In this paper we introduce the smallest equivalence relation ξ∗ on a finite fuzzy hypergroup S such that the quotient group S/ξ∗, the set of all equivalence classes, is a solvable group. The characterization of solvable groups via strongly regular relation is investigated and several results on the topic are presented.

New (96,20,4,4) and (96,19,2,4) regular partial difference sets are constructed, together with the corresponding strongly regular graphs. Our source are (96,20,4) regular symmetric designs.

We examine the problem of finite-state representability of infinitestate processes w.r.t. certain behavioural equivalences. We show that the classical notion of regularity becomes insufficient in case of all equivalences of van Glabbeek’s hierarchy except bisimilarity, and we design and justify a generalization in the form of strong regularity and finite characterizations. We show that the cond...

We show for many strongly regular graphs ∆, and for all Taylor graphs ∆ except the hexagon, that locally ∆ graphs have bounded diameter.

This paper begins the project of adapting the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups, to padic groups, continuing in the direction charted by Vogan in his 1993 paper on the Langlands correspondence. This paper presents three theorems in that direction. The first theorem shows how Lusztig’s work on perverse sheaves aris...

A quasi-symmetric design (QSD) is a (v, k, λ) design with two intersection numbers x, y, where 0 ≤ x < y < k. The block graph of a QSD is a strongly regular graph (SRG), whereas the converse is not true. Using Neumaier’s classification of SRGs related to the smallest eigenvalue, a complete parametric classification of QSDs whose block graph is an SRG with smallest eigenvalue −3, or second large...

let g be a graph with p vertices and q edges and a = {0, 1, 2, . . . , [q/2]}. a vertex labeling f : v (g) → a induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv. for a ∈ a, let vf (a) be the number of vertices v with f(v) = a. a graph g is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in a, |vf (a) − vf (b)| ≤ 1 and the in...

where J is the all ones matrix and I is the identity. These matrix conditions are equivalent to the combinatorial conditions that the graph is both inand out-regular, and that the number of directed 2-paths from a vertex x to a vertex y is t if x = y, λ if x → y, and μ otherwise. Recently, these graphs were studied by Klin et al. [2], including some new constructions and a list of feasible para...