A 
-cyclic graph, or 
-graph, is a connected simple graph admitting a closed walk
of length 
that traverses every graph edge at least once. The
numbers of 
-graphs
for 
,
4, ... are 1, 3, 3, 10, 12, 35, 58, 160, 341, 958, 2444, 7242, 21190, 67217, 217335,
... (OEIS A081809) (FlowProblem), the first
few of which are illustrated above.
Every finite connected simple graph with at least one graph edge is 
for some 
, since doubling each graph edge
gives a multigraph with an Eulerian
cycle. For example, every connected graph
on two to six nodes with the exception of the complete graph 
is 
for some 
.
These graphs are useful when counting graph cycles. In a graph with adjacency
matrix 
,
counts closed 
-walks, distinguishing their starting
vertex and traversal direction, where 
denotes the matrix trace.
To obtain the number of undirected 
-cycles for 
, subtract the closed walks
that are not cycles and divide by 
, accounting for the 
starting vertices and two
traversal directions of each cycle.
-Graphs."