An important property of strongly regular graphs is that the second subconstituent of any primitive strongly regular graph is always connected. Brouwer asked to what extent this statement can be generalized to distanceregular graphs. In this paper, we show that if γ is any vertex of a distanceregular graph Γ and t is the index where the standard sequence corresponding to the second largest eige...

It is proved that a graph K has an embedding as a regular map on some closed surface if and only if its automorphism group contains a subgroup G which acts transitively on the oriented edges of K such that the stabiliser G e of every edge e is dihedral of order 4 and the stabiliser G v of each vertex Š is a dihedral group the cyclic subgroup of index 2 of which acts regularly on the edges incid...

We consider in this paper rooted labeled directed graphs as models for semistructured data. A path inclusion constraint, in such graphs, is an expression p q where p and q are regular expressions over the alphabet of labels. An path inclusion constraint p q is satisfied by a rooted labeled directed graph if the set of nodes reached, from the root, by the words of the language described by p is ...

A spin model is a square matrix that encodes the basic data for a statistical mechanical construction of link invariants due to V.F.R. Jones. Every spin model W is contained in a canonical Bose-Mesner algebra N (W ). In this paper we study the distance-regular graphs whose Bose-Mesner algebra M satisfies W ∈ M ⊆ N (W ). Suppose W has at least three distinct entries. We show that is 1-homogeneou...

We consider a distance-regular graph ? with diameter d 3 and eigenvalues k = 0 > 1 > > d. We show the intersection numbers a 1 ; b 1 satisfy (a 1 + 1) 2 : We say ? is tight whenever ? is not bipartite, and equality holds above. We characterize the tight property in a number of ways. For example, we show ? is tight if and only if the intersection numbers are given by certain rational expressions...

We give a sufficient condition for a distance-regular graph to be Hamiltonian. In particular, the Petersen graph is the only connected nonHamiltonian strongly regular graph on fewer than 99 vertices.