A semi-complete bigraph \(G\) has adjacency matrix
\[A = \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix},\]
where \(B + B^T = J – I\), so \(B\) is the adjacency matrix of a tournament \(T\) corresponding to \(G\). We determine algebraic and structural properties of this class of graphs. In particular, we obtain a condition equivalent to the connectedness of a semi-complete bigraph; moreover we determine characterizations of semi-complete bigraphs having 4 distinct eigenvalues in the case of \(G\) regular or \(A\) irreducible. In particular, a regular semi-complete bigraph has 4 distinct eigenvalues if and only if it corresponds to a doubly regular tournament.

Let \(D\) be an asymmetric digraph and \(A\) a finite group. We give a formula for the characteristic polynomial of a cyclic \(A\)-cover of \(D\). This is a generalization of a formula for the characteristic polynomial of a regular covering of a graph. Furthermore, we discuss cyclic abelian covers of \(D\).

We define an almost-convex polygon as a non-convex polygon in which any two vertices see each other inside the polygon unless they are not adjacent and belong to a chain of consecutive concave vertices. Using inclusion-exclusion techniques, we find formulas for the number of triangulations of almost-convex polygons in terms of the number and position of the concave vertices. We translate these formulas into the language of generating functions and provide several simple asymptotic estimates. We also prove that certain balanced configurations yield the maximum number of triangulations.

Let \(n,s\) be positive integers, and let \(r = 1 + \frac{1}{s}\). In this note we prove that if the sequence \(\{a_n(r)\}_{n=1}^{\infty}\) satisfies \(ra_n(r)= \sum_{k=1}^{n}\binom{n}{k}a_k(r), n> 1\), then \(a_n(r) = na_1(r)\left[(n -1)! / {(s+1)}(log r)^n+{{1/r(s+1)}} \right]\). Moreover, we obtain a related combinatorial identity.

A Mendelsohn triple system, \(MTS(v) = (X, \mathcal{B})\), is called self-converse if it and its converse \((X, \mathcal{B}^{-1})\) are isomorphic, where \(\mathcal{B}^{-1 } = \{\langle z,y,x\rangle; \langle x,y,z\rangle \in \mathcal{B}\}\). In this paper, the existence spectrum of self-converse \(MTS(v)\) is given, which is \(v \equiv 0\) or \(1 \pmod{3}\) and \(v \neq 6\).

In this paper, we discuss the automorphism groups of circulant digraphs. Our main purpose is to determine the full automorphism groups of circulant digraphs of degree \(3\).

The spectrum for the decomposition of \(\lambda K_v\) into \(3\)-perfect \(9\)-cycles is found for all \(\lambda > 1\). (The case \(\lambda = 1\) was dealt with in an earlier paper by the authors and Lindner.) The necessary conditions for the existence of a suitable decomposition turn out to be sufficient.

A directed triple system of order \(v\), denoted by \(DTS(v)\), is called \((f,k)\)-rotational if it has an automorphism consisting of \(f\) fixed points and \(k\) cycles each of length \((v-f)/k\). In this paper, we obtain a necessary and sufficient condition for the existence of \((f,k)\)-rotational \(DTS(v)\) for any arbitrary positive integer \(k\).

Let \( {R} = (r_1, r_2, \ldots, r_m)\) and \( {S} = (s_1, s_2, \ldots, s_n)\) be nonnegative integral vectors. Denote by \( {A}( {R}, {S})\) the class of \((0,1)\) matrices with row sum vector \( {R}\) and column sum vector \( {S}\). We study a generalization of invariant positions called locally invariant positions of a class \( {A}( {R}, {S})\). For a normalized class, locally invariant positions have in common with invariant positions the property that they lie above and to the left of some simple rook path through the set of positions.