We study a generalization of strongly regular graphs. We call a graph strongly walkregular if there is an ` > 1 such that the number of walks of length ` from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint un...

Given a k-graph Λ and an element p of N, we define the dual k-graph, pΛ. We show that when Λ is row-finite and has no sources, the C∗-algebras C∗(Λ) and C∗(pΛ) coincide. We use this isomorphism to apply Robertson and Steger’s results to calculate the K-theory of C∗(Λ) when Λ is finite and strongly connected and satisfies the aperiodicity condition.

The values t = 1, 3, 5, 6, 9 satisfy the standard necessary conditions for existence of a generalised quadrangle of order (3, t). This gives the following possible parameter sets for strongly regular graphs that are pseudo-geometric for such a generalised quadrangle: (v, k, λ, μ) = (16, 6, 2, 2), (40, 12, 2, 4), (64, 18, 2, 6), (76, 31, 2, 7) and (112, 30, 2, 10). It is well-known that there ar...

Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence matrix of a symmetric design and the all-ones matrix. Amorphous pseudocyclic association schemes are examples of such association schemes whose associated sym...

For given nonnegative integers k, s an upper bound on the minimum number of vertices of a strongly connected digraph with exactly k kernels and s solutions is presented.

Because of irreversibility of movements, motion planning on directed graphs is much more intricate than that on graphs. Recently we showed that the feasibility of motion planning on acyclic and strongly connected directed graphs can be decided in time O(nm) (n, m are respectively the number of vertices and arcs of the directed graph), but left the feasibility of motion planning on (general) dir...

Recently the author introduced a semidefinite upper bound on the square of the stability number of a graph, the inverse theta number, which is proved here to be multiplicative with respect to the strong graph product, hence to be an upper bound for the square of the Shannon capacity of the graph. We also describe a heuristic algorithm for the stable set problem based on semidefinite programming...

In this paper, we examine the structure of vertexand edge-transitive strongly regular graphs, using normal quotient reduction. We show that the irreducible graphs in this family have quasiprimitive automorphism groups, and prove (using the Classification of Finite Simple Groups) that no graph in this family has a holomorphic simple automorphism group. We also find some constraints on the parame...

A graph is n-e.c. (n-existentially closed) if for every pair of subsets U , W of the vertex set V of the graph such that U ∩W = / 0 and |U |+ |W | = n, there is a vertex v ∈ V − (U ∪W ) such all edges between v and U are present and no edges between v and W are present. A graph is strongly regular if it is a regular graph such that the number of vertices mutually adjacent to a pair of vertices ...

We give a construction of strongly regular Cayley graphs on finite fields Fq by using union of cyclotomic classes and index 4 Gauss sums. In particular, we obtain two infinite families of strongly regular graphs with new parameters.