Periodic graphs with coincident edges: folding-ladder and related graphs

Area of Science:

• Graph Theory
• Geometric Graph Theory
• Combinatorial Geometry

Background:

• Ladder graphs possess maximum-symmetry embeddings where edges can coincide.
• Folding ladders are a specific type of ladder graph without zero-length edges.

Purpose of the Study:

• To investigate high-symmetry 3-periodic folding ladders.
• To analyze the structural properties of 3-periodic vertex- and edge-transitive folding ladders.
• To determine the volume characteristics of coincident-edge configurations in these ladders.

Main Methods:

• Exploration of graph embeddings with maximum symmetry.
• Analysis of 3-periodic structures, focusing on vertex- and edge-transitivity.
• Geometric analysis of folding ladder configurations.

Main Results:

• Identification of high-symmetry 3-periodic folding ladders.
• Demonstration that the coincident-edge configuration in these ladders maximizes volume for a fixed edge length.
• Establishment that these configurations are isomegetic to higher-symmetry 3-periodic graphs.

Conclusions:

• The study provides concrete examples of high-symmetry 3-periodic folding ladders.
• The maximum volume configuration for folding ladders has significant geometric implications.
• These findings connect folding ladders to broader concepts in graph theory and geometry.