In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix, or discrete Laplacian, is a matrix representation of a graph.

The Laplacian allows a natural link between discrete representations, such as graphs, and continuous representations, such as vector spaces and manifolds. The most important …

For any oriented graph G obtained from the underlying graph of G, the rank of the incidence matrix B is equal to m c, where c is the number of connected components of the underlying …

Graph Laplacian is used to study the structure and properties of graphs. It provides information about various aspects of a graph, such as connectivity, diffusion processes, and spectral …

Nov 11, 2020 · In this post, I’ll walk through the intuition behind the graph Laplacian and describe how it represents the discrete analogue to the Laplacian operator on continuous multivariate …

The goal of this set of notes is to demonstrate a simple way to divide a graph into two “strongly connected” subgraphs when possible. For example consider the graph:

HANCHEN LI f graphs with results from linear algebra. This paper aims to introduce properties of the graph Laplacian and show how these properties can be utilized to help generate insights …

Feb 9, 2025 · This document explores three core aspects: the basic mathematical foundation of Graph Laplacians, their application in semi-supervised learning through smoothing techniques, …

Before we can define the Laplacian matrix of a graph we need the notion of an orientation on a graph. An orientation of is an assignment of a direction to each edge by declaring one vertex …

The Laplacian matrix of a graph carries the same information as the adjacency matrix obvi-ously, but has different useful and important properties, many relating to its spectrum. We start with a …