Structure of thin irreducible modules of a Q-polynomial distance-regular graph
نویسندگان
چکیده
منابع مشابه
On bipartite Q-polynomial distance-regular graphs
Let Γ denote a bipartite Q-polynomial distance-regular graph with vertex set X, diameter d ≥ 3 and valency k ≥ 3. Let RX denote the vector space over R consisting of column vectors with entries in R and rows indexed by X. For z ∈ X, let ẑ denote the vector in RX with a 1 in the z-coordinate, and 0 in all other coordinates. Fix x, y ∈ X such that ∂(x, y) = 2, where ∂ denotes path-length distance...
متن کاملA duality between pairs of split decompositions for a Q-polynomial distance-regular graph
Let Γ denote a Q-polynomial distance-regular graph with diameter D ≥ 3 and standard module V . Recently Ito and Terwilliger introduced four direct sum decompositions of V ; we call these the (μ, ν)–split decompositions of V , where μ, ν ∈ {↓, ↑}. In this paper we show that the (↓, ↓)–split decomposition and the (↑, ↑)–split decomposition are dual with respect to the standard Hermitian form on V...
متن کاملThe subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two
We consider a bipartite distance-regular graph Γ with diameter D ≥ 4, valency k ≥ 3, intersection numbers bi, ci, distance matrices Ai, and eigenvalues θ0 > θ1 > · · · > θD. Let X denote the vertex set of Γ and fix x ∈ X. Let T = T (x) denote the subalgebra of MatX(C) generated by A,E ∗ 0 , E ∗ 1 , . . . , E ∗ D, where A = A1 and E ∗ i denotes the projection onto the i th subconstituent of Γ wi...
متن کاملA characterization of Q-polynomial distance-regular graphs
We obtain the following characterization of Q-polynomial distance-regular graphs. Let Γ denote a distance-regular graph with diameter d ≥ 3. Let E denote a minimal idempotent of Γ which is not the trivial idempotent E0. Let {θ∗ i }i=0 denote the dual eigenvalue sequence for E. We show that E is Q-polynomial if and only if (i) the entry-wise product E ◦ E is a linear combination of E0, E, and at...
متن کاملThe Matching Polynomial of a Distance-regular Graph
A distance-regular graph of diameter d has 2d intersection numbers that determinemany properties of graph (e.g., its spectrum). We show that the first six coefficients of the matching polynomial of a distance-regular graph can also be determined from its intersection array, and that this is the maximum number of coefficients so determined. Also, the converse is true for distance-regular graphs ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Linear Algebra and its Applications
سال: 2010
ISSN: 0024-3795
DOI: 10.1016/j.laa.2010.06.005