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We show that if a sequence of dense graphs Gn has the property that for every fixed graph F , the density of copies of F in Gn tends to a limit, then there is a natural " limit object " , namely a symmetric measurable function W : [0, 1] 2 → [0, 1]. This limit object determines all the limits of subgraph densities. Conversely, every such function arises as… (More)

The <i>exponent of matrix multiplication</i> is the smallest real number ω such that for all ε>0, <i>O</i>(n<sup>ω+ε</sup>) arithmetic operations suffice to multiply two <i>n×n</i> matrices. The standard algorithm for matrix multiplication shows that ω≤3. Strassen's remarkable result [5] shows that ω≤2.81,… (More)

We study generalizations of the " contraction-deletion " relation of the Tutte polynomial, and other similar simple operations, to other graph parameters. The question can be set in the framework of graph algebras introduced by Freedman, Lovász and Schrijver in [2], and it relates to their behavior under basic graph operations like contraction and… (More)

We define a distance of two graphs that reflects the closeness of both local and global properties. We also define convergence of a sequence of graphs, and show that a graph sequence is convergent if and only if it is Cauchy in this distance. Every convergent graph sequence has a limit in the form of a symmetric measurable function in two variables. We use… (More)

We investigate families of graphs and graphons (graph limits) that are defined by a finite number of prescribed subgraph densities. Our main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities. Generalizing results of Turán, Erd˝ os–Simonovits and Chung–Graham–Wilson, we… (More)

In an earlier paper the authors proved that limits of convergent graph sequences can be described by various structures, including certain 2-variable real functions called graphons, random graph models satisfying certain consistency conditions, and normalized, multiplica-tive and reflection positive graph parameters. In this paper we show that each of these… (More)

Matroids allowing an odd ear decomposition can be viewed as natural generalizations of factor-critical graphs. We show that a matroid representable over a eld of characteristic 2 allows an odd ear decomposition if and only if it can be represented by a space on which the induced scalar product is a nondegenerate symplectic form. We will also show that for a… (More)

We study the question how many subgroups, cosets or subspaces are needed to cover a finite Abelian group or a vector space if we have some natural restrictions on the structure of the covering system. For example we determine, how many cosets we need, if we want to cover all but one element of an Abelian group. This result is a group theoretical extension… (More)