We consider the problem of minimizing the number of triangles in a graph of given order and size and describe the asymptotic structure of extremal graphs. This is achieved by characterizing the set of flag algebra homomorphisms that minimize the triangle density.

A non-uniform hypergraph H = (V,E) consists of a vertex set V and an edge set E ⊆ 2V ; the edges in E are not required to all have the same cardinality. The set of all cardinalities of edges in H is denoted by R(H), the set of edge types. For a fixed hypergraph H, the Turán density π(H) is defined to be limn→∞maxGn hn(Gn), where the maximum is taken over all H-free hypergraphs Gn on n vertices ...

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A graph G = (V, E) is matching immune if there is no matching cut in G. We show that for any matching immune graph G, |E| ≥ ⌈3(|V | − 1)/2⌉. This bound is tight, as we define operations that construct, from a given vertex, exactly the class of matching immune graphs that attain the bound.

Usual graph classes, such as complete graphs, paths, cycles and stars, frequently appear as extremal graphs in graph theory problems. Here we want to turn the reader’s attention to two novel, simply defined, graph classes that appear as extremal graphs in several graph theory problems. We call them bags and bugs. As examples of problems where bags and bugs appear, we show that balanced bugs max...

My research interests lie in spectral and extremal graph theory, as well as in the analysis of complex networks. A primary goal is to deduce fundamental structural graph properties from the spectrum of the graph or a few easily-computable invariants. In order to gain control over these invariants, extremal graph theory aims to tightly bound them and characterize the graphs achieving these bound...

This paper proves limit theorems for the number of monochromatic edges in uniform random colorings of general random graphs. The limit theorems are universal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of colors, and works for any graph sequence deterministic or random. The proofs are based on moment calculations which relates to res...