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 showed that ϕ + 1 = ϕ^2 cannot be a chromatic root of any chromatic polynomial where, ϕ is the golden ratio, a result extended to ϕ^n for positive integer n. In contrast, ϕ + 2 is a possible chromatic root (Harvey and Royle 2020; e.g., the graphs depicted above), a result which can be extended to ϕ + n for integer n>=2 (Alikhani and Hasni 2012, Alikhani and Ghanbar 2024) using the result that if α is a chromatic root, then for any natural number n, α + n is also a chromatic root.
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