A Cayley digraph G = C(Γ, X) for a group Γ and a generating set X is the digraph with vertex set V (G) = Γ and arcs (g, gx) where g ∈ Γ and x ∈ X. The reverse of C(Γ, X) is the Cayley digraph G−1 = C(Γ, X−1) where X−1 = {x−1;x ∈ X}. We are interested in sufficient conditions for a Cayley digraph not to be isomorphic to its reverse and focus on Cayley digraphs of metacyclic groups with small gen...

In this paper we introduce two polytopes that respect a digraph in the sense that for every vector in the polytope every component corresponds to a node and is at least equal to the component corresponding to each successor of this node. The sharing polytope is the set of all elements from the unit simplex that respect the digraph. The fuzzy polytope is the set of all elements of the unit cube ...

A minimum feedback arc set of a digraph is a smallest sized set of arcs that when reversed makes the resulting digraph acyclic. Given an acyclic digraph D, we seek a smallest sized tournament T that has D as a minimum feedback arc set. The reversing number of a digraph was defined by Barthélemy et. al. to be r(D) = |V (T )| − |V (D)|. We will completely determine the reversing number for a disj...

The silyl enol ether derivatives of ketones or esters tethered by a hydrocarbon or ether linkage to the 3-position of a pyridine ring undergo dearomatising nucleophilic attack on the ring once it is activated (as an acylpyridinium species) by the addition of methyl chloroformate. The bicyclic dihydropyridine products are in some cases unstable, but may be isolated after hydrogenation as fused b...

We show that the monoid M2(T) of 2 × 2 tropical matrices is a regular semigroup satisfying the semigroup identity A2B4A2A2B2A2B4A2 =A2B4A2B2A2A2B4A2. Studying reduced identities for subsemigroups of M2(T), and introducing a faithful semigroup representation for the bicyclic monoid by 2 × 2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.

The support of a matrix M is the (0, 1)-matrix with ij-th entry equal to 1 if the ij-th entry of M is non-zero, and equal to 0, otherwise. The digraph whose adjacency matrix is the support of M is said to be the digraph of M . This paper observes some structural properties of digraphs and Cayley digraphs, of unitary matrices. We prove that a group generated by two elements has a set of generato...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A kernel N of D is an independent set of vertices such that for every w∈V (D) − N there exists an arc from w to N . A digraph D is called right-pretransitive (resp. left-pretransitive) when (u; v)∈A(D) and (v; w)∈A(D) implies (u; w)∈A(D) or (w; v)∈A(D) (resp. (u; v)∈A(D) and (v; w)∈A(D) implies (u; ...

A digraph obtained by replacing each edge of a complete n-partite graph by an arc or a pair of mutually opposite arcs is called a semicomplete n-partite digraph. We call (D)=max16 i6 n{|Vi|} the independence number of the semicomplete n-partite digraph D, where V1; V2; : : : ; Vn are the partite sets of D. Let p and c, respectively, denote the number of vertices in a longest directed path and t...