Given a directed graph G, the K node-disjoint paths problem consists in finding a partition of G into K node-disjoint paths, such that each path ends up in a given subset of nodes in G. This article provides a necessary condition for the K node-disjoint paths problem which combines (1) the structure of the reduced graph associated with G, (2) the structure of each strongly connected component o...

The notion of a C-ultrahomogeneous graph, due to Isaksen et al., is adapted for digraphs, and then a strongly connected ~ C4-ultrahomogeneous oriented graph on 168 vertices and 126 pairwise arc-disjoint 4-cycles is presented, with regular indegree and outdegree 3 and no circuits of lengths 2 and 3, by altering a definition of the Coxeter graph via pencils of ordered lines of the Fano plane in w...

Forty years ago, Goethals and Seidel showed that if the adjacency algebra of a strongly regular graph X contains a Hadamard matrix then X is of Latin square type or of negative Latin square type [8]. We extend their result to complex Hadamard matrices and find only three additional families of parameters for which the strongly regular graphs have complex Hadamard matrices in their adjacency alg...

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...

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...