5,019 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
Listening to Garhwali Popular Music in and out of Place
Listening to popular music is a central means by which people construct their place in the world, both literally and figuratively. For Garhwalis living inside and outside of the Himalayas, listening to vernacular popular music has been one way in which they imagine themselves to be part of a specific place and a larger cultural region. Displacement is a major theme of these songs, and practices of listening underline the mobile and trans-local aspects of life for many Garhwalis. In order to assess the impact of popular music consumption on notions of place, and vice versa, this article provides ethnographic vignettes of musical consumption in Garhwali villages and small towns, Garhwali pilgrimage sites, and migrant contexts outside of Garhwal. I suggest that much of the emotional salience and enduring popularity of Garhwali gīt derive from the emotional and physical displacement of married women and male migrants
On the critical points of the entropic principle
In a recent paper, hep-th/0509109, Gukov et al. introduced an entropy
functional on the moduli space of Calabi-Yau compactifications. The maxima of
this functional are then interpreted as "preferred" Calabi-Yau
compactifications. In this note we show that for compact Calabi-Yaus, all
regular critical points of this entropic principle are maxima.Comment: 11 page
Pseudo-distance-regularised graphs are distance-regular or distance-biregular
The concept of pseudo-distance-regularity around a vertex of a graph is a
natural generalization, for non-regular graphs, of the standard
distance-regularity around a vertex. In this note, we prove that a
pseudo-distance-regular graph around each of its vertices is either
distance-regular or distance-biregular. By using a combinatorial approach, the
same conclusion was reached by Godsil and Shawe-Taylor for a distance-regular
graph around each of its vertices. Thus, our proof, which is of an algebraic
nature, can also be seen as an alternative demonstration of Godsil and
Shawe-Taylor's theorem
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