We consider the signless $p$-Laplacian $Q_p$ of a graph, a generalisation of the quadratic form of the signless Laplacian matrix (the case $p=2$). In analogy to Rayleigh's principle the minimum and maximum of $Q_p$ on the $p$-norm unit sphere are called its smallest and largest eigenvalues, respectively. We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds for the smallest eigenvalue in terms of a graph parameter related to the bipartiteness. The latter result generalises bounds by Desai and Rao and, interestingly, at $p=1$ upper and lower bounds coincide.
@article{10_37236_6683,
author = {Elizandro Max Borba and Uwe Schwerdtfeger},
title = {Eigenvalue bounds for the signless {\(p\)-Laplacian}},
journal = {The electronic journal of combinatorics},
year = {2018},
volume = {25},
number = {2},
doi = {10.37236/6683},
zbl = {1392.05070},
url = {http://geodesic.mathdoc.fr/articles/10.37236/6683/}
}
TY - JOUR
AU - Elizandro Max Borba
AU - Uwe Schwerdtfeger
TI - Eigenvalue bounds for the signless \(p\)-Laplacian
JO - The electronic journal of combinatorics
PY - 2018
VL - 25
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/6683/
DO - 10.37236/6683
ID - 10_37236_6683
ER -
%0 Journal Article
%A Elizandro Max Borba
%A Uwe Schwerdtfeger
%T Eigenvalue bounds for the signless \(p\)-Laplacian
%J The electronic journal of combinatorics
%D 2018
%V 25
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/6683/
%R 10.37236/6683
%F 10_37236_6683
Elizandro Max Borba; Uwe Schwerdtfeger. Eigenvalue bounds for the signless \(p\)-Laplacian. The electronic journal of combinatorics, Tome 25 (2018) no. 2. doi: 10.37236/6683
