Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Eigenvalue interlacing and weight parameters of graphs   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 show 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 Lov\'asz, it is shown that the weight Shannon capacity
$\Theta^*$ of a connected graph $\G$, with $n$ vertices and (adjacency matrix)
eigenvalues $\lambda_1>\lambda_2\ge\...\ge \lambda_n$, satisfies $$ \Theta\le
\Theta^* \le \frac{\|\vecnu\|^2}{1-\frac{\lambda_1}{\lambda_n}} $$ where
$\Theta$ is the (standard) Shannon capacity and $\vecnu$ 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. spectrum are derived.

29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
25
The categories of this paper are: patt-sol
0
The categories of this paper are: hep-ph
12
The categories of this paper are: nucl-th
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
37
The categories of this paper are: supr-con
16
The categories of this paper are: stat
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
32
The categories of this paper are: atom-ph