73 research outputs found
Beyond graph energy: norms of graphs and matrices
In 1978 Gutman introduced the energy of a graph as the sum of the absolute
values of graph eigenvalues, and ever since then graph energy has been
intensively studied.
Since graph energy is the trace norm of the adjacency matrix, matrix norms
provide a natural background for its study. Thus, this paper surveys research
on matrix norms that aims to expand and advance the study of graph energy.
The focus is exclusively on the Ky Fan and the Schatten norms, both
generalizing and enriching the trace norm. As it turns out, the study of
extremal properties of these norms leads to numerous analytic problems with
deep roots in combinatorics.
The survey brings to the fore the exceptional role of Hadamard matrices,
conference matrices, and conference graphs in matrix norms. In addition, a vast
new matrix class is studied, a relaxation of symmetric Hadamard matrices.
The survey presents solutions to just a fraction of a larger body of similar
problems bonding analysis to combinatorics. Thus, open problems and questions
are raised to outline topics for further investigation.Comment: 54 pages. V2 fixes many typos, and gives some new materia
Balanced supersaturation for some degenerate hypergraphs
A classical theorem of Simonovits from the 1980s asserts that every graph
satisfying must contain copies of . Recently, Morris and
Saxton established a balanced version of Simonovits' theorem, showing that such
has copies of , which
are `uniformly distributed' over the edges of . Moreover, they used this
result to obtain a sharp bound on the number of -free graphs via the
container method. In this paper, we generalise Morris-Saxton's results for even
cycles to -graphs. We also prove analogous results for complete
-partite -graphs.Comment: Changed title, abstract and introduction were rewritte
Some remarks on the Zarankiewicz problem
The Zarankiewicz problem asks for an estimate on , the largest
number of 's in an matrix with all entries or
containing no submatrix consisting entirely of 's. We show that
a classical upper bound for due to K\H{o}v\'ari, S\'os and
Tur\'an is tight up to the constant for a broad range of parameters. The proof
relies on a new quantitative variant of the random algebraic method.Comment: 6 page
Some remarks on the Zarankiewicz problem
The Zarankiewicz problem asks for an estimate on z(m,n;s,t), the largest number of 1's in an m×n matrix with all entries 0 or 1 containing no s×t submatrix consisting entirely of 1's. We show that a classical upper bound for z(m,n;s,t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method
Extremal Graph Theory and Dimension Theory for Partial Orders
This dissertation analyses several problems in extremal combinatorics.In Part I, we study the following problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger. Given a graph H and an integer t, what is the minimum number of coloured edges in a t-edge-coloured graph G on n vertices such that G does not contain a rainbow copy of H, but adding a new edge to G in any colour creates a rainbow copy of H? We determine the growth rates of these numbers for almost all graphs H and all t e(H).In Part II, we study dimension theory for finite partial orders. In Chapter 1, we introduce and define the concepts we use in the succeeding chapters.In Chapter 2, we determine the dimension of the divisibility order on [n] up to a factor of (log log n).In Chapter 3, we answer a question of Kim, Martin, Masak, Shull, Smith, Uzzell, and Wang on the local bipartite covering numbers of difference graphs.In Chapter 4, we prove some bounds on the local dimension of any pair of layers of the Boolean lattice. In particular, we show that the local dimension of the first and middle layers is asymptotically n / log n.In Chapter 5, we introduce a new poset parameter called local t-dimension. We also discuss the fractional variants of this and other dimension-like parameters.All of Part I is joint work with Antnio Giro of the University of Cambridge and Kamil Popielarz of the University of Memphis.Chapter 2 of Part II is joint work with Victor Souza of IMPA (Instituto de Matemtica Pura e Aplicada, Rio de Janeiro).Chapter 3 of Part II is joint work with Antnio Giro
Graph properties, graph limits and entropy
We study the relation between the growth rate of a graph property and the
entropy of the graph limits that arise from graphs with that property. In
particular, for hereditary classes we obtain a new description of the colouring
number, which by well-known results describes the rate of growth.
We study also random graphs and their entropies. We show, for example, that
if a hereditary property has a unique limiting graphon with maximal entropy,
then a random graph with this property, selected uniformly at random from all
such graphs with a given order, converges to this maximizing graphon as the
order tends to infinity.Comment: 24 page
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