1,817 research outputs found
Nodal domain theorems for -Laplacians on signed graphs
We establish various nodal domain theorems for generalized -Laplacians on
signed graphs, which unify most of the results on nodal domains of graph
-Laplacians and arbitrary symmetric matrices. Based on our nodal domain
estimates, we also obtain a higher order Cheeger inequality that relates the
variational eigenvalues of -Laplacians and Atay-Liu's multi-way Cheeger
constants on signed graphs. Moreover, we show new results for 1-Laplacian on
signed graphs, including a sharp upper bound of the number of nonzeros of
certain eigenfunction corresponding to the -th variational eigenvalues, and
some identities relating variational eigenvalues and -way Cheeger constants.Comment: 31 pages. Comments are welcome
Graphs with many strong orientations
We establish mild conditions under which a possibly irregular, sparse graph
has "many" strong orientations. Given a graph on vertices, orient
each edge in either direction with probability independently. We show
that if satisfies a minimum degree condition of and has
Cheeger constant at least , then the
resulting randomly oriented directed graph is strongly connected with high
probability. This Cheeger constant bound can be replaced by an analogous
spectral condition via the Cheeger inequality. Additionally, we provide an
explicit construction to show our minimum degree condition is tight while the
Cheeger constant bound is tight up to a factor.Comment: 14 pages, 4 figures; revised version includes more background and
minor changes that better clarify the expositio
Isoperimetric Inequalities in Simplicial Complexes
In graph theory there are intimate connections between the expansion
properties of a graph and the spectrum of its Laplacian. In this paper we
define a notion of combinatorial expansion for simplicial complexes of general
dimension, and prove that similar connections exist between the combinatorial
expansion of a complex, and the spectrum of the high dimensional Laplacian
defined by Eckmann. In particular, we present a Cheeger-type inequality, and a
high-dimensional Expander Mixing Lemma. As a corollary, using the work of Pach,
we obtain a connection between spectral properties of complexes and Gromov's
notion of geometric overlap. Using the work of Gunder and Wagner, we give an
estimate for the combinatorial expansion and geometric overlap of random
Linial-Meshulam complexes
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