On strongly closed subgraphs of highly regular graphs

AbstractA geodetically closed induced subgraph Δ of a graph Γ is defined to be strongly closed if Γi(α) ∩ Γ1(β) stays in Δ for every i and α, β ϵ Δ with ∂(α, β) = i. We study the existence conditions of strongly closed subgraphs in highly regular graphs such as distance-regular graphs or distance-biregular graphs

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

A characterization of the Hamming graph by strongly closed subgraphs

AbstractThe Hamming graph H(d,q) satisfies the following conditions: (i)For any pair (u,v) of vertices there exists a strongly closed subgraph containing them whose diameter is the distance between u and v. In particular, any strongly closed subgraph is distance-regular.(ii)For any pair (x,y) of vertices at distance d−1 the subgraph induced by the neighbors of y at distance d from x is a clique of size a1+1.In this paper we prove that a distance-regular graph which satisfies these conditions is a Hamming graph

Width and dual width of subsets in polynomial association schemes

AbstractThe width of a subset C of the vertices of a distance-regular graph is the maximum distance which occurs between elements of C. Dually, the dual width of a subset in a cometric association scheme is the index of the “last” eigenspace in the Q-polynomial ordering to which the characteristic vector of C is not orthogonal. Elementary bounds are derived on these two new parameters. We show that any subset of minimal width is a completely regular code and that any subset of minimal dual width induces a cometric association scheme in the original. A variety of examples and applications are considered

Distance-Biregular Graphs and Orthogonal Polynomials

This thesis is about distance-biregular graphs– when they exist, what algebraic and structural properties they have, and how they arise in extremal problems. We develop a set of necessary conditions for a distance-biregular graph to exist. Using these conditions and a computer, we develop tables of possible parameter sets for distancebiregular graphs. We extend results of Fiol, Garriga, and Yebra characterizing distance-regular graphs to characterizations of distance-biregular graphs, and highlight some new results using these characterizations. We also extend the spectral Moore bounds of Cioaba et al. to semiregular bipartite graphs, and show that distance-biregular graphs arise as extremal examples of graphs meeting the spectral Moore bound