Eigenvalue interlacing and weight parameters of graphs 1 (original) (raw)

Eigenvalue interlacing is a versatile technique for deriving results in algebraic combinatorics. In particular, it has been successfully used for proving a number of results about the relation between the (adjacency matrix or Laplacian) spectrum of a graph and some of its properties. For instance, some characterizations of regular partitions, and bounds for some parameters, such as the independence and chromatic numbers, the diameter, the bandwidth, etc., have been obtained. For each parameter of a graph involving the cardinality of some vertex sets, we can define its corresponding weight parameter by giving some "weights" (that is, the entries of the positive eigenvector) to the vertices and replacing cardinalities by square norms. The key point is that such weights "regularize" the graph, and hence allow us to define a kind of regular partition, called "pseudo-regular," intended for general graphs. Here we s~aow how to use interlacing for proving resu...