Analog modulation has served us very well over the years. Digital modulation is an improvement over analog modulation because it provides better bandwidth utilization over analog modulation, less power for signal propagation, it is natural for packet transmission, forward error correction, automatic repeat request, encryption, compression, and signal transformation so that it looks like noise to the adversary. Digital wireless communication is an enormous area that is rapidly growing. Digital communication is a field in which theoretical ideas have had an unusually powerful impact on system design and practice. In this research paper we provide a digital modulation algorithm for efficient transmission based on circular probability distribution theory.

A primitive hypergraph is a hypergraph with maximum cardinality three and maximum degree three such that every \(3\)-edge is adjacent only to \(2\)-edges and is incident only to vertices of degree two. Deciding the bicolorability of a primitive hypergraph is NP-complete (a straightforward consequence of results in [14]). We provide sufficient conditions, similar to the Sterboul conditions proved by Défossez [5], for the existence of a bicoloring of a primitive hypergraph, and we provide a polynomial algorithm for bicoloring a primitive hypergraph if those conditions hold. We then draw a connection between this algorithm and the well-known necessary and sufficient conditions given by Berge [1] for maximal matchings in graphs, which leads to a characterization of bicolorability of primitive hypergraphs.

Let \( D \) be a strongly connected oriented graph with vertex-set \( V \) and arc-set \( A \). The distance from a vertex \( u \) to another vertex \( v \), \( d(u,v) \), is the minimum length of oriented paths from \( u \) to \( v \). Suppose \( B = \{b_1, b_2, b_3, \ldots, b_k\} \) is a nonempty ordered subset of \( V \). The representation of a vertex \( v \) with respect to \( B \), \( r(v|B) \), is defined as a vector \( (d(v,b_1), d(v,b_2), \ldots, d(v,b_k)) \). If any two distinct vertices \( u,v \) satisfy \( r(u|B) \neq r(v|B) \), then \( B \) is said to be a resolving set of \( D \). If the cardinality of \( B \) is minimum, then \( B \) is said to be a basis of \( D \), and the cardinality of \( B \) is called the directed metric dimension of \( D \).

Let \( G \) be the underlying graph of \( D \) admitting a \( C_n \)-covering. A \( C_n \)-simple orientation is an orientation on \( G \) such that every \( C_n \) in \( D \) is strongly connected. This paper deals with metric dimensions of oriented wheels, oriented fans, and amalgamation of oriented cycles, all of which admit \( C_n \)-simple orientations.

A Stanton-type graph \( S(n, m) \) is a connected multigraph on \( n \) vertices such that for a fixed integer \( m \) with \( n – 1 \leq m \leq \binom{n}{2} \), there is exactly one edge of multiplicity \( i \) (and no others) for each \( i = 1, 2, \ldots, m \). In a recent paper, the authors decomposed \( \lambda K_{n} \) (for the appropriate minimal values of \( \lambda \)) into two of the four possible types of \( S(4, 3) \)’s. In this note, decompositions of \( \lambda K_{n} \) (for the appropriate minimal values of \( \lambda \)) into the remaining two types of \( S(4, 3) \)’s are given.