On the limit value of compactness of some graph classes

Area of Science:

• Graph theory
• Network analysis
• Hypermedia systems

Background:

• The graph index, limit of compactness (CB), measures hypermedia structure.
• Previous studies observed CB approaching 1 for large-scale small-world graphs.
• The origin of this limit behavior (specific to small-world graphs or general) was unclear.

Purpose of the Study:

• To determine the necessary and sufficient conditions for any sequence of connected graphs to exhibit a limit of compactness (CB) of 1.
• To generalize these findings for disconnected graph classes.
• To provide a theoretical framework for calculating graph compactness limits efficiently.

Main Methods:

• Proof-theoretical analysis of graph sequences.
• Derivation of necessary and sufficient conditions for CB=1 limit.
• Generalization of results for disconnected graphs.

Main Results:

• Established the precise conditions for a sequence of connected graphs to achieve a limit of compactness (CB) of 1.
• Extended these conditions to encompass disconnected graph classes.
• Demonstrated the applicability of the findings through four illustrative graph class examples.

Conclusions:

• The limit behavior of graph compactness (CB=1) is not exclusive to small-world graphs.
• The derived conditions provide a general theoretical framework for understanding graph compactness limits.
• This proof-theoretical approach offers computational savings for analyzing graph classes.