Matrix Energy as a Measure of Topological Complexity of a Graph

TL;DR

This paper introduces matrix energy as a novel measure of topological complexity in graphs, providing a formal metric that captures global connectivity features and identifying a critical boundary point in graph density.

Contribution

It develops the matrix energy measure for graph complexity, demonstrates its validity, and introduces the P point concept as a connectivity regime boundary.

Findings

Matrix energy satisfies Weyuker criteria as a complexity measure.

The P point delineates different connectivity regimes in finite graphs.

The metric offers insights into global connectivity patterns.

Abstract

The complexity of highly interconnected systems is rooted in the interwoven architecture defined by its connectivity structure. In this paper, we develop matrix energy of the underlying connectivity structure as a measure of topological complexity and highlight interpretations about certain global features of underlying system connectivity patterns. The proposed complexity metric is shown to satisfy the Weyuker criteria as a measure of its validity as a formal complexity metric. We also introduce the notion of P point in the graph density space. The P point acts as a boundary between multiple connectivity regimes for finite-size graphs.