Statistics of planar graphs viewed from a vertex: A study via labeled trees

TL;DR

This paper analyzes the local structure of random planar graphs near a vertex using bijections with labeled trees, providing exact formulas for local statistics and probabilities in large graphs.

Contribution

It introduces a novel method to derive exact local statistics of planar graphs via bijections with labeled trees, extending understanding of their local structure.

Findings

Exact generating functions for local graph statistics

Explicit formulas for probabilities of local configurations

Moments of local degree distributions computed

Abstract

We study the statistics of edges and vertices in the vicinity of a reference vertex (origin) within random planar quadrangulations and Eulerian triangulations. Exact generating functions are obtained for theses graphs with fixed numbers of edges and vertices at given geodesic distances from the origin. Our analysis relies on bijections with labeled trees, in which the labels encode the information on the geodesic distance from the origin. In the case of infinitely large graphs, we give in particular explicit formulas for the probabilities that the origin have given numbers of neighboring edges and/or vertices, as well as explicit values for the corresponding moments.