The Rise of the Algorithmic Medium
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# Laplacian matrix - Wikipedia, the free encyclopedia

In the mathematical field of graph theory the Laplacian matrix, sometimes called admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Together with Kirchhoff's theorem it can be used to calculate the number of spanning trees for a given graph. The Laplacian matrix can be used to find many other properties of the graph; see spectral graph theory. Cheeger's inequality from Riemannian geometry has a discrete analogue involving the Laplacian Matrix; this is perhaps the most important theorem in Spectral Graph theory and one of the most useful facts in algorithmic applications. It approximates the sparsest cut of a graph through the second eigenvalue of its Laplacian.

Given a simple graph G with n vertices, its Laplacian matrix $L:=(\ell_{i,j})_{n \times n}$ is defined as:[1]

That is, it is the difference of the degree matrix D and the adjacency matrix A of the graph. In the case of directed graphs, either the indegree or outdegree might be used, depending on the application.

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# The Rise of the Algorithmic Medium

Algorithms are the new medium between people and people, people and data, data and data. Artificial intelligence, cognitive computing, machine learning, neural networks, internet of things, smart cities, cryptocurrencies...
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