A chromatically unique graph 
is a finite simple graph for which 
is determined by its chromatic
polynomial 
. Equivalently, 
implies that 
and 
are isomorphic graphs,
so the graph 
is uniquely determined up to isomorphism by its chromatic polynomial. If 
and 
are nonisomorphic but share the same chromatic
polynomial, they are said to be chromatically
equivalent.
Cycle graphs are chromatically unique (Chao and Whitehead 1978), as are Turán graphs (Chao and Novacky 1982).
Named graphs that are chromatically nonunique include the 3- and 4-barbell graph, bislit cube, bull graph, claw graph, 3-matchstick graph, Moser spindle, 2-Sierpiński gasket graph, star graphs, triakis tetrahedral graph, and 6- and 8-wheel graphs.
The numbers of chromatically nonunique simple graphs on 
nodes for 
, 2, ... are 0, 0, 0, 4, 18, 115, 905, 11642, 267398, ...
(OEIS A137567), while the corresponding numbers
of chromatically unique graphs are 1, 2, 4, 7, 16, 41, 139, 704, 7270, ... (OEIS
A137568).
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Malays. Math. Sci. Soc. 24, 3-103, 2001.Chao, C.-Y.; Guo,
Z. Y.; and Li, N. Z. "Some Families of Chromatically Equivalent Graphs."
Bull. Malays. Math. Soc. 15, 77-82, 1992.Chao, C.-Y.;
Guo, Z.-Y.; Li, N.-Z. "On 
-Graphs. Chromatic Polynomials and Related Topics (Shanghai,
1994)." Discr. Math. 172, 9-16, 1997.Chao, C. Y.
and Novacky, G. A. "On Maximally Saturated Graphs." Disc. Math. 41,
139-143, 1982.Chao, C. Y. and Whitehead, E. G. Jr. "On
Chromatic Equivalence of Graphs." In 
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1987.Sloane, N. J. A. Sequences