1,586 research outputs found
Transversal designs and induced decompositions of graphs
We prove that for every complete multipartite graph there exist very
dense graphs on vertices, namely with as many as
edges for all , for some constant , such that can be
decomposed into edge-disjoint induced subgraphs isomorphic to~. This result
identifies and structurally explains a gap between the growth rates and
on the minimum number of non-edges in graphs admitting an
induced -decomposition
Geometric aspects of 2-walk-regular graphs
A -walk-regular graph is a graph for which the number of walks of given
length between two vertices depends only on the distance between these two
vertices, as long as this distance is at most . Such graphs generalize
distance-regular graphs and -arc-transitive graphs. In this paper, we will
focus on 1- and in particular 2-walk-regular graphs, and study analogues of
certain results that are important for distance regular graphs. We will
generalize Delsarte's clique bound to 1-walk-regular graphs, Godsil's
multiplicity bound and Terwilliger's analysis of the local structure to
2-walk-regular graphs. We will show that 2-walk-regular graphs have a much
richer combinatorial structure than 1-walk-regular graphs, for example by
proving that there are finitely many non-geometric 2-walk-regular graphs with
given smallest eigenvalue and given diameter (a geometric graph is the point
graph of a special partial linear space); a result that is analogous to a
result on distance-regular graphs. Such a result does not hold for
1-walk-regular graphs, as our construction methods will show
Quantum Experiments and Graphs III: High-Dimensional and Multi-Particle Entanglement
Quantum entanglement plays an important role in quantum information
processes, such as quantum computation and quantum communication. Experiments
in laboratories are unquestionably crucial to increase our understanding of
quantum systems and inspire new insights into future applications. However,
there are no general recipes for the creation of arbitrary quantum states with
many particles entangled in high dimensions. Here, we exploit a recent
connection between quantum experiments and graph theory and answer this
question for a plethora of classes of entangled states. We find experimental
setups for Greenberger-Horne-Zeilinger states, W states, general Dicke states,
and asymmetrically high-dimensional multipartite entangled states. This result
sheds light on the producibility of arbitrary quantum states using photonic
technology with probabilistic pair sources and allows us to understand the
underlying technological and fundamental properties of entanglement.Comment: 7 pages, 7 figures; Appendix 3 pages, 5 figure
On the eigenvalues of Cayley graphs on the symmetric group generated by a complete multipartite set of transpositions
Given a finite simple graph \cG with vertices, we can construct the
Cayley graph on the symmetric group generated by the edges of \cG,
interpreted as transpositions. We show that, if \cG is complete multipartite,
the eigenvalues of the Laplacian of \Cay(\cG) have a simple expression in
terms of the irreducible characters of transpositions, and of the
Littlewood-Richardson coefficients. As a consequence we can prove that the
Laplacians of \cG and of \Cay(\cG) have the same first nontrivial
eigenvalue. This is equivalent to saying that Aldous's conjecture, asserting
that the random walk and the interchange process have the same spectral gap,
holds for complete multipartite graphs.Comment: 29 pages. Includes modification which appear on the published version
in J. Algebraic Combi
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