English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Parallel to the signless Laplacian spectral theory, we introduce and developthe nonlinear spectral theory of signless $1$-Laplacian on graphs. Again, thefirst eigenvalue $\mu^+_1$ of the signless 1-Laplacian precisely characterizesthe bipartiteness of a graph and naturally connects to the maxcut problem.However, the dual Cheeger constant $h^+$, which has only some upper and lowerbounds in the Laplacian spectral theory, is proved to be $1-\mu^+_1$. Thestructure of the eigenvectors and the graphic feature of eigenvalues are alsostudied. The Courant nodal domain theorem for graphs is extended to thesignless 1-Laplacian. A set-pair version of the Lov\'asz extension, which aimsat the equivalence between discrete combination optimization and continuousfunction optimization, is established to recover the relationship$h^+=1-\mu^+_1$. A local analysis of the associated functional yields aninverse power method to determine $h^+$ and then produces an efficientimplementation of the recursive spectral cut algorithm for the maxcut problem.

Spectrum of the Signless 1-Laplacian and the Dual Cheeger Constant on Graphs