Eigenvalue interlacing and weight parameters of graphs 1

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~<o*~< I lv l l 1 ; l / " ; n ' Work supported in part by the Spanish Research Council (Comisi6n Interministerial de Ciencia y Tecnologfa, CICYT) under projects TIC 94-0592 and TIC 97-0963. 2 E-mail: [email protected] 0024-3795/99/$ see front matter © 1999 Published by Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 3 79 5(9 8) 1 0 2 3 8 0 276 M.A. Fiol / Linear Algebra and its Applications 290 (1999) 275~01 where O is the (standard) Shannon capacity and v is the positive eigenvector normalized to have smallest entry 1. In the special case of regular graphs, the results obtained have some interesting corollaries, such as an upper bound for some of the multiplicities of the eigenvalues of a distance-regular graph. Finally, some results involving the Laplacian spectrum are derived. © 1999 Published by Elsevier Science Inc. All rights reserved. A M S classOqcation. 05C50