Ramsey theory is a highly active research area in mathematics that studies the emergence of order in large disordered structures. Ramsey numbers mark the threshold at which order first appears and are extremely difficult to calculate due to their explosive rate of growth. Recently, an algorithm that can be implemented using adiabatic quantum evolution has… (More)
In the Graph Isomorphism (GI) problem two N-vertex graphs G and G are given and the task is to determine whether there exists a permutation of the vertices of G that preserves adjacency and transforms G → G. If yes, then G and G are said to be isomorphic; otherwise they are non-isomorphic. The GI problem is an important problem in computer science and is… (More)
We prove a two-point concentration for the independent domination number of the random graph G n,p provided p 2 ln(n) ≥ 64ln((ln n)/p).
Properties of (connected) graphs whose closed or open neighborhood families are Sperner, anti-Sperner, distinct or none of the proceeding have been extensively examined. In this paper we examine 24 properties of the neighborhood family of a graph. We give asymptotic formulas for the number of (connected) labelled graphs for 12 of these properties. For the… (More)
We prove that the number of integers in the interval [0, x] that are non-trivial Ramsey numbers r(k, n) (3 ≤ k ≤ n) has order of magnitude √ x ln x .
The number of local maxima (resp., local minima) in a tree T ∈ ᐀ n rooted at r ∈ [n] is denoted by M r (T) (resp., by m r (T)). We find exact formulas as rational functions of n for the expectation and variance of M 1 (T) and m n (T) when T ∈ ᐀ n is chosen randomly according to a uniform distribution. As a consequence, a.a.s. M 1 (T) and m n (T) belong to a… (More)