Laplacian matrices of graphs: a survey

doi:10.1016/0024-3795(94)90486-3

Linear Algebra and its Applications

Volumes 197–198, January–February 1994, Pages 143-176

Dedicated to Miroslav Fiedler in commemoration of his retirement.

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Abstract

Let G be a graph on n vertices. Its Laplacian matrix is the n-by-n matrix L(G)D(G)−A(G), where A(G) is the familiar (0,1) adjacency matrix, and D(G) is the diagonal matrix of vertex degrees. This is primarily an expository article surveying some of the many results known for Laplacian matrices. Its six sections are: Introduction, The Spectrum, The Algebraic Connectivity, Congruence and Equivalence, Chemical Applications, and Immanants.

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^∗: This article was prepared in conjunction with the author's lecture at the 1992 conference of the International Linear Algebra Society in Lisbon. Its preparation was supported by the National Security Agency under Grant MDA904-90-H-4024. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon.