By modifying a construction for Hadamard (Menon) difference sets we construct two infinite families of negative Latin square type partial difference sets in groups of the form Z3 × Zp where p is any odd prime. One of these families has the wellknown Paley parameters, which had previously only been constructed in p-groups. This provides new constructions of Hadamard matrices and implies the exis...

in this article we study two different generalizations of von neumann regularity, namely strong topological regularity and weak regularity, in the banach algebra context. we show that both are hereditary properties and under certain assumptions, weak regularity implies strong topological regularity. then we consider strong topological regularity of certain concrete algebras. moreover we obtain ...

Let Γ denote a D-bounded distance-regular graph, where D ≥ 3 is the diameter of Γ. For 0 ≤ s ≤ D − 3 and a weak-geodetically closed subgraph ∆ of Γ with diameter s, define a graph G(∆) whose vertex set is the collection of all weak-geodetically closed subgraphs of diameter s+2 containing ∆, and vertex Ω is adjacent to vertex Ω′ in G if and only if Ω∩Ω′ as a subgraph of Γ has diameter s+1. We sh...

We analyze when the Moore–Penrose inverse of the combinatorial Laplacian of a distance– regular graph is a M–matrix; that is, it has non–positive off–diagonal elements. In particular, our results include some previously known results on strongly regular graphs.

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...

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.

Antipodal covers of strongly regular graphs which are not necessarily distance-regular are studied. The structure of short cycles in an antipodal cover is considered. In most cases, this provides a tool to determine if a strongly regular graph has an antipodal cover. In these cases, covers cannot be distance-regular except when they cover a complete bipartite graph. A relationship between antip...

Let be an antipodal distance-regular graph of diameter 4, with eigenvalues θ0 > θ1 > θ2 > θ3 > θ4. Then its Krein parameter q4 11 vanishes precisely when is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when is locally strongly regular with nontrivial eigenvalues p := θ2 and −q := θ3. When this is the case, the intersection parameters of can be parametrized b...

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...