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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
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Geodesic Distance in Planar Graphs
We derive the exact generating function for planar maps (genus zero
fatgraphs) with vertices of arbitrary even valence and with two marked points
at a fixed geodesic distance. This is done in a purely combinatorial way based
on a bijection with decorated trees, leading to a recursion relation on the
geodesic distance. The latter is solved exactly in terms of discrete
soliton-like expressions, suggesting an underlying integrable structure. We
extract from this solution the fractal dimensions at the various
(multi)-critical points, as well as the precise scaling forms of the continuum
two-point functions and the probability distributions for the geodesic distance
in (multi)-critical random surfaces. The two-point functions are shown to obey
differential equations involving the residues of the KdV hierarchy.Comment: 38 pages, 8 figures, tex, harvmac, eps
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
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