Xk-Digraphs

AbstractLet G be a directed graph on n vertices (single loops allowed) such that there are λ directed paths of length k from P to Q for any distinct pair of vertices (P, Q). We prove that if n > 2 and k > 2, G is regular. The regular case is also discussed

On the matrix equation Al + Al+k = Jn

AbstractThis paper studies the matrix equation Al + Al+k = Jn, where l,k are nonnegative integers, Jn is the n × n matrix of all l's, and A is an unknown (0,1)-matrix. We shall provide a solution for every odd k and every n which is feasible, i.e. n = dl + dl+k for some nonnegative integer d, and show that the equation has no solution in other cases with some trivial cases excluded. We also show that for any solution A to this equation there must be a (0,1)-matrix C satisfying I + Ck = Jdk+1, and Γ(A), the associated digraph of A, is the lth iterated line digraph of Γ(C). In particular, the well-known Kautz digraph K(d, l + 1) can be characterized as Γ(A), where A satisfies Al + Al+1 = Jn for n = dl + dl+1

Directed strongly walk-regular graphs

We generalize the concept of strong walk-regularity to directed graphs. We call a digraph strongly $\ell$-walk-regular with $\ell >1$ if the number of walks of length $\ell$ from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. This generalizes also the well-studied strongly regular digraphs and a problem posed by Hoffman. Our main tools are eigenvalue methods. The case that the adjacency matrix is diagonalizable with only real eigenvalues resembles the undirected case. We show that a digraph $\Gamma$ with only real eigenvalues whose adjacency matrix is not diagonalizable has at most two values of $\ell$ for which $\Gamma$ can be strongly $\ell$-walk-regular, and we also construct examples of such strongly walk-regular digraphs. We also consider digraphs with nonreal eigenvalues. We give such examples and characterize those digraphs $\Gamma$ for which there are infinitely many $\ell$ for which $\Gamma$ is strongly $\ell$-walk-regular

Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

AbstractFor a nonnegative n×n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A)=n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n×n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials

Directed graphs with unique paths of fixed length

AbstractWe present a solution of the matrix equation Ak = −I + J, where A is a (0, 1)-matrix