# Random graphs induced by Catalan pairs

## TL;DR

This paper investigates the properties of random Catalan-pair graphs, focusing on their expected edges, isolated vertices, and subgraph counts, revealing insights into their structural characteristics.

## Contribution

It introduces the study of random Catalan-pair graphs and derives asymptotic formulas for key graph properties, a novel exploration in this graph family.

## Key findings

- Expected number of edges determined asymptotically
- Expected number of isolated vertices calculated
- Expected counts of specific subgraphs derived

## Abstract

We consider Catalan-pair graphs, a family of graphs that can be viewed as representing certain interactions between pairs of objects which are enumerated by the Catalan numbers. In this paper we study random Catalan-pair graphs and deduce various properties of these random graphs. In particular, we asymptotically determine the expected number of edges and isolated vertices, and more generally we determine the expected number of (induced) subgraphs isomorphic to a given connected graph.