As defined by DeAlba et al. (2009) and references on the American Institute of Mathematics Minimum Rank Graph Catalogs, an 
-necklace graph is a cubic graph
on 
vertices obtained from a cycle
graph 
by adding an additional vertex associated with every third vertex 
of the cycle and connecting each new vertex with vertices
, 
, and 
of the cycle. In other words, it is the graph obtained by
connecting 
diamond graphs placed around a circle with a total
of 
edges, where edges are used to connect
vertices of degree 2 in neighboring diamonds. The graph might therefore even more
appropriately be termed the "diamond necklace graph." The first few necklace
graphs for 
,
2, ... are illustrated above. The (degenerate) 
case gives the tetrahedral
graph 
.
With the exception of 
,
-necklace graphs are unit-distance
and matchstick graphs. In fact, the 2-necklace
graph is the minimum 3-regular matchstick graph,
illustrated above in a matchstick embedding.
Amazingly and amusingly, the 
-necklace graph has exactly 
independent edge sets
(E. Weisstein Feb. 10, 2026).
The 
-necklace
graph is the 
special case of a graph based on the truncted square
lattice, known in this work as the truncated
square lattice graph.
