446 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
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
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
A new general family of mixed graphs
A new general family of mixed graphs is presented, which generalizes both the pancake graphs and the cycle prefix digraphs. The obtained graphs are vertex transitive and, for some values of the parameters, they constitute the best infinite families with asymptotically optimal (or quasi-optimal) diameter for their number of verticesPeer ReviewedPostprint (author's final draft
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