Lecture 4 : Turán ’ s theorem and Erdős - Stone theorem

The lower bound is the same as before, due to the fact that Tr(n) is Kr+1-free. Define the adjacency matrix A = A(G) = (aij) for a graph G of order n. Let V = {v1, · · · , vn}. Then A is a n-by-n 0 − 1 matrix such that aij = 1 if and only if vivj ∈ E. Thus A is symmetric. We will be interested in a quadratic form ⟨Ax,x⟩ where x denotes a vector of length n. This is often called the Lagrangian of G. Define