The blow-up of a graph H is the graph obtained from replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different. Erdős et al. and Chen et al. determined the extremal number of blow-ups of stars. Glebov determined the extremal number and found all extremal graphs for blowups of paths. We determine the extremal number and find the extremal graphs ...

The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of ho...

Let G be a simple undirected graph. Denote by mi(G) (respectively, xi(G)) the number of maximal (respectively, maximum) independent sets in G. Erdős and Moser raised the problem of determining the maximum value of mi(G) among all graphs of order n and the extremal graphs achieving this maximum value. This problem was solved by Moon and Moser. Then it was studied for many special classes of grap...

The minimum size of a k-connected graph with given order and stability number is investigated. If no connectivity is required, the answer is given by Turán’s Theorem. For connected graphs, the problem has been solved recently independently by Christophe et al., and by Gitler and Valencia. In this paper, we give a short proof of their result and determine the extremal graphs. We settle the case ...