81 research outputs found
A short proof of the odd-girth theorem
Recently, it has been shown that a connected graph with
distinct eigenvalues and odd-girth is distance-regular. The proof of
this result was based on the spectral excess theorem. In this note we present
an alternative and more direct proof which does not rely on the spectral excess
theorem, but on a known characterization of distance-regular graphs in terms of
the predistance polynomial of degree
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
A characterization and an application of weight-regular partitions of graphs
A natural generalization of a regular (or equitable) partition of a graph,
which makes sense also for non-regular graphs, is the so-called weight-regular
partition, which gives to each vertex a weight that equals the
corresponding entry of the Perron eigenvector . This
paper contains three main results related to weight-regular partitions of a
graph. The first is a characterization of weight-regular partitions in terms of
double stochastic matrices. Inspired by a characterization of regular graphs by
Hoffman, we also provide a new characterization of weight-regularity by using a
Hoffman-like polynomial. As a corollary, we obtain Hoffman's result for regular
graphs. In addition, we show an application of weight-regular partitions to
study graphs that attain equality in the classical Hoffman's lower bound for
the chromatic number of a graph, and we show that weight-regularity provides a
condition under which Hoffman's bound can be improved
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