Skip to main content

Eigenfunctions and Nodal Domains

  • Chapter
  • 2608 Accesses

Part of the Lecture Notes in Mathematics book series (LNM,volume 1915)

In the previous chapter we have seen that (due to the Perron-Frobenius Theorem) the eigenfunctions of the first eigenvalue λ1 have all entries positive (or negative) for a generalized Laplacian matrix M of a connected graph G. Fiedler [67] has shown that for eigenfunctions of the smallest nonzero eigenvalue of a graph the subgraph induced by nonpositive vertices (i.e., vertices with nonpositive function values) and the subgraph induced by nonnegative vertices are both connected. In other words, an eigenfunction of the second eigenvalue has exactly two weak nodal domains (also called weak sign graphs).

Keywords

  • Connected Graph
  • Sign Pattern
  • Boundary Edge
  • Characteristic Edge
  • Sign Graph

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   34.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   44.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Rights and permissions

Reprints and Permissions

Copyright information

© 2007 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

(2007). Eigenfunctions and Nodal Domains. In: Laplacian Eigenvectors of Graphs. Lecture Notes in Mathematics, vol 1915. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73510-6_3

Download citation