A root of a chromatic polynomial is known as a chromatic root (Dong et al. 2005, Alikhani and Ghanbari 2024). Chromatic roots are potentially
complex.

Tutte (1970) showed that
cannot be a chromatic root of any chromatic polynomial
where,
is the golden ratio, a result extended to for positive integer (Alikhani and Peng 2009).

In contrast,
is a possible chromatic root (Harvey and Royle 2020; e.g., the graphs depicted above),
a result which can be extended to for integer (Alikhani and Hasni 2012, Alikhani and Ghanbar 2024)
using the result that if
is a chromatic root, then for any natural number ,
is also a chromatic root.

Sokal (2004) showed that chromatic roots are dense in the complex
plane (Cameron and Morgan 2016).

An interval in which no chromatic roots are possible is known as a chromatic root-free interval. The plots above show a histogram of chromatic roots along
the real axis and the positions of chromatic roots in
the complex plane for graphs in GraphData
(the latter of which shows clear deviations from uniformity).

Alikhani, S. and Ghanbari, N. "Golden Ratio in Graph Theory: A Survey." 9 Jul 2024. https://arxiv.org/abs/2407.15860.Alikhani,
S., Hasni, R. "Algebraic Integers as Chromatic and Domination Roots." Int.
J. Combin., Article ID 780765, 2012.Alikhani, S. Peng, Y. H.
"Chromatic Zeros and the Golden Ratio." Appl. Anal. Disc. Math.3,
120-122, 2009.Cameron, P. J. and Morgan, K. "Algebraic Properties
of Chromatic Roots." 3 Oct 2016. https://arxiv.org/abs/1610.00424.Dong,
F. M., Koh, K. M.; and Teo, K. L. Chromatic
Polynomials and Chromaticity of Graphs. Singapore: World Scientific, 2005.Harvey,
D. J. and Royle, G. F. "Chromatic Roots at 2 and the Beraha Number
." J. Graph Th.95,
445-456, 2020.Sokal, A. D. "Chromatic Roots Are Dense in the
Whole Complex Plane." Combin. Probab. and Comput.13, 221-261,
2004.Tutte, W. T. "On Chromatic Polynomials and the Golden
Ratio." J. Combin. Theory, Ser. B9, 289-296, 1970.