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Journals and Conferences
The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a well-defined integer μ0(g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value μ0(g) is attained by a graph whose girth is exactly g. The values of μ0(g) when 3 ≤ g… (More)
A proof that a concept is learnable provided the Vapnik-Chervonenkis dimension is finite is given. The proof is more explicit than previous proofs and introduces two new parameters which allow bounds on the sample size obtained to be improved by a factor of approximately 4log2(e).
The ‘dollar game’ represents a kind of diffusion process on a graph. Under the rules of the game some configurations are both stable and recurrent, and these are known as critical configurations. The set of critical configurations can be given the structure of an abelian group, and it turns out that the order of the group is the tree-number of the graph.… (More)
The critical group of a graph is an abelian group that arises in several contexts, and there are some similarities with the groups that are used in cryptography. We construct a family of graphs with critical groups that are cyclic, and discuss the associated computational problems using algorithms based on the theory of ‘chip-firing’.
This paper is motivated by a problem that arises in the study of partition functions of antiferromagnetic Potts models, including as a special case the chromatic polynomial. It relies on a theorem of Beraha, Kahane and Weiss, which asserts that the zeros of certain sequences of polynomials approach the curves on which a matrix has two eigenvalues with equal… (More)