Determinant identities for Laplace matrices

DOI10.1016/j.laa.2009.08.028zbMath1184.15006OpenAlexW2068861748MaRDI QIDQ1044622

Stephan G. Wagner, Elmar Teufl

Publication date: 18 December 2009

Published in: Linear Algebra and its Applications (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.laa.2009.08.028

zbMATH Keywords

minorselectrical networksLaplace matrixspanning tree enumerationSylvester's determinantal identity

Mathematics Subject Classification ID

Determinants, permanents, traces, other special matrix functions (15A15) Enumeration in graph theory (05C30) Matrix equations and identities (15A24) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Applications of graph theory to circuits and networks (94C15)

Related Items (15)

Enumeration of spanning trees on contact graphs of disk packings ⋮ Counting spanning trees of a type of generalized Farey graphs ⋮ On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs ⋮ Enumeration of spanning trees of 2-separable networks ⋮ Enumeration of spanning trees of complete multipartite graphs containing a fixed spanning forest ⋮ Combinatorial explanation of the weighted Laplacian characteristic polynomial of a graph and applications ⋮ Enumeration of spanning trees in the sequence of Dürer graphs ⋮ Counting spanning trees of \((1, N\))-periodic graphs ⋮ Resistance scaling and the number of spanning trees in self-similar lattices ⋮ COUNTING SPANNING TREES IN PRISM AND ANTI-PRISM GRAPHS ⋮ Enumeration of spanning trees of middle graphs ⋮ The number of spanning trees of a family of 2-separable weighted graphs ⋮ The enumeration of spanning tree of weighted graphs ⋮ Number of spanning trees in the sequence of some graphs ⋮ Counting spanning trees in almost complete multipartite graphs

Cites Work

• The number of spanning trees in self-similar graphs
• Spanning trees on the Sierpinski gasket
• Lectures on probability theory and statistics. Ecole d'Eté de probabilités de Saint-Flour XXV - 1995. Lectures given at the summer school in Saint-Flour, France, July 10-26, 1995
• Matrix tree theorems
• On identities satisfied by minors of a matrix
• Some determinant expansions and the matrix-tree theorem
• The short-cut test
• A Combinatorial Proof of the All Minors Matrix Tree Theorem
• Spectral properties of self-similar lattices and iteration of rational maps
• Four-terminal reducibility and projective-planar wye-delta-wye-reducible graphs
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