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 $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$ and diameter $D$. Let $A$ denote the adjacency matrix of $\Gamma$. For a vertex $x\in X$ and for $0 \leq i \leq D$, let $E^*_i(x)$ denote the projection matrix to the $i$th subconstituent space of $\Gamma$ with respect to $x$. The Terwilliger algebra $T(x)$ of $\Gamma$ with respect to $x$ is the semisimple subalgebra of $\mathrm{Mat}_X(\mathbb{C})$ generated by $A, E^*_0(x), \ldots, E^*_D(x)$. Let $V=\mathbb{C}^X$ denote a $\mathbb{C}$-vector space consisting of complex column vectors with rows indexed by $X$. We say $\Gamma$ is pseudo-vertex-transitive whenever for any vertices $x,y \in X$, both (i) the Terwilliger algebras $T(x)$ and $T(y)$ of $\Gamma$ are isomorphic; and (ii) there exists a $\mathbb{C}$-vector space isomorphism $\rho:V\to V$ such that $(\rho A - A \rho)V=0$ and $(\rho E^*_i(x) - E^*_i(y)\rho)V=0$ for all $0\leq i \leq D$. In this paper we discuss pseudo-vertex transitivity for distance-regular graphs with diameter $D\in \{2,3,4\}$. In the case of diameter two, a strongly regular graph $\Gamma$ is thin, and $\Gamma$ is pseudo-vertex-transitive if and only if every local graph of $\Gamma$ 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