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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 k-regular graph on v vertices and let τ denote the least eigenvalue of its adjacency matrix A(X). If α(X) denotes the maximum size of an independent set in X, we have the following well known bound: α(X) ≤ v 1 − k τ. It is less well known that if equality holds here and S is a maximum independent set in X with characteristic vector x, then the… (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)

A k x n Latin rectangle is a k x n matrix with entries from {I, 2, ..., n } such that no entry occurs more than once in any row or column. Equivalently, it is an ordered set of k disjoint perfect matchings of K,,". We prove that the number of k x n Latin rectangles is asymptotically as n + oo with k = o(n6I7). This improves substantially on previous work by… (More)

We use difference sets to construct interesting sets of lines in complex space. Using (v, k, 1)-difference sets, we obtain k 2 − k + 1 equian-gular lines in C k when k − 1 is a prime power. Using semiregular relative difference sets with parameters (k, n, k, λ) we construct sets of n + 1 mutually unbiased bases in C k. We show how to construct these… (More)