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

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

Krein parameters and antipodal tight graphs with diameter 3 and 4

AbstractWe determine which Krein parameters of nonbipartite antipodal distance-regular graphs of diameter 3 and 4 can vanish, and give combinatorial interpretations of their vanishing. We also study tight distance-regular graphs of diameter 3 and 4. In the case of diameter 3, tight graphs are precisely the Taylor graphs. In the case of antipodal distance-regular graphs of diameter 4, tight graphs are precisely the graphs for which the Krein parameter q114 vanishes

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

Incidence geometry from an algebraic graph theory point of view

The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense. The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte. Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them. Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight. A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5. Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems. The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch