Inducibility of the Net Graph

TL;DR

This paper investigates the maximum number of induced net graphs in larger graphs, showing near-fractal behavior and identifying the extremal structures for large sizes using advanced combinatorial techniques.

Contribution

It demonstrates that the net graph is nearly a fractalizer and characterizes the extremal graphs maximizing induced copies for large sizes using flag algebra and stability methods.

Findings

Maximum induced net graphs follow a recursive formula for large n.

The extremal graphs are balanced iterated blow ups of the net.

The unique extremal structure for sizes 6^k is identified.

Abstract

A graph $F$ is called a fractalizer if for all $n$ the only graphs which maximize the number of induced copies of $F$ on $n$ vertices are the balanced iterated blow ups of $F$. While the net graph is not a fractalizer, we show that the net is nearly a fractalizer. Let $N(n)$ be the maximum number of induced copies of the net graph among all graphs on $n$ vertices. For sufficiently large $n$ we show that, $N(n) = x_1\cdot x_2 \cdot x_3 \cdot x_4 \cdot x_5 \cdot x_6 + N(x_1) + N(x_2) + N(x_3) + N(x_4) + N(x_5) + N(x_6)$ where $\sigma x_i = n$ and all $x_i$ are as equal as possible. Furthermore, we show that the unique graph which maximizes $N(6^k)$ is the balanced iterated blow up of the net for $k$ sufficiently large. We expand on the standard flag algebra and stability techniques through more careful counting and numerical optimization techniques.