452 research outputs found
The extremal spectral radii of -uniform supertrees
In this paper, we study some extremal problems of three kinds of spectral
radii of -uniform hypergraphs (the adjacency spectral radius, the signless
Laplacian spectral radius and the incidence -spectral radius).
We call a connected and acyclic -uniform hypergraph a supertree. We
introduce the operation of "moving edges" for hypergraphs, together with the
two special cases of this operation: the edge-releasing operation and the total
grafting operation. By studying the perturbation of these kinds of spectral
radii of hypergraphs under these operations, we prove that for all these three
kinds of spectral radii, the hyperstar attains uniquely the
maximum spectral radius among all -uniform supertrees on vertices. We
also determine the unique -uniform supertree on vertices with the second
largest spectral radius (for these three kinds of spectral radii). We also
prove that for all these three kinds of spectral radii, the loose path
attains uniquely the minimum spectral radius among all
-th power hypertrees of vertices. Some bounds on the incidence
-spectral radius are given. The relation between the incidence -spectral
radius and the spectral radius of the matrix product of the incidence matrix
and its transpose is discussed
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
Extremal problems for the p-spectral radius of graphs
The -spectral radius of a graph of order is defined for any real
number as
The most remarkable feature of is that it
seamlessly joins several other graph parameters, e.g., is the Lagrangian, is the spectral
radius and is the number of edges. This
paper presents solutions to some extremal problems about , which are common generalizations of corresponding edge and
spectral extremal problems.
Let be the -partite Tur\'{a}n graph of order
Two of the main results in the paper are:
(I) Let and If is a -free graph of order
then unless
(II) Let and If is a graph of order with then has an edge contained in at least
cliques of order where is a positive number depending
only on and Comment: 21 pages. Some minor corrections in v
Intersecting families of discrete structures are typically trivial
The study of intersecting structures is central to extremal combinatorics. A
family of permutations is \emph{-intersecting} if
any two permutations in agree on some indices, and is
\emph{trivial} if all permutations in agree on the same
indices. A -uniform hypergraph is \emph{-intersecting} if any two of its
edges have vertices in common, and \emph{trivial} if all its edges share
the same vertices.
The fundamental problem is to determine how large an intersecting family can
be. Ellis, Friedgut and Pilpel proved that for sufficiently large with
respect to , the largest -intersecting families in are the trivial
ones. The classic Erd\H{o}s--Ko--Rado theorem shows that the largest
-intersecting -uniform hypergraphs are also trivial when is large. We
determine the \emph{typical} structure of -intersecting families, extending
these results to show that almost all intersecting families are trivial. We
also obtain sparse analogues of these extremal results, showing that they hold
in random settings.
Our proofs use the Bollob\'as set-pairs inequality to bound the number of
maximal intersecting families, which can then be combined with known stability
theorems. We also obtain similar results for vector spaces.Comment: 19 pages. Update 1: better citation of the Gauy--H\`an--Oliveira
result. Update 2: corrected statement of the unpublished Hamm--Kahn result,
and slightly modified notation in Theorem 1.6 Update 3: new title, updated
citations, and some minor correction
Some extremal problems for hereditary properties of graphs
This note answers extremal questions like: what is the maximum number of
edges in a graph of order n, which belongs to some hereditary property. The
same question is answered also for the spectral radius and other similar
parameters
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