9 research outputs found
An inequality on the cosines of a tight distance-regular graph
AbstractLet Γ denote a distance-regular graph with diameter d⩾3, and assume Γ is tight in the sense of Jurišić et al. [J. Algebraic Combin. 12 (2000) 163–197]. Let θ denote the second largest or the smallest eigenvalue of Γ. We obtain an inequality involving the first, second and third cosines associated with θ. We investigate the relationship between equality being attained and the existence of dual bipartite Q-polynomial structures on Γ
A duality between pairs of split decompositions for a Q-polynomial distance-regular graph
AbstractLet 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. We also show that the (↓,↑)-split decomposition and the (↑,↓)-split decomposition are dual with respect to the standard Hermitian form on V
Remarks on pseudo-vertex-transitive graphs with small diameter
Let denote a -polynomial distance-regular graph with vertex set
and diameter . Let denote the adjacency matrix of . For a
vertex and for , let denote the projection
matrix to the th subconstituent space of with respect to . The
Terwilliger algebra of with respect to is the semisimple
subalgebra of generated by . Let denote a -vector space consisting
of complex column vectors with rows indexed by . We say is
pseudo-vertex-transitive whenever for any vertices , both (i) the
Terwilliger algebras and of are isomorphic; and (ii)
there exists a -vector space isomorphism such that
and for all . In this paper we discuss pseudo-vertex transitivity for
distance-regular graphs with diameter . In the case of diameter
two, a strongly regular graph is thin, and is
pseudo-vertex-transitive if and only if every local graph of has the
same spectrum. In the case of diameter three, Taylor graphs are thin and
pseudo-vertex-transitive. In the case of diameter four, antipodal tight graphs
are thin and pseudo-vertex-transitive.Comment: 29 page
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Tight distance-regular graphs and the Q-polynomial property
Let Γ denote a distance-regular graph with diameter d ≥ 3, and assume Γ is tight (in the sense of Jurišić, Koolen and Terwilliger). Let θ denote the second largest or smallest eigenvalue of Γ, and let σ0, σ1, . . . , σd denote the associated cosine sequence. We obtain an inequality involving σ0, σ1, . . . , σd for each integer i (1 ≤ i ≤ d - 1), and we show equality for all i is closely related to Γ being Q-polynomial with respect to θ. We use this idea to investigate the Q-polynomial structures in tight distance-regular graphs. © Springer-Verlag 2001