A Lucas cube graph of order 
is a graph that can be defined based
on the 
-Fibonacci cube graph by forbidding graph
vertex strings that have a 1 both in the first and last positions. Explicitly,
it is a graph on the subset
of 
-tuples that are cyclically free of adjacent
1's (i.e., consecutive 1's cannot occur in the middle of the string and 1's cannot
be present in both first and last positions of the string), with vertices
joined by edges iff they differ
in exactly one position.
Munarini et al. (2001) determined many structural and enumerative properties of the Lucas cubes.
The 
th
Lucas cube graph is denoted 
(Munarini et al. 2001) or 
(Ilić et al. 2012, Ilić and Milošević
2017).
Special cases are summarized in the following table.
-
has vertex
count equal to the Lucas number 
.
The Lucas cube graphs are median graphs (Klavžar 2005, Došlić and Podrug 2023). They are also unit-distance.
