For arbitrary odd prime power q and s ∈ (0, 1] such that q is an integer, we construct a doubly–infinite series of (q, q)–bipartite graphs which are biregular of degrees q and q and of girth 8. These graphs have the greatest number of edges among all known (n,m)–bipartite graphs with the same asymptotics of lognm, n → ∞. For s = 1/3, our graphs provide an explicit counterexample to a conjecture...

The order of every finite group G can be expressed as a product of coprime positive integers m1, . . . ,mt such that π(mi) is a connected component of the prime graph of G. The integers m1, . . . ,mt are called the order components of G. Some nonabelian simple groups are known to be uniquely determined by their order components. As the main result of this paper, we show that the projective symp...

A graph with a vertex set V is said to have a prime labeling if its vertices can be labeled with distinct integers 1; 2; ; jV j such that for every edge fx; yg, the labels assigned to x and y are relatively prime. A tree is prime if it has at least one prime labeling. Around 1980, Entringer conjectured that every tree is prime. After three decades, this conjecture remains open. Nevertheless, a ...

Twin primes are primes of the form (p, p + 2). There are many proofs for the infinitude of prime numbers, but it is very difficult to prove whether there are an infinite number of pairs of twin primes. Most mathemati­ cians agree that the evidence points toward this conclusion, but numerous attempts at a proof have been falsified by subsequent review. The prob­ lem itself, one of the most famou...

Call a set of integers {b1, b2, . . . , bk} admissible if for any prime p, at least one congruence class modulo p does not contain any of the bi. Let ρ ∗(x) be the size of the largest admissible set in [1, x]. The Prime k-tuples Conjecture states that any for any admissible set, there are infinitely many n such that n+b1, n+b2, . . . n+bk are simultaneously prime. In 1974, Hensley and Richards ...

The goal of this paper is two-fold. We first focus on the problem of deciding whether two monomial rotation symmetric (MRS) Boolean functions are affine equivalent via a permutation. Using a correspondence between such functions and circulant matrices, we give a simple necessary and sufficient condition. We connect this problem with the well known Ádám’s conjecture from graph theory. As applica...