Digraphs with a fixed number of edges and vertices, having a maximal number of walks of length 2

TL;DR

This paper investigates the maximum number of directed walks of length 2 in digraphs with fixed edges and vertices, linking graph theory with algebraic growth concepts, and identifies optimal structures under certain conditions.

Contribution

It introduces a novel graph-theoretic problem inspired by algebraic growth, characterizes optimal digraphs as 'stars of saturated stars', and connects combinatorial optimization with algebraic concepts.

Findings

Maximal walks of length 2 are achieved by 'stars of saturated stars' structures.

The problem is equivalent to maximizing a quadratic form over partitions within an n-by-n box.

Optimal digraphs depend on mild restrictions on the number of vertices.

Abstract

Inspired by the work of Backelin on non-commutative correspondences to Macaulay's theorem of the growth of the Hilbert series of affine algebras, we study embedding dimension dependant versions of his degree 2 to degree 3 result. In graph-theoretical terms, we study the following question: what is the maximal number of directed walks of length 2 in a digraph with (k) edges and (n) vertices? The problem can also be formulated as follows: maximize (< \lambda, \lambda^T >) when (\lambda) is a partition of (k), contained in an (n \times n) box. We show that for mild restrictions on (n), optimal digraphs are the ``stars of saturated stars''.