The spectrum of an infinite directed graph. (English) Zbl 0754.47004
Summary: Mohar established the principle on spectra of infinite graphs:
(i) the spectrum of a graph is symmetric for the real axis (actually contained in it),
(ii) the spectral radius of a graph is a spectre, and
(iii) the spectral radius is monotone as a function of graphs.
In this note, we show Mohar’s principles for infinite directed graphs. In addition, we estimate the isoperimetric number and the chromatic one by the spectral radius.
(i) the spectrum of a graph is symmetric for the real axis (actually contained in it),
(ii) the spectral radius of a graph is a spectre, and
(iii) the spectral radius is monotone as a function of graphs.
In this note, we show Mohar’s principles for infinite directed graphs. In addition, we estimate the isoperimetric number and the chromatic one by the spectral radius.
MSC:
| 47A10 | Spectrum, resolvent |
| 47A12 | Numerical range, numerical radius |
| 05C20 | Directed graphs (digraphs), tournaments |
| 94D05 | Fuzzy sets and logic (in connection with information, communication, or circuits theory) |
| 47B15 | Hermitian and normal operators (spectral measures, functional calculus, etc.) |
