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)
We derive bounds on the size of an independent set based on eigen-values. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erd˝ os-Rényi graphs. We investigate further properties of our bounds, and show how our results on the Erd˝ os-Rényi graphs can be extended to other polarity… (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)
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)
We introduce and discuss Jones pairs. These provide a generalization and a new approach to the four-weight spin models of Bannai and Bannai. We show that each four-weight spin model determines a " dual " pair of association schemes.
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)