Title: Spectrum of signless 1-Laplacian on simplicial complexes,
Abstract: We first develop a general framework for signless 1-Laplacian defined in
terms of the combinatorial structure of a simplicial complex. The structure of
the eigenvectors and the complex feature of eigenvalues are studied. The
Courant nodal domain theorem for partial differential equation is extended to
the signless 1-Laplacian on complex. We also study the effects of a wedge sum
and a duplication of a motif on the spectrum of the signless 1-Laplacian, and
identify some of the combinatorial features of a simplicial complex that are
encoded in its spectrum. A special result is that the independent number and
clique covering number on a complex provide lower and upper bounds of the
multiplicity of the largest eigenvalue of signless 1-Laplacian, respectively,
which has no counterpart of $p$-Laplacian for any $p>1$.