928 research outputs found
Cospectral digraphs from locally line digraphs
A digraph \G=(V,E) is a line digraph when every pair of vertices
have either equal or disjoint in-neighborhoods. When this condition only
applies for vertices in a given subset (with at least two elements), we say
that \G is a locally line digraph. In this paper we give a new method to
obtain a digraph \G' cospectral with a given locally line digraph \G with
diameter , where the diameter of \G' is in the interval .
In particular, when the method is applied to De Bruijn or Kautz digraphs, we
obtain cospectral digraphs with the same algebraic properties that characterize
the formers
Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groups
In this paper, we present a method to obtain regular (or equitable)
partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of
permutation groups on letters. We prove that every partition of the number
gives rise to a regular partition of the Cayley graph. By using
representation theory, we also obtain the complete spectra and the eigenspaces
of the corresponding quotient (di)graphs. More precisely, we provide a method
to find all the eigenvalues and eigenvectors of such (di)graphs, based on their
irreducible representations. As examples, we apply this method to the pancake
graphs and to a recent known family of mixed graphs
(having edges with and without direction). As a byproduct, the existence of
perfect codes in allows us to give a lower bound for the multiplicity of
its eigenvalue
On Middle Cube Graphs
We study a family of graphs related to the -cube. The middle cube graph of parameter k is the subgraph of induced by the set of vertices whose binary representation has either or number of ones. The middle cube graphs can be obtained from the well-known odd graphs by doubling their vertex set. Here we study some of the properties of the middle cube graphs in the light of the theory of distance-regular graphs. In particular, we completely determine their spectra (eigenvalues and their multiplicities, and associated eigenvectors)
The spectra of Manhattan street networks
AbstractThe multidimensional Manhattan street networks constitute a family of digraphs with many interesting properties, such as vertex symmetry (in fact they are Cayley digraphs), easy routing, Hamiltonicity, and modular structure. From the known structural properties of these digraphs, we determine their spectra, which always contain the spectra of hypercubes. In particular, in the standard (two-dimensional) case it is shown that their line digraph structure imposes the presence of the zero eigenvalue with a large multiplicity
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