Abstract
In the survey paper by Van Dam, Koolen and Tanaka (2016), they asked to classify the thin $Q$-polynomial distance-regular graphs. In this paper, we show that a thin distance-regular graph with the same intersection numbers as a Grassmann graph $J_q(n, D)~ (n \geqslant 2D)$ is the Grassmann graph if $D$ is large enough.
Highlights
A finite connected graph Γ with vertex set V (Γ) and path-length distance function ∂ is called distance-regular if, for any vertices x, y ∈ V (Γ) and any non-negative integers i, j, the number phij of vertices at distance i from x and distance j from y depends only on i, j and h := ∂(x, y), and does not depend on the particular choice of x and y
We show that a thin distance-regular graph with the same intersection numbers as a Grassmann graph Jq(n, D) (n 2D) is the Grassmann graph if D is large enough
A distance-regular graph Γ of diameter D (D := max{∂(x, y) | x, y ∈ V (Γ)}) is said to have classical parameters (D, b, α, β) if its intersection numbers can be expressed in terms of these four classical parameters
Summary
A finite connected graph Γ with vertex set V (Γ) and path-length distance function ∂ is called distance-regular if, for any vertices x, y ∈ V (Γ) and any non-negative integers i, j, the number phij of vertices at distance i from x and distance j from y depends only on i, j and h := ∂(x, y), and does not depend on the particular choice of x and y. A distance-regular graph Γ of diameter D (D := max{∂(x, y) | x, y ∈ V (Γ)}) is said to have classical parameters (D, b, α, β) if its intersection numbers can be expressed in terms of these four classical parameters (see Subsection 2.4). The Grassmann graph Jq(n, D) is a distance-regular graph with classical parameters This paper shows that the Grassmann graphs with large diameter are characterized by their intersection numbers as thin distance-regular graphs. Metsch [14], Gavrilyuk and Koolen [8] showed that the Grassmann graph Jq(n, D) is uniquely determined by its intersection numbers in many cases. We will use those results later in the paper to show the main result.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
