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We study the maximum number $ex(n,e,H)$ of copies of a graph $H$ in graphs with given number of vertices and edges. We show that for any fixed graph $H$, $ex(n,e,H)$ is asymptotically realized by the quasi-clique provided that the edge density is sufficiently large. We also investigate a variant of this problem, when the host graph is bipartite.
Let k_r(n,m) denote the minimum number of r-cliques in graphs with n vertices and m edges. For r=3,4 we give a lower bound on k_r(n,m) that approximates k_r(n,m) with an error smaller than n^r/(n^2-2m). The solution is based on a constraint minimizat
What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turan, Rademacher solved the first non-trivial case of this problem in 1941. The problem was revived by ErdH{o}s in 1955; it is now
Motivated by work of ErdH{o}s, Ota determined the maximum size $g(n,k)$ of a $k$-connected nonhamiltonian graph of order $n$ in 1995. But for some pairs $n,k,$ the maximum size is not attained by a graph of connectivity $k.$ For example, $g(15,3)=77$
The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced triangle-free graphs with a
Given a digraph $D$ with $m$ arcs and a bijection $tau: A(D)rightarrow {1, 2, ldots, m}$, we say $(D, tau)$ is an antimagic orientation of a graph $G$ if $D$ is an orientation of $G$ and no two vertices in $D$ have the same vertex-sum under $tau$, wh