27,720 research outputs found
On Products and Line Graphs of Signed Graphs, their Eigenvalues and Energy
In this article we examine the adjacency and Laplacian matrices and their
eigenvalues and energies of the general product (non-complete extended -sum,
or NEPS) of signed graphs. We express the adjacency matrix of the product in
terms of the Kronecker matrix product and the eigenvalues and energy of the
product in terms of those of the factor signed graphs. For the Cartesian
product we characterize balance and compute expressions for the Laplacian
eigenvalues and Laplacian energy. We give exact results for those signed
planar, cylindrical and toroidal grids which are Cartesian products of signed
paths and cycles.
We also treat the eigenvalues and energy of the line graphs of signed graphs,
and the Laplacian eigenvalues and Laplacian energy in the regular case, with
application to the line graphs of signed grids that are Cartesian products and
to the line graphs of all-positive and all-negative complete graphs.Comment: 30 page
An application of Hoffman graphs for spectral characterizations of graphs
In this paper, we present the first application of Hoffman graphs for
spectral characterizations of graphs. In particular, we show that the
-clique extension of the -grid is determined by its
spectrum when is large enough. This result will help to show that the
Grassmann graph is determined by its intersection numbers as a
distance regular graph, if is large enough
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
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