We show various upper bounds for the order of a digraph (or mixed graph) whose Hermitian adjacency matrix has an eigenspace prescribed codimension. In particular, this generalizes so-called absolute bound (simple) graphs first shown by Delsarte, Goethals, and Seidel (1977) extended Bell Rowlinson (2003). doing so, we also adapt Blokhuis' theory (1983) harmonic analysis in real hyperbolic spaces...

We call the digraph D an m-coloured digraph if its arcs are coloured with m colours. If D is an m-coloured digraph and a ∈ A(D), colour(a) will denote the colour has been used on a. A path (or a cycle) is called monochromatic if all of its arcs are coloured alike. A γ-cycle in D is a sequence of vertices, say γ = (u0, u1, . . . , un), such that ui 6= uj if i 6= j and for every i ∈ {0, 1, . . . ...

Like the Coxeter graph became reattached into the Klein graph in [3], the Levi graphs of the 93 and 103 self-dual configurations, known as the Pappus and Desargues (k-transitive) graphs P and D (where k = 3), also admit reattachments of the distance-(k − 1) graphs of half of their oriented shortest cycles via orientation assignments on their common (k−1)-arcs, concurrent for P and opposite for ...

We introduce and study four optimization problems that generalize the well-known subset sum problem. Given a node-weighted digraph, select a subset of vertices whose total weight does not exceed a given budget. Some additional constraints need to be satisfied. The (weak resp.) digraph constraint imposes that if (all incoming nodes of resp.) a node x belongs to the solution, then the latter comp...

The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and combinatorial bounds on the number of leaves in out-branchings. We show that – every strongly connected digraph D of order n with minimum indegree at least 3 has an ou...

A digraph G is a difference digraph iff there exists an S ⊂ IN such that G is isomorphic to the digraph DD(S) = (V,A), where V = S and A = {(i, j) : i, j ∈ V ∧ i− j ∈ V }. For some classes of digraphs, e.g. alternating trees, oriented cycles, tournaments etc., it is known, under which conditions these digraphs are difference digraphs (cf. [5]). We generalize the so-called sourcejoin (a construc...

In this paper we study the problem of existence of a crossing-free acyclic hamiltonian path completion (for short, HP-completion) set for embedded upward planar digraphs. In the context of book embeddings, this question becomes: given an embedded upward planar digraph G, determine whether there exists an upward 2-page book embedding of G preserving the given planar embedding. Given an embedded ...

A k-digraph is a digraph in which every vertex has outdegree at most k. A (k ∨ l)digraph is a digraph in which a vertex has either outdegree at most k or indegree at most l. Motivated by function theory, we study the maximum value Φ(k) (resp. Φ(k, l)) of the arc-chromatic number over the k-digraphs (resp. (k ∨ l)-digraphs). El-Sahili [3] showed that Φ(k, k) ≤ 2k+1. After giving a simple proof o...