73 research outputs found

    Beyond graph energy: norms of graphs and matrices

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    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

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    A classical theorem of Simonovits from the 1980s asserts that every graph GG satisfying e(G)≫v(G)1+1/k{e(G) \gg v(G)^{1+1/k}} must contain ≳(e(G)v(G))2k\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k} copies of C2kC_{2k}. Recently, Morris and Saxton established a balanced version of Simonovits' theorem, showing that such GG has ≳(e(G)v(G))2k\gtrsim \left(\frac{e(G)}{v(G)}\right)^{2k} copies of C2kC_{2k}, which are `uniformly distributed' over the edges of GG. Moreover, they used this result to obtain a sharp bound on the number of C2kC_{2k}-free graphs via the container method. In this paper, we generalise Morris-Saxton's results for even cycles to Θ\Theta-graphs. We also prove analogous results for complete rr-partite rr-graphs.Comment: Changed title, abstract and introduction were rewritte

    Some remarks on the Zarankiewicz problem

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    The Zarankiewicz problem asks for an estimate on z(m,n;s,t)z(m, n; s, t), the largest number of 11's in an m×nm \times n matrix with all entries 00 or 11 containing no s×ts \times t submatrix consisting entirely of 11's. We show that a classical upper bound for z(m,n;s,t)z(m, n; s, t) 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

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    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

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    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

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    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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