A Latin square of order n is an n × n array of n symbols, in which each symbol occurs exactly once in each row and column. A transversal is a set of n entries, one selected from each row and each column of a Latin square of order n such that no two entries contain the same symbol. Define T (n) to be the maximum number of transversals over all Latin squares of order n. We show that b n ≤ T (n) ≤ c n √ n n! for n ≥ 5, where b ≈ 1.719 and c ≈ 0.614. A corollary of this result is an upper bound on the number of placements of n non-attacking queens on an n × n toroidal chess board. Some divisibility properties of the number of transversals in Latin squares based on finite groups are established. We also provide data from a computer enumeration of transversals in all Latin squares of order at most 9, all groups of order at most 23 and all possible turn-squares of order 14.

Change detection in dynamic networks is an important problem in many areas, such as fraud detection, cyber intrusion detection and health care monitoring. It is a challenging problem because it involves a time sequence of graphs, each of which is usually very large and sparse with heterogeneous vertex degrees, resulting in a complex, high dimensional mathematical object. Spectral embedding methods provide an effective way to transform a graph to a lower dimensional latent Euclidean space that preserves the underlying structure of the network. Although change detection methods that use spectral embedding are available, they do not address sparsity and degree heterogeneity that usually occur in noisy real-world graphs and a majority of these methods focus on changes in the behaviour of the overall network.In this paper, we adapt previously developed techniques in spectral graph theory and propose a novel concept of applying Procrustes techniques to embedded points for vertices in a graph to detect changes in entity behaviour. Our spectral embedding approach not only addresses sparsity and degree heterogeneity issues, but also obtains an estimate of the appropriate embedding dimension. We call this method CDP (change detection using Procrustes analysis). We demonstrate the performance of CDP through extensive simulation experiments and a real-world application. CDP successfully detects various types of vertex-based changes including (i) changes in vertex degree, (ii) changes in community membership of vertices, and (iii) unusual increase or decrease in edge weight between vertices. The change detection performance 1.2 Hewapathirana et al.of CDP is compared with two other baseline methods that employ alternative spectral embedding approaches. In both cases, CDP generally shows superior performance.A network is a collection of entities, that have inherent relationships. Some examples include a social network of friendships among people, a communication network of company employees connected by phone calls, emails or text messages, and a biological network of neurons connected by their synapses. A network can be mathematically conceptualized as a graph by associating entities with vertices, and relationships with edges connecting vertices in the graph. For example, in the graph representation of a social network like Facebook, vertices may represent friends and edges represent friendship connections.Most real-world networks evolve as time progresses. That is, the entities and their relationships keep evolving with time. This type of relational data can be represented as a dynamic network. For example, a communication network of a company is a dynamic network because new employees (entities) join the network and communication patterns (relationships) are modified continuously. Although both the entities and the relationships in a network can vary over time, in this paper, we assume that a dynamic network consists of a fixed set of entities with time varying relationships between them. A dynamic network can be represente...

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Let G = G(n) be a randomly chosen k-edge-coloured k-regular graph with 2n vertices, where k = k(n). Equivalently, G is the union of a random set of k disjoint perfect matchings. Let h = h(n) be a graph with m = m(n) edges such that m 2 + mk = o(n). Using a switching argument, we find an asymptotic estimate of the expected number of subgraphs of G isomorphic to h. Isomorphisms may or may not respect the edge colouring, and other generalisations are also presented. Special attention is paid to matchings and cycles.The results in this paper are essential to a forthcoming paper of McLeod in which an asymptotic estimate for the number of k-edge-coloured k-regular graphs for k = o(n 5/6 ) is found.

The reconstruction of evolutionary trees from data sets on overlapping sets of species is a central problem in phylogenetics. Provided that the tree reconstructed for each subset of species is rooted and that these trees fit together consistently, the space of all parent trees that 'display' these trees was recently shown to satisfy the following strong property: there exists a path from any one parent tree to any other parent tree by a sequence of local rearrangements (nearest neighbour interchanges) so that each intermediate tree also lies in this same tree space. However, the proof of this result uses a non-constructive argument. In this paper we describe a specific, polynomial-time procedure for navigating from any given parent tree to another while remaining in this tree space. The results are of particular relevance to the recent study of 'phylogenetic terraces'.