1,456 research outputs found
Spectral Properties of Oriented Hypergraphs
An oriented hypergraph is a hypergraph where each vertex-edge incidence is
given a label of or . The adjacency and Laplacian eigenvalues of an
oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and
Laplacian matrices of an oriented hypergraph which depend on structural
parameters of the oriented hypergraph are found. An oriented hypergraph and its
incidence dual are shown to have the same nonzero Laplacian eigenvalues. A
family of oriented hypergraphs with uniformally labeled incidences is also
studied. This family provides a hypergraphic generalization of the signless
Laplacian of a graph and also suggests a natural way to define the adjacency
and Laplacian matrices of a hypergraph. Some results presented generalize both
graph and signed graph results to a hypergraphic setting.Comment: For the published version of the article see
http://repository.uwyo.edu/ela/vol27/iss1/24
Spectral Properties of Complex Unit Gain Graphs
A complex unit gain graph is a graph where each orientation of an edge is
given a complex unit, which is the inverse of the complex unit assigned to the
opposite orientation. We extend some fundamental concepts from spectral graph
theory to complex unit gain graphs. We define the adjacency, incidence and
Laplacian matrices, and study each of them. The main results of the paper are
eigenvalue bounds for the adjacency and Laplacian matrices.Comment: 13 pages, 1 figure, to appear in Linear Algebra App
Spectral convergence of non-compact quasi-one-dimensional spaces
We consider a family of non-compact manifolds X_\eps (``graph-like
manifolds'') approaching a metric graph and establish convergence results
of the related natural operators, namely the (Neumann) Laplacian \laplacian
{X_\eps} and the generalised Neumann (Kirchhoff) Laplacian \laplacian {X_0}
on the metric graph. In particular, we show the norm convergence of the
resolvents, spectral projections and eigenfunctions. As a consequence, the
essential and the discrete spectrum converge as well. Neither the manifolds nor
the metric graph need to be compact, we only need some natural uniformity
assumptions. We provide examples of manifolds having spectral gaps in the
essential spectrum, discrete eigenvalues in the gaps or even manifolds
approaching a fractal spectrum. The convergence results will be given in a
completely abstract setting dealing with operators acting in different spaces,
applicable also in other geometric situations.Comment: some references added, still 36 pages, 4 figure
Unsolved Problems in Spectral Graph Theory
Spectral graph theory is a captivating area of graph theory that employs the
eigenvalues and eigenvectors of matrices associated with graphs to study them.
In this paper, we present a collection of topics in spectral graph theory,
covering a range of open problems and conjectures. Our focus is primarily on
the adjacency matrix of graphs, and for each topic, we provide a brief
historical overview.Comment: v3, 30 pages, 1 figure, include comments from Clive Elphick, Xiaofeng
Gu, William Linz, and Dragan Stevanovi\'c, respectively. Thanks! This paper
will be published in Operations Research Transaction
The inertia of weighted unicyclic graphs
Let be a weighted graph. The \textit{inertia} of is the triple
, where
are the number of the positive, negative and zero
eigenvalues of the adjacency matrix of including their
multiplicities, respectively. , is called the
\textit{positive, negative index of inertia} of , respectively. In this
paper we present a lower bound for the positive, negative index of weighted
unicyclic graphs of order with fixed girth and characterize all weighted
unicyclic graphs attaining this lower bound. Moreover, we characterize the
weighted unicyclic graphs of order with two positive, two negative and at
least zero eigenvalues, respectively.Comment: 23 pages, 8figure
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