## Illinois Journal of Mathematics

### The spectrum of the $p$-Laplacian and $p$-harmonic morphisms on graphs

Hiroshi Takeuchi

#### Abstract

For a real number $p$ with $1<p<\infty$ we consider the spectrum of the $p$-Laplacian on graphs, $p$-harmonic morphisms between two graphs, and estimates for the solutions of $p$-Laplace equations on graphs. More precisely, we prove a Cheeger type inequality and a Brooks type inequality for infinite graphs. We also define $p$-harmonic morphisms and horizontally conformal maps between two graphs and prove that these two concepts are equivalent. Finally, we give some estimates for the solutions of $p$-Laplace equations, which coincide with Green kernels in the case $p=2$.

#### Article information

Source
Illinois J. Math., Volume 47, Number 3 (2003), 939-955.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258138202

Digital Object Identifier
doi:10.1215/ijm/1258138202

Mathematical Reviews number (MathSciNet)
MR2007245

Zentralblatt MATH identifier
1122.31003

#### Citation

Takeuchi, Hiroshi. The spectrum of the $p$-Laplacian and $p$-harmonic morphisms on graphs. Illinois J. Math. 47 (2003), no. 3, 939--955. doi:10.1215/ijm/1258138202. https://projecteuclid.org/euclid.ijm/1258138202