7,535 research outputs found
Extremals of Functions on Graphs with Applications to Graphs and Hypergraphs
AbstractThe method used in an article by T. S. Matzkin and E. G. Straus [Canad. J. Math. 17 (1965), 533–540] is generalized by attaching nonnegative weights to t-tuples of vertices in a hypergraph subject to a suitable normalization condition. The edges of the hypergraph are given weights which are functions of the weights of its t-tuples and the graph is given the sum of the weights of its edges. The extremal values and the extremal points of these functions are determined. The results can be applied to various extremal problems on graphs and hypergraphs which are analogous to P. Turán's Theorem [Colloq. Math. 3 (1954), 19–30: (Hungarian) Mat. Fiz. Lapok 48 (1941), 436–452]
Normalized graph Laplacians for directed graphs
We consider the normalized Laplace operator for directed graphs with positive
and negative edge weights. This generalization of the normalized Laplace
operator for undirected graphs is used to characterize directed acyclic graphs.
Moreover, we identify certain structural properties of the underlying graph
with extremal eigenvalues of the normalized Laplace operator. We prove
comparison theorems that establish a relationship between the eigenvalues of
directed graphs and certain undirected graphs. This relationship is used to
derive eigenvalue estimates for directed graphs. Finally we introduce the
concept of neighborhood graphs for directed graphs and use it to obtain further
eigenvalue estimates.Comment: 40 pages, 3 figure
The minimal density of triangles in tripartite graphs
We determine the minimal density of triangles in a tripartite graph with
prescribed edge densities. This extends a previous result of Bondy, Shen,
Thomass\'e and Thomassen characterizing those edge densities guaranteeing the
existence of a triangle in a tripartite graph.
To be precise we show that a suitably weighted copy of the graph formed by
deleting a certain 9-cycle from has minimal triangle density among
all weighted tripartite graphs with prescribed edge densities.Comment: 44 pages including 12 page appendix of C++ cod
Extremal spectral behavior of weighted random -regular graphs
Analyzing the spectral behavior of random matrices with dependency among
entries is a challenging problem. The adjacency matrix of the random
-regular graph is a prominent example that has attracted immense interest. A
crucial spectral observable is the extremal eigenvalue, which reveals useful
geometric properties of the graph. According to the Alon's conjecture, which
was verified by Friedman, the (nontrivial) extremal eigenvalue of the random
-regular graph is approximately .
In the present paper, we analyze the extremal spectrum of the random
-regular graph (with fixed) equipped with random edge-weights, and
precisely describe its phase transition behavior with respect to the tail of
edge-weights. In addition, we establish that the extremal eigenvector is always
localized, showing a sharp contrast to the unweighted case where all
eigenvectors are delocalized. Our method is robust and inspired by a
sparsification technique developed in the context of Erd\H{o}s-R\'{e}nyi graphs
(Ganguly and Nam, '22), which can also be applied to analyze the spectrum of
general random matrices whose entries are dependent.Comment: 36 page
Box Graphs and Singular Fibers
We determine the higher codimension fibers of elliptically fibered Calabi-Yau
fourfolds with section by studying the three-dimensional N=2 supersymmetric
gauge theory with matter which describes the low energy effective theory of
M-theory compactified on the associated Weierstrass model, a singular model of
the fourfold. Each phase of the Coulomb branch of this theory corresponds to a
particular resolution of the Weierstrass model, and we show that these have a
concise description in terms of decorated box graphs based on the
representation graph of the matter multiplets, or alternatively by a class of
convex paths on said graph. Transitions between phases have a simple
interpretation as `flopping' of the path, and in the geometry correspond to
actual flop transitions. This description of the phases enables us to enumerate
and determine the entire network between them, with various matter
representations for all reductive Lie groups. Furthermore, we observe that each
network of phases carries the structure of a (quasi-)minuscule representation
of a specific Lie algebra. Interpreted from a geometric point of view, this
analysis determines the generators of the cone of effective curves as well as
the network of flop transitions between crepant resolutions of singular
elliptic Calabi-Yau fourfolds. From the box graphs we determine all fiber types
in codimensions two and three, and we find new, non-Kodaira, fiber types for
E_6, E_7 and E_8.Comment: 107 pages, 44 figures, v2: added case of E7 monodromy-reduced fiber
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