An exact closed walks series formula for the complexity of regular graphs and some related bounds

TL;DR

This paper derives an exact formula for the complexity of regular graphs using closed walks, providing bounds and applications to biological neuronal activity.

Contribution

It introduces a novel formula linking graph complexity to closed walks, expanding understanding of regular graphs and their applications.

Findings

Exact formula for regular graph complexity established

Infinite bounds on complexity derived from the formula

Applications to neuronal activity in biology

Abstract

The complexity of a graph is the number of its labeled spanning trees. In this work complexity is studied in settings that admit regular graphs. An exact formula is established linking complexity of the complement of a regular graph to numbers of closed walks in the graph by way of an infinite alternating series. Some consequences of this result yield infinite classes of lower and upper bounds on the complexity of such graphs. Applications of these mathematical results to biological problems on neuronal activity are described.