Lecture

Laplacian Matrix: Properties and Examples

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In course

ME-427: Networked control systems

This course offers an introduction to control systems using communication networks for interfacing sensors, actuators, controllers, and processes. Challenges due to network non-idealities and opportunities offered by communication will be analyzed. ...

Description

This lecture covers the Laplacian matrix, its definition, properties, and examples associated with undirected graphs. It discusses time-varying consensus theorems, globally reachable nodes, and achieving average consensus in networked control systems. The lecture also explores the Laplacian matrix, consensus in continuous time, and balanced graphs. Properties of the Laplacian matrix, such as off-diagonal elements, zero row-sums, and eigenvalues, are explained. The association between digraphs and undirected graphs, Laplacian matrices, and adjacency matrices is detailed, along with examples and propositions related to graph connectivity.

Instructor

Giancarlo Ferrari Trecate

Expertise Automatic control, Dependable control, Neural-network Control, Machine learning, Microgrids, Hybrid systems Mission See the webpage of the DECODE (Dependable Control and Decision) group. Teaching & PhD PhD Students Laura Meroi, Julien Pallage, Nicolas Léo Pierre Kirsch, Simone Baratto, Daniele Martinelli, Riccardo Cescon, Jakob Nylöf, Sabri El Amrani Past EPFL PhD Students Pulkit Nahata, Mustafa Sahin Turan, Baiwei Guo, Jean-Sébastien Brouillon, Clara Galimberti, Andrea Martin, Muhammad Zakwan,

Official source

Ontological neighbourhood

Mathematics

Discrete mathematics: Graph theory

Related concepts (32)

Connectivity (graph theory)

In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. In an undirected graph G, two vertices u and v are called connected if G contains a path from u to v.

Graph (discrete mathematics)

In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges.

Mixed graph

In graph theory, a mixed graph G = (V, E, A) is a graph consisting of a set of vertices V, a set of (undirected) edges E, and a set of directed edges (or arcs) A. Consider adjacent vertices . A directed edge, called an arc, is an edge with an orientation and can be denoted as or (note that is the tail and is the head of the arc). Also, an undirected edge, or edge, is an edge with no orientation and can be denoted as or . For the purpose of our application example we will not be considering loops or multiple edges of mixed graphs.

Signed graph

In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign. A signed graph is balanced if the product of edge signs around every cycle is positive. The name "signed graph" and the notion of balance appeared first in a mathematical paper of Frank Harary in 1953. Dénes Kőnig had already studied equivalent notions in 1936 under a different terminology but without recognizing the relevance of the sign group.

K-vertex-connected graph

In graph theory, a connected graph G is said to be k-vertex-connected (or k-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest k for which the graph is k-vertex-connected. A graph (other than a complete graph) has connectivity k if k is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them.

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