Lane H. Clark

Experimental determination of Ramsey numbers.

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 been proposed that calculates the two-color Ramsey numbers R(m,n). Here we present results of an experimental implementation of this algorithm and show that it correctly determines the Ramsey numbers R(3,3) and R(m,2) for 4≤m≤8. The R(8,2) computation used 84 qubits of which 28 were computational qubits. This computation is the largest experimental implementation of a scientifically meaningful adiabatic evolution algorithm that has been done to date.

Parity dimension for graphs

Abstract For a graph G = ( V , E ), by N = A + I , we denote the closed neighborhood matrix of G where A and I are the adjacency matrix of G and identity matrix, respectively. The parity dimension of G , denoted PD( G ), is the dimension of the null space of N over the field Z 2 . We investigate parity dimension for trees, graphs and random graphs.

Graph isomorphism and adiabatic quantum computing

problem in computer science and is thought to be of comparable difficulty to integer factorization. In this paper we present a quantum algorithm that solves arbitrary instances of GI and which also provides an approach to determiningallautomorphismsofagivengraph.WeshowhowtheGIproblemcanbeconvertedtoacombinatorial optimization problem that can be solved using adiabatic quantum evolution. We numerically simulate the algorithm’s quantum dynamics and show that it correctly (i) distinguishes nonisomorphic graphs; (ii) recognizes isomorphic graphs and determines the permutation(s) that connect them; and (iii) finds the automorphism group of a given graph G. We then discuss the GI quantum algorithm’s experimental implementation, and close by showing how it can be leveraged to give a quantum algorithm that solves arbitrary instances of the NP-complete subgraph isomorphism problem. The computational complexity of an adiabatic quantum algorithm is largely determined by the minimum energy gap � (N) separating the ground and first-excited states in the limit of large problem size N � 1. Calculating � (N) in this limit is a fundamental open problem in adiabatic quantum computing, and so it is not possible to determine the computational complexity of adiabatic quantum algorithms in general, nor consequently, of the specific adiabatic quantum algorithms presented here. Adiabatic quantum computing has been shown to be equivalent to the circuit model of quantum computing, and so development of adiabatic quantum algorithms continues to be of great interest.

Efficient domination of the orientations of a graph

Abstract For an orientation G of a simple graph G, N G [x] denotes the vertex x together with all those vertices in G for which there are arcs directed toward x. A set S of vertices of G is an efficient dominating set (EDS) of G provided that IFld | N G [x]∩ S| = 1 for every x in G . An efficiency of G is an ordered pair ( G , S), where S is an EDS of the orientation G of G. The number of distinct efficiencies of G is denoted is denoted by η(G). We give a formula for η(G) which allows us to calculate it for complete graphs, complete bipartite graphs, cycles, and paths. We find the minimum and maximum value of η(G) among all graphs with a fixed number of edges. We also find the minimum and maximum value of η(G), as well as the external graphs, among all graphs with a fixed number of vertices. Finally, we show that the probability a random oriented graph has an EDS is exponentially small when such graph is chosen according to a uniform distribution.

The minimum number of subgraphs in a graph and its complement

For a graphb F without isolated vertices, let M(F; n) denote the minimum number of monochromatic copies of F in any 2-coloring of the edges of Kn. Burr and Rosta conjectured that when F has order t, size u, and a automorphisms. Independently, Sidorenko and Thomason have shown that the conjecture is false. We give families of graphs F of order t, of size u, and with a automorphisms where . We show also that the asymptotic value of M(F; n) is not solely a function of the order, size and number of automorphisms of F.

Asymptotic normality of the Ward numbers

We prove a central limit theorem for the Ward numbers. c 1999 Elsevier Science B.V. All rights reserved.