65 research outputs found
On the spectrum of hypergraphs
Here we study the spectral properties of an underlying weighted graph of a
non-uniform hypergraph by introducing different connectivity matrices, such as
adjacency, Laplacian and normalized Laplacian matrices. We show that different
structural properties of a hypergrpah, can be well studied using spectral
properties of these matrices. Connectivity of a hypergraph is also investigated
by the eigenvalues of these operators. Spectral radii of the same are bounded
by the degrees of a hypergraph. The diameter of a hypergraph is also bounded by
the eigenvalues of its connectivity matrices. We characterize different
properties of a regular hypergraph characterized by the spectrum. Strong
(vertex) chromatic number of a hypergraph is bounded by the eigenvalues.
Cheeger constant on a hypergraph is defined and we show that it can be bounded
by the smallest nontrivial eigenvalues of Laplacian matrix and normalized
Laplacian matrix, respectively, of a connected hypergraph. We also show an
approach to study random walk on a (non-uniform) hypergraph that can be
performed by analyzing the spectrum of transition probability operator which is
defined on that hypergraph. Ricci curvature on hypergraphs is introduced in two
different ways. We show that if the Laplace operator, , on a hypergraph
satisfies a curvature-dimension type inequality
with and then any non-zero eigenvalue of can be bounded below by . Eigenvalues of a normalized Laplacian operator defined on a connected
hypergraph can be bounded by the Ollivier's Ricci curvature of the hypergraph
On the -spectral radius of hypergraphs
For real and a hypergraph , the -spectral radius
of is the largest eigenvalue of the matrix , where is the adjacency matrix of , which is a
symmetric matrix with zero diagonal such that for distinct vertices of
, the -entry of is exactly the number of edges containing both
and , and is the diagonal matrix of row sums of . We study
the -spectral radius of a hypergraph that is uniform or not necessarily
uniform. We propose some local grafting operations that increase or decrease
the -spectral radius of a hypergraph. We determine the unique
hypergraphs with maximum -spectral radius among -uniform hypertrees,
among -uniform unicyclic hypergraphs, and among -uniform hypergraphs with
fixed number of pendant edges. We also determine the unique hypertrees with
maximum -spectral radius among hypertrees with given number of vertices
and edges, the unique hypertrees with the first three largest (two smallest,
respectively) -spectral radii among hypertrees with given number of
vertices, the unique hypertrees with minimum -spectral radius among the
hypertrees that are not -uniform, the unique hypergraphs with the first two
largest (smallest, respectively) -spectral radii among unicyclic
hypergraphs with given number of vertices, and the unique hypergraphs with
maximum -spectral radius among hypergraphs with fixed number of pendant
edges
Note on power hypergraphs with equal domination and matching numbers
We present some examples that refute two recent results in the literature
concerning the equality of the domination and matching numbers for power and
generalized power hypergraphs. In this note we pinpoint the flaws in the proofs
and suggest how they may be mended.Comment: 7 pages, 1 figure, XIII Encuentro Andaluz de Matem\'atica Discreta,
(C\'adiz) Spain, july, 202
Principal eigenvectors and principal ratios in hypergraph Tur\'an problems
For a general class of hypergraph Tur\'an problems with uniformity , we
investigate the principal eigenvector for the -spectral radius (in the sense
of Keevash--Lenz--Mubayi and Nikiforov) for the extremal graphs, showing in a
strong sense that these eigenvectors have close to equal weight on each vertex
(equivalently, showing that the principal ratio is close to ). We
investigate the sharpness of our result; it is likely sharp for the Tur\'an
tetrahedron problem.
In the course of this latter discussion, we establish a lower bound on the
-spectral radius of an arbitrary -graph in terms of the degrees of the
graph. This builds on earlier work of Cardoso--Trevisan, Li--Zhou--Bu,
Cioab\u{a}--Gregory, and Zhang.
The case of our results leads to some subtleties connected to
Nikiforov's notion of -tightness, arising from the Perron-Frobenius theory
for the -spectral radius. We raise a conjecture about these issues, and
provide some preliminary evidence for our conjecture.Comment: 21 pages, 1 figure. Dedicated to the memory of Vladimir Nikiforo
Connections Between Extremal Combinatorics, Probabilistic Methods, Ricci Curvature of Graphs, and Linear Algebra
This thesis studies some problems in extremal and probabilistic combinatorics, Ricci curvature of graphs, spectral hypergraph theory and the interplay between these areas. The first main focus of this thesis is to investigate several Ramsey-type problems on graphs, hypergraphs and sequences using probabilistic, combinatorial, algorithmic and spectral techniques: The size-Ramsey number RË(G, r) is defined as the minimum number of edges in a hypergraph H such that every r-edge-coloring of H contains a monochromatic copy of G in H. We improved a result of Dudek, La Fleur, Mubayi and Rödl [ J. Graph Theory 2017 ] on the size-Ramsey number of tight paths and extended it to more colors.
An edge-colored graph G is called rainbow if every edge of G receives a different color. The anti-Ramsey number of t edge-disjoint rainbow spanning trees, denoted by r(n, t), is defined as the maximum number of colors in an edge-coloring of Kn containing no t edge-disjoint rainbow spanning trees. Confirming a conjecture of Jahanbekam and West [J. Graph Theory 2016], we determine the anti-Ramsey number of t edge-disjoint rainbow spanning trees for all values of n and t.
We study the extremal problems on Berge hypergraphs. Given a graph G = (V, E), a hypergraph H is called a Berge-G, denoted by BG, if there exists an injection i ⶠV (G) â V (H) and a bijection f ⶠE(G) â E(H) such that for every e = uv â E(G), (i(u), i(v)) â f(e). We investigate the hypergraph Ramsey number of Berge cliques, the cover-Ramsey number of Berge hypergraphs, the cover-TurĂĄn desity of Berge hypergraphs as well as Hamiltonian Berge cycles in 3-uniform hypergraphs.
The second part of the thesis uses the âgeometryâ of graphs to derive concentration inequalities in probabilities spaces. We prove an Azuma-Hoeffding-type inequality in several classical models of random configurations, including the ErdĆs-RĂ©nyi random graph models G(n, p) and G(n,M), the random d-out(in)-regular directed graphs, and the space of random permutations. The main idea is using Ollivierâs work on the Ricci curvature of Markov chairs on metric spaces. We give a cleaner form of such concentration inequality in graphs. Namely, we show that for any Lipschitz function f on any graph (equipped with an ergodic random walk and thus an invariant distribution Îœ) with Ricci curvature at least Îș \u3e 0, we have
Îœ (âŁf â EÎœf⣠℠t) †2 exp (-t 2Îș/7).
The third part of this thesis studies a problem in spectral hypergraph theory, which is the interplay between graph theory and linear algebra. In particular, we study the maximum spectral radius of outerplanar 3-uniform hypergraphs. Given a hypergraph H, the shadow of H is a graph G with V (G) = V (H) and E(G) = {uv ⶠuv â h for some h â E(H)}. A 3-uniform hypergraph H is called outerplanar if its shadow is outerplanar and all faces except the outer face are triangles, and the edge set of H is the set of triangle faces of its shadow. We show that the outerplanar 3-uniform hypergraph on n vertices of maximum spectral radius is the unique hypergraph with shadow K1 + Pnâ1
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