The number of possibilities for random dating

TL;DR

This paper derives compact formulas to estimate the expected number of subgraph copies within a random subgraph of a regular graph, aiding understanding of graph structure in probabilistic settings.

Contribution

It provides new, surprisingly compact formulas for counting subgraphs in random subgraphs of regular graphs, advancing combinatorial and probabilistic graph theory.

Findings

Derived compact formulas for subgraph counts in random regular graphs

Enhanced understanding of subgraph distribution in probabilistic graph models

Applicable to various problems in graph theory and network analysis

Abstract

Let $G$ be a regular graph and $H$ a subgraph on the same vertex set. We give surprisingly compact formulas for the number of copies of $H$ one expects to find in a random subgraph of $G$.