353,243 research outputs found
Group twin coloring of graphs
For a given graph , the least integer such that for every
Abelian group of order there exists a proper edge labeling
so that for each edge is called the \textit{group twin
chromatic index} of and denoted by . This graph invariant is
related to a few well-known problems in the field of neighbor distinguishing
graph colorings. We conjecture that for all graphs
without isolated edges, where is the maximum degree of , and
provide an infinite family of connected graph (trees) for which the equality
holds. We prove that this conjecture is valid for all trees, and then apply
this result as the base case for proving a general upper bound for all graphs
without isolated edges: , where
denotes the coloring number of . This improves the best known
upper bound known previously only for the case of cyclic groups
The Milnor degree of a three-manifold
The Milnor degree of a 3-manifold is an invariant that records the maximum
simplicity, in terms of higher order linking, of any link in the 3-sphere that
can be surgered to give the manifold. This invariant is investigated in the
context of torsion linking forms, nilpotent quotients of the fundamental group,
Massey products and quantum invariants, and the existence of 3-manifolds with
any prescribed Milnor degree and first Betti number is established.
Along the way, it is shown that the number M(k,r) of linearly independent
Milnor invariants of degree k, distinguishing r-component links in the 3-sphere
whose lower degree invariants vanish, is positive except in the classically
known cases (when r = 1, and when r = 2 with k = 2, 4 or 6).Comment: This version, to appear in Journal of Topology, includes only minor
revisions, and added proof of Corollary 1.
Topological Data Analysis of Task-Based fMRI Data from Experiments on Schizophrenia
We use methods from computational algebraic topology to study functional
brain networks, in which nodes represent brain regions and weighted edges
encode the similarity of fMRI time series from each region. With these tools,
which allow one to characterize topological invariants such as loops in
high-dimensional data, we are able to gain understanding into low-dimensional
structures in networks in a way that complements traditional approaches that
are based on pairwise interactions. In the present paper, we use persistent
homology to analyze networks that we construct from task-based fMRI data from
schizophrenia patients, healthy controls, and healthy siblings of schizophrenia
patients. We thereby explore the persistence of topological structures such as
loops at different scales in these networks. We use persistence landscapes and
persistence images to create output summaries from our persistent-homology
calculations, and we study the persistence landscapes and images using
-means clustering and community detection. Based on our analysis of
persistence landscapes, we find that the members of the sibling cohort have
topological features (specifically, their 1-dimensional loops) that are
distinct from the other two cohorts. From the persistence images, we are able
to distinguish all three subject groups and to determine the brain regions in
the loops (with four or more edges) that allow us to make these distinctions
Distinguishing partitions of complete multipartite graphs
A \textit{distinguishing partition} of a group with automorphism group
is a partition of that is fixed by no nontrivial element of
. In the event that is a complete multipartite graph with its
automorphism group, the existence of a distinguishing partition is equivalent
to the existence of an asymmetric hypergraph with prescribed edge sizes. An
asymptotic result is proven on the existence of a distinguishing partition when
is a complete multipartite graph with parts of size and
parts of size for small , and large , . A key tool
in making the estimate is counting the number of trees of particular classes
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