627 research outputs found
Integer symmetric matrices having all their eigenvalues in the interval [-2,2]
We completely describe all integer symmetric matrices that have all their
eigenvalues in the interval [-2,2]. Along the way we classify all signed
graphs, and then all charged signed graphs, having all their eigenvalues in
this same interval. We then classify subsets of the above for which the integer
symmetric matrices, signed graphs and charged signed graphs have all their
eigenvalues in the open interval (-2,2).Comment: 33 pages, 18 figure
Modular Decomposition and the Reconstruction Conjecture
We prove that a large family of graphs which are decomposable with respect to
the modular decomposition can be reconstructed from their collection of
vertex-deleted subgraphs.Comment: 9 pages, 2 figure
On the Classification of All Self-Dual Additive Codes over GF(4) of Length up to 12
We consider additive codes over GF(4) that are self-dual with respect to the
Hermitian trace inner product. Such codes have a well-known interpretation as
quantum codes and correspond to isotropic systems. It has also been shown that
these codes can be represented as graphs, and that two codes are equivalent if
and only if the corresponding graphs are equivalent with respect to local
complementation and graph isomorphism. We use these facts to classify all codes
of length up to 12, where previously only all codes of length up to 9 were
known. We also classify all extremal Type II codes of length 14. Finally, we
find that the smallest Type I and Type II codes with trivial automorphism group
have length 9 and 12, respectively.Comment: 18 pages, 4 figure
On realization graphs of degree sequences
Given the degree sequence of a graph, the realization graph of is the
graph having as its vertices the labeled realizations of , with two vertices
adjacent if one realization may be obtained from the other via an
edge-switching operation. We describe a connection between Cartesian products
in realization graphs and the canonical decomposition of degree sequences
described by R.I. Tyshkevich and others. As applications, we characterize the
degree sequences whose realization graphs are triangle-free graphs or
hypercubes.Comment: 10 pages, 5 figure
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