Super-Walk Formulae for Even and Odd Laplacians in Finite Graphs

TL;DR

This paper introduces new formulas linking graph Laplacians to specific walk counts in finite graphs, enabling efficient computation of walk signs based on graph orientation.

Contribution

It presents novel theorems connecting Laplacians with walk enumeration, including methods to count walk signs in oriented finite graphs.

Findings

Established formulas for counting walks using Laplacians

Defined two types of walks with orientation-based sign counting

Provided theoretical tools for analyzing walk structures in graphs

Abstract

The number of walks from one vertex to another in a finite graph can be counted by the adjacency matrix. In this paper, we prove two theorems that connect the graph Laplacian with two types of walks in a graph. By defining two types of walks and giving orientation to a finite graph, one can easily count the number of the total signs of each kind of walk from one element to another of a fixed length.