1,794 research outputs found
Classical properties of algebras using a new graph association
We study the relation between algebraic structures and Graph Theory. We have
defined five different weighted digraphs associated to a finite dimensional
algebra over a field in order to tackle important properties of the associated
algebras, mainly the nilpotency and solvability in the case of Leibniz
algebras
Vertices with the Second Neighborhood Property in Eulerian Digraphs
The Second Neighborhood Conjecture states that every simple digraph has a
vertex whose second out-neighborhood is at least as large as its first
out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle
intersection graph of an even graph is a new graph whose vertices are the
cycles in a cycle decomposition of the original graph and whose edges represent
vertex intersections of the cycles. By using a digraph variant of this concept,
we prove that Eulerian digraphs which admit a simple dicycle intersection graph
have not only adhere to the Second Neighborhood Conjecture, but have a vertex
of minimum outdegree that has the Second Neighborhood Property.Comment: fixed an error in an earlier version and made structural change
Which Digraphs with Ring Structure are Essentially Cyclic?
We say that a digraph is essentially cyclic if its Laplacian spectrum is not
completely real. The essential cyclicity implies the presence of directed
cycles, but not vice versa. The problem of characterizing essential cyclicity
in terms of graph topology is difficult and yet unsolved. Its solution is
important for some applications of graph theory, including that in
decentralized control. In the present paper, this problem is solved with
respect to the class of digraphs with ring structure, which models some typical
communication networks. It is shown that the digraphs in this class are
essentially cyclic, except for certain specified digraphs. The main technical
tool we employ is the Chebyshev polynomials of the second kind. A by-product of
this study is a theorem on the zeros of polynomials that differ by one from the
products of Chebyshev polynomials of the second kind. We also consider the
problem of essential cyclicity for weighted digraphs and enumerate the spanning
trees in some digraphs with ring structure.Comment: 19 pages, 8 figures, Advances in Applied Mathematics: accepted for
publication (2010) http://dx.doi.org/10.1016/j.aam.2010.01.00
Vertices with the Second Neighborhood Property in Eulerian Digraphs
The Second Neighborhood Conjecture states that every simple digraph has a
vertex whose second out-neighborhood is at least as large as its first
out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle
intersection graph of an even graph is a new graph whose vertices are the
cycles in a cycle decomposition of the original graph and whose edges represent
vertex intersections of the cycles. By using a digraph variant of this concept,
we prove that Eulerian digraphs which admit a simple cycle intersection graph
have not only adhere to the Second Neighborhood Conjecture, but that local
simplicity can, in some cases, also imply the existence of a Seymour vertex in
the original digraph.Comment: This is the version accepted for publication in Opuscula Mathematic
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