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 o...
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 results about some weight parameters and pseudo-regular
partitions of a graph. For instance, generalizing a well-known result of Lovfisz, it is
shown that the weight Shannon capacity 6)* of a connected graph F, with n vertices and
(adjacency matrix) eigenvalues 2j > )~2 ~> '" ~> 2,, satisfies
o~
Citation
Fiol, M. A. Eigenvalue interlacing and weight parameters of graphs. "Linear algebra and its applications", Març 1999, vol. 290, p. 275-301.