The set of vertices with positive curvature in a planar graph with nonnegative curvature

TL;DR

This paper establishes the maximum number of vertices with positive curvature in planar graphs with nonnegative curvature and demonstrates that such graphs have finite automorphism groups with bounded order.

Contribution

It provides a sharp upper bound for positive curvature vertices and proves finiteness and bounds for automorphism groups in these graphs.

Findings

Maximum number of positive curvature vertices established

Automorphism group of such graphs is finite

Upper bound for the order of automorphism group provided

Abstract

In this paper, we give the sharp upper bound for the number of vertices with positive curvature in a planar graph with nonnegative combinatorial curvature. Based on this, we show that the automorphism group of a planar---possibly infinite---graph with nonnegative combinatorial curvature and positive total curvature is a finite group, and give an upper bound estimate for the order of the group.