We discuss a class of models for the evolution of tree networks in which new nodes are recruited into the network at random times, and nodes already in the network may die at random times. Stochastic mechanisms for growth and death of the network that are either sensitive or insensitive to the coordination number or degree of nodes are studied using simulations and mean-field approximations. Cr...

Let Gn be a class of graphs on n vertices. For an integer c, let ex(Gn, c) be the smallest integer such that if G is a graph in Gn with more than ex(Gn, c) edges, then G contains a cycle of length more than c. A classical result of Erdös and Gallai is that if Gn is the class of all simple graphs on n vertices, then ex(Gn, c) = c 2 (n − 1). The result is best possible when n − 1 is divisible by ...

Throughout this paper Gg" will denote a graph with n vertices and k edges where circuits consisting of two edges and loops (i . e. circuits of one edge) are not permitted and G'" will denote a graph of n vertices and k edges where loops and circuits with two edges are permitted . v(G) (respectively v(G)) will denote the number of edges of G (respectively G) . If x,, x" . . ., x,, are some of th...

We prove that if p ≡ 4, 7 (mod 9) is prime and 3 is not a cube modulo p, then both of the equations x + y = p and x + y = p have a solution with x, y ∈ Q.

The theorem in question asserts that, if n > k, then, in the set of integers n, n+1, n-J-2, . . ., n +k-1, there is a number containing a prime divisor greater than k . If n = k+ 1, we obtain the well-known theorem of Chebyshev . The theorem was first asserted and proved by Sylvester t about forty-five years ago . Recently Schurt has rediscovered and again proved the theorem . The following pro...