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Shilla distance-regular graphs
A Shilla distance-regular graph G (say with valency k) is a distance-regular
graph with diameter 3 such that its second largest eigenvalue equals to a3. We
will show that a3 divides k for a Shilla distance-regular graph G, and for G we
define b=b(G):=k/a3. In this paper we will show that there are finitely many
Shilla distance-regular graphs G with fixed b(G)>=2. Also, we will classify
Shilla distance-regular graphs with b(G)=2 and b(G)=3. Furthermore, we will
give a new existence condition for distance-regular graphs, in general.Comment: 14 page
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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