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...

We consider a strongly regular graph, G, and associate a three dimensional Euclidean Jordan algebra, V, to the adjacency matrix A of G. Then, by considering convergent series of Hadamard powers of the idempotents of the unique complete system of orthogonal idempotents of V, we establish new feasibility conditions for the existence of strongly regular graphs.

A distributed algorithm is developed that can be used to compute the topology of a network, given that each site starts with information about sites it is adjacent to, the network is strongly connected, and communication channels are unidirectional. The program is derived using assertional reasoning.

I point out that an “infinite-dimensional, homogeneous, pregeometry” on a structure M is the same thing as a generically stable type p(x) such that (p(x), x = x) is strongly regular. I also discuss the case of quasi-minimal structures and groups.

In 1959 S.S. Shrikhande wrote a paper concerning L2 association schemes [11]. Out of this paper arose a strongly regular graph with parameters (16, 6, 2, 2) that was not isomorphic to L2(4). This graph turned out to be important in the study of strongly regular graphs as a whole. In this paper, we survey the various constructions and properties of this graph.

We study the half-integral k-Directed Disjoint Paths Problem ( 2kDDPP) in highly strongly connected digraphs. The integral kDDPP is NP-complete even when restricted to instances where k = 2, and the input graph is L-strongly connected, for any L ≥ 1. We show that when the integrality condition is relaxed to allow each vertex to be used in two paths, the problem becomes efficiently solvable in h...

Let V be a real finite dimensional representation of a compact Lie group G. It is well-known that the algebra R[V ] of G-invariant polynomials on V is finitely generated, say by σ1, . . . , σp. Schwarz [38] proved that each G-invariant C-function f on V has the form f = F (σ1, . . . , σp) for a Cfunction F on R. We investigate this representation within the framework of Denjoy–Carleman classes....

The group PGL(2, q) has an embedding into PGL(3, q) such that it acts as the group fixing a nonsingular conic in PG(2, q). This action affords a coherent configuration R(q) on the set L(q) of non-tangent lines of the conic. We show that the relations can be described by using the cross-ratio. Our results imply that the restrictions R+(q) and R−(q) of R(q) to the set L+(q) of secant (hyperbolic)...

Hosik Lee,1 Young-Woo Son,1 Noejung Park,2 Seungwu Han,3 and Jaejun Yu1,* 1School of Physics and Center for Strongly Correlated Materials Research, Seoul National University, Seoul 151-747, Korea 2Department of Applied Physics, Dankook University, Seoul 140-714, Korea 3Department of Physics, Ewha Womans University, Seoul 120-750, Korea Received 23 May 2005; revised manuscript received 14 Septem...