Twisted edge Laplacians on finite graphs from a K\"{a}hler structure

TL;DR

This paper introduces a Kahler structure on finite graphs, explores the twisted edge Laplacian, and analyzes metric properties using noncommutative geometry tools, revealing finite diameter with respect to Connes' distance.

Contribution

It presents a novel Kahler structure on finite graphs and studies the associated twisted edge Laplacian and metric properties using spectral triples.

Findings

Finite diameter of points with respect to Connes' distance

Introduction of a Kahler structure on finite graphs

Analysis of the twisted holomorphic Dolbeault-Dirac spectral triple

Abstract

In this paper we study a Kahler structure on finite points. In particular, we study the edge Laplacian of a graph twisted by the Kahler structure introduced in this paper. We also discuss a metric aspect from a twisted holomorphic Dolbeault-Dirac spectral triple and show that the points have a finite diameter with respect to Connes' distance.