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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)
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)
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)
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)