Towards a classification of $1$-homogeneous distance-regular graphs with positive intersection number $a_1$
Jack H. Koolen, Mamoon Abdullah, Brhane Gebremichel, Jae-Ho Lee
TL;DR
The paper addresses the classification of $1$-homogeneous distance-regular graphs with $a_1>0$ and diameter $D\ge5$ by introducing the key parameter $b=\dfrac{b_1}{\theta_1+1}$ and proving that if the local intersection number $c_2\ge2$, then $b\ge1$ and the graph must belong to one of six structural families or have valency bounded by a polynomial in $b$. The approach combines the CAB$_i$ framework, analysis of local graphs (often strongly regular), and detailed casework on local connectivity vs. disconnectedness to derive a finite list of possibilities, including regular near $2D$-gons, Johnson graphs $J(2D,D)$, halved $\ell$-cubes, folded Johnson graphs $\bar{J}(4D,2D)$, folded halved $4D$-cubes, or a valency bound $k\le F(b)$ with an explicit polynomial $F$. The work further yields classifications for 1-homogeneous graphs with classical parameters and for tight distance-regular graphs, thereby advancing the broader program of understanding these highly symmetric graphs and supporting conjectures about tight graphs. The results identify explicit infinite families (Johnson graphs, halved/folded cubes) and provide sharp parametric bounds that constrain the landscape of such graphs in higher diameter regimes.
Abstract
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.