Dual Cheeger constant for weighted graphs over ordered fields

TL;DR

This paper introduces a dual Cheeger constant for weighted graphs over arbitrary real-closed ordered fields, providing estimates and eigenvalue bounds, including for graphs over non-Archimedean fields, with demonstrated sharpness.

Contribution

It extends the concept of the Cheeger constant to dual form for weighted graphs over arbitrary ordered fields, with new estimates and eigenvalue bounds, including non-Archimedean cases.

Findings

Derived bounds for the dual Cheeger constant in terms of graph vertices.

Estimated the largest eigenvalue of the Laplace operator using the dual Cheeger constant.

Showed the sharpness of the derived estimates.

Abstract

We consider a dual Cheeger constant $\overline h$ for finite graphs with edge weights from an arbitrary real-closed ordered field. We obtain estimates of $\overline h$ in terms of number of vertices in graph. Further, we estimate the largest eigenvalue for the discrete Laplace operator in terms of $\overline h$ and show the sharpness of estimates. As an example we consider graphs over non-Archimedean field of the Levi-Civita numbers.