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The number of vertices missed by a maximum matching in a graph G is the multiplicity of zero as a root of the matchings polynomial (G; x) of G, and hence many results in matching theory can be expressed in terms of this multiplicity. Thus, if mult(; G) denotes the multiplicity of as a zero of (G; x), then Gallai's lemma is equivalent to the assertion that(More)
Let X be a graph on n vertices with with adjacency matrix A and let H(t) denote the matrix-valued function exp(iAt). If u and v are distinct vertices in X, we say perfect state transfer from u to v occurs if there is a time τ such that |H(τ) u,v | = 1. If u ∈ V (X) and there is a time σ such that |H(σ) u,u | = 1, we say X is periodic at u with period σ. We(More)
ii Preface The main theme of these notes is the study of various algebraic bounds for the size of independent sets, cliques and colourings and, in particular, the application of these bounds to graphs coming from association schemes. In some cases my aim is to determine (say) the chromatic number of a particular graph. In other cases my interest is in the(More)
Let X be a graph on n vertices with adjacency matrix A and let H(t) denote the matrix-valued function exp(iAt). If u and v are distinct vertices in X, we say perfect state transfer from u to v occurs if there is a time τ such that |H(τ)u,v| = 1. The chief problem is to characterize the cases where perfect state transfer occurs. In this paper, it is shown(More)