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 \(D\).

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.

This paper examines the numbers of lattice paths of length \(n\) from the origin to integer points along the line \((a,b,c,d) + t(1,-1,1,-1)\). These numbers form a sequence which this paper shows is log concave, and for sufficiently large values of \(n\), the location of the maximum of this sequence is shown. This paper also shows unimodality of such sequences for other lines provided that \(n\) is sufficiently large.

A cover of a finite set \(N\) is a collection of subsets of \(N\) whose union is \(N\). We determine the number of such covers whose blocks all have distinct sizes. The cases of unordered and ordered blocks are each considered.