthermore, a pseudo primitive idempotent for $\theta $ is $E _{ \theta }= s \sum _{ i=0} ^{ d } \sigma _{ i } A _{ i }$, where $s$ is any nonzero scalar. Let [^$( v)$] hatv be the characteristic vector of a vertex $v\in V\Gamma $. For an edge $xy$ of $\Gamma $and the characteristic vector $w$ of the set of common neighbours of $x$ and $y$, we say that the edge $xy$ is tight with respect to $\theta $ whenever $\theta \neq k$ and a nontrivial linear combination of vectors $E[^( x)]$ Ehatx , $E[^( y)]$ Ehaty and $Ew$ is contained in Span[^$( z)] | z$ Ĩ $V$ G, P$( z, x)= d$= P( $z, y$) mathrmSpan{hatz$\mid z\in $VGamma, $partial(z,x)=$d=$partial(z,y)$} . When an edge of $\Gamma $is tight with respect to two distinct real numbers, a parameterization with $d+1$ parameters of the members of the intersection array of $\Gamma $is given (using the pseudo cosines $\sigma _{1},\cdots , \sigma _{ d }$, and an auxiliary parameter $\epsilon $). Let $S$ be the set of all the vertices of $\Gamma $that are not at distance $d$ from both vertices $x$ and $y$ that are adjacent. The graph $\Gamma $is pseudo 1 -homogeneous with respect to $xy$ whenever the distance partition of $S$ corresponding to the distances from $x$ and $y$ is equitable in the subgraph induced on $S$. We show $\Gamma $is pseudo 1-homogeneous with respect to the edge $xy$ if and only if the edge $xy$ is tight with respect to two distinct real numbers. Finally, let us fix a vertex $x$ of $\Gamma $. Then the graph $\Gamma $is pseudo 1-homogeneous with respect to any edge $xy$, and the local graph of $x$ is connected if and only if there is the above parameterization with $d+1$ parameters $\sigma _{1},\cdots , \sigma _{ d }, \epsilon $ and the local graph of $x$ is strongly regular with nontrivial eigenvalues $a _{1} \sigma /(1+ \sigma )$ and $( \sigma _{2} - 1)/( \sigma - \sigma _{2})$.


keywords distance-regular graphs, primitive idempotents, cosine sequence, locally strongly regular, 1-homogeneous property, tight distance-regular graph, pseudo primitive idempotent, tight edges, pseudo 1-homogeneous