1,254 research outputs found
Dual concepts of almost distance-regularity and the spectral excess theorem
Generally speaking, `almost distance-regular' graphs share some, but not
necessarily all, of the regularity properties that characterize
distance-regular graphs. In this paper we propose two new dual concepts of
almost distance-regularity, thus giving a better understanding of the
properties of distance-regular graphs. More precisely, we characterize
-partially distance-regular graphs and -punctually eigenspace
distance-regular graphs by using their spectra. Our results can also be seen as
a generalization of the so-called spectral excess theorem for distance-regular
graphs, and they lead to a dual version of it
On almost distance-regular graphs
Distance-regular graphs are a key concept in Algebraic Combinatorics and have
given rise to several generalizations, such as association schemes. Motivated
by spectral and other algebraic characterizations of distance-regular graphs,
we study `almost distance-regular graphs'. We use this name informally for
graphs that share some regularity properties that are related to distance in
the graph. For example, a known characterization of a distance-regular graph is
the invariance of the number of walks of given length between vertices at a
given distance, while a graph is called walk-regular if the number of closed
walks of given length rooted at any given vertex is a constant. One of the
concepts studied here is a generalization of both distance-regularity and
walk-regularity called -walk-regularity. Another studied concept is that of
-partial distance-regularity or, informally, distance-regularity up to
distance . Using eigenvalues of graphs and the predistance polynomials, we
discuss and relate these and other concepts of almost distance-regularity, such
as their common generalization of -walk-regularity. We introduce the
concepts of punctual distance-regularity and punctual walk-regularity as a
fundament upon which almost distance-regular graphs are built. We provide
examples that are mostly taken from the Foster census, a collection of
symmetric cubic graphs. Two problems are posed that are related to the question
of when almost distance-regular becomes whole distance-regular. We also give
several characterizations of punctually distance-regular graphs that are
generalizations of the spectral excess theorem
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
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