Towards a classification of $1$-homogeneous distance-regular graphs with positive intersection number $a_1$

Let $Γ$ be a graph with diameter at least two. Then $Γ$ is said to be $1$-homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices $x$ and $y$ in $Γ$, the distance partition of the vertex set of $Γ$ with respect to both $x$ and $y$ is equitable, and the parameters corresponding to equitable partitions are independent of the choice of $x$ and $y$. Assume that $Γ$ is $1$-homogeneous distance-regular with intersection number $a_1>0$ and diameter $D\geqslant 5$. Define $b=b_1/(θ_1+1)$, where $b_1$ is the intersection number and $θ_1$ is the second largest eigenvalue of $Γ$. We show that if intersection number $c_2$ is at least $2$, then $b\geqslant 1$ and one of the following (i)--(vi) holds: (i) $Γ$ is a regular near $2D$-gon, (ii) $Γ$ is a Johnson graph $J(2D,D)$, (iii) $Γ$ is a halved $\ell$-cube with $\ell \in \{2D,2D+1\}$, (iv) $Γ$ is a folded Johnson graph $\bar{J}(4D,2D)$, (v) $Γ$ is a folded halved $4D$-cube, (vi) the valency of $Γ$ is bounded by a function of $b$. Using this result, we characterize $1$-homogeneous graphs with classical parameters and $a_1>0$, as well as tight distance-regular graphs.

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