A construction of a minimum cycle basis for the wreath product of a star by a path, two stars and a star by a wheel is given. Moreover, the basis numbers of these products are determined.

For any \( h \in \mathbb{N} \), a graph \( G = (V, E) \) is said to be \( h \)-magic if there exists a labeling \( l: E(G) \to \mathbb{Z}_h \setminus \{0\} \) such that the induced vertex labeling \( l^+: V(G) \to \mathbb{Z}_h \), defined by

\[ l^+(v) = \sum_{uv \in E(G)} l(uv), \]

is a constant map. When this constant is \( 0 \), we call \( G \) a zero-sum \( h \)-magic graph. The null set of \( G \) is the set of all natural numbers \( h \in \mathbb{N} \) for which \( G \) admits a zero-sum \( h \)-magic labeling. A graph \( G \) is said to be uniformly null if every magic labeling of \( G \) induces a zero sum. In this paper, we will identify the null sets of certain planar graphs such as wheels and fans.

In this paper, we find six new weighing matrices of order \( 2n \) and weight \( 2n-9 \) constructed from two circulants, by establishing various patterns on the locations of the nine zeros in a potential solution.

In the past few years, several studies have appeared that relate to the existence of \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments. In these studies, the number of players in the tournament is taken to be a prime \( p \) of the form \( p \equiv 2^k + 1 \pmod{2^k+1} \), where \( k \geq 2 \). For the cases \( k = 2, 3, 4 \) it has been shown [6,4,5,12] that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all such primes except for the impossible cases \( p = 5, 13, 17 \). For the cases \( k = 5, 6, 7 \) it has been shown [13] that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments exist for all such primes less than \( 3{,}200{,}000 \) and that \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all such primes less than \( 3{,}200{,}000 \) with the exception that existence or non-existence of these designs for \( p = 97, 193, 449, 577, 641, 1409 \) is an open question. Here the case \( k = 8 \) is considered. It is established that \( \mathbb{Z} \)-cyclic directed-triplewhist tournaments and \( \mathbb{Z} \)-cyclic ordered-triplewhist tournaments exist for all primes \( p \equiv 257 \pmod{512} \), \( p \leq 6{,}944{,}177 \), except possibly for \( p = 257, 769, 3329 \). For \( p = 3329 \) we are able to construct a \( \mathbb{Z} \)-cyclic directed-triplewhist tournament, but the existence of a \( \mathbb{Z} \)-cyclic ordered-triplewhist tournament remains an open question. Furthermore, for each type of design it is conjectured that our basic constructions will produce these designs whenever \( p > 5{,}299{,}457 \).

The domination graph of a digraph \( D \), denoted \( \text{dom}(D) \), is created using the vertex set of \( D \) and edge \( uv \in E(\text{dom}(D)) \) whenever \( (u,z) \in A(D) \) or \( (v,z) \in A(D) \) for any other vertex \( z \in V(D) \). Specifically, we consider directed graphs whose underlying graphs are isomorphic to their domination graphs. In particular, digraphs are completely characterized where \( UG^c(D) \) is the union of two disjoint paths.