1,063 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
Moments in graphs
Let be a connected graph with vertex set and a {\em weight function}
that assigns a nonnegative number to each of its vertices. Then, the
{\em -moment} of at vertex is defined to be
M_G^{\rho}(u)=\sum_{v\in V} \rho(v)\dist (u,v) , where \dist(\cdot,\cdot)
stands for the distance function. Adding up all these numbers, we obtain the
{\em -moment of }: M_G^{\rho}=\sum_{u\in
V}M_G^{\rho}(u)=1/2\sum_{u,v\in V}\dist(u,v)[\rho(u)+\rho(v)]. This
parameter generalizes, or it is closely related to, some well-known graph
invariants, such as the {\em Wiener index} , when for every
, and the {\em degree distance} , obtained when
, the degree of vertex . In this paper we derive some
exact formulas for computing the -moment of a graph obtained by a general
operation called graft product, which can be seen as a generalization of the
hierarchical product, in terms of the corresponding -moments of its
factors. As a consequence, we provide a method for obtaining nonisomorphic
graphs with the same -moment for every (and hence with equal mean
distance, Wiener index, degree distance, etc.). In the case when the factors
are trees and/or cycles, techniques from linear algebra allow us to give
formulas for the degree distance of their product
Deterministic hierarchical networks
It has been shown that many networks associated with complex systems are
small-world (they have both a large local clustering coefficient and a small
diameter) and they are also scale-free (the degrees are distributed according
to a power law). Moreover, these networks are very often hierarchical, as they
describe the modularity of the systems that are modeled. Most of the studies
for complex networks are based on stochastic methods. However, a deterministic
method, with an exact determination of the main relevant parameters of the
networks, has proven useful. Indeed, this approach complements and enhances the
probabilistic and simulation techniques and, therefore, it provides a better
understanding of the systems modeled. In this paper we find the radius,
diameter, clustering coefficient and degree distribution of a generic family of
deterministic hierarchical small-world scale-free networks that has been
considered for modeling real-life complex systems
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)
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