The triangular snake graph 
is the graph on 
vertices with 
odd defined by starting with the path
graph 
and adding edges 
for 
,
..., 
.
The first few are illustrated above, and special cases are summarized in the following
table.
Triangular snakes are unit-distance and matchstick by construction, perfect.
They are graceful when the number of triangles
is congruent to 0 or 1 (mod 4) (Moulton 1989, Gallian 2018), which is equivalent
to when 
.
Triangular snakes are also geodetic.
." Path and Cycle. Int. J. Comp. Appl. 44,
20-22, 2012.