Counting Cycles and Finite Dimensional L Norms

We obtain sharp bounds for the number of n–cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. We prove sharp estimates on both ∑n i=1 x k i and ∑n i=1 |xi|, subject to the constraints that ∑n i=1 x 2 i = C and ∑n i=1 xi = 0. Introduction This note was inspired by the following question, which had been asked at the oral entrance exams, see [5], to the Moscow State University Mathematics Department (MekhMat) to certain applicants: Question 1. Let G be a graph with E edges. Let T be the number of triangles of G. Show that there exists a constant C, such that T ≤ CE for all G. Before proceeding any further, let us answer the question. We will assume that G is a simple, loopless, undirected graph — that is, there is exactly one edge connecting two vertices v and w of G, and there are no edges whose two endpoints are actually the same vertex. We will need the following Definition 1. The adjacency matrix A(G) is the matrix with entries