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- David Bryant, D. Bryant
- 2004

A phylogenetic tree represents historical evolutionary relationships between different species or organisms. The space of possible phylogenetic trees is both complex and exponentially large. Here we study combinatorial features of neighbourhoods within this space, with respect to four standard tree metrics. We focus on the splits of a tree: the bipartitions… (More)

We prove that all Paley graphs can be decomposed into Hamilton cycles. This paper is dedicated to Gert Sabidussi in celebration of his 80th birthday.

- David Bryant, D. Bryant
- 2006

In this chapter we take statistical models designed for trees and adapt them for split networks, a more general class of mathematical structures. The models we propose provide natural swing-bridges between trees, filling in gaps in the probability simplex. There are many reasons why we might want to do this. Firstly, the split networks provide a graphical… (More)

Repacking in cycle decompositions. A method which takes a decomposition of a graph into edge-disjoint cycles and produces a decomposition of a new graph into edge-disjoint cycles of the same lengths will be discussed. This method, called repacking, underpins much of the recent progress on cycle decomposition problems. Orthogonally resolvable cycle… (More)

It is shown that if F 1 , F 2 ,. .. , F t are bipartite 2-regular graphs of order n and α 1 , α 2 ,. .. , α t are non-negative integers such that α 1 +α 2 +· · ·+α t = n−2 2 , α 1 ≥ 3 is odd, and α i is even for i = 2, 3,. .. , t, then there exists a 2-factorisation of K n −I in which there are exactly α i 2-factors isomorphic to F i for i = 1, 2,. .. , t.… (More)

For any rational number h and all sufficiently large n we give a deterministic construction for an n × hn compressed sensing matrix with (1 , t)-recoverability where t = O(√ n). Our method uses pairwise balanced designs and complex Hadamard matrices in the construction of-equiangular frames, which we introduce as a generalisation of equiangular tight… (More)

The next article by Darryn and Daniel provides an introduction to the Lindner conjecture and its solution.

The circulant graph of order n with connection set S is denoted by Circ(n, S). Several results on decompositions of Circ(n, {1, 2}) and Circ(n, {1, 2, 3}) are proved here. The existence problems for decom-positions into paths of arbitrary specified lengths and for decomposi-tions into cycles of arbitrary specified lengths are completely solved for Circ(n,… (More)

We prove that all connected vertex-transitive graphs of order p 2 , p a prime, can be decomposed into Hamilton cycles.