Consider a star graph 
consisting of a central hub vertex and 
spokes, but instead of placing a single point at the end of
each spoke, place 
points along it (in addition to the shared central point). While this graph has been
considered by various authors, it apparently has not been given a standard name.
In this work, to avoid confusion with star graphs and permutation
star graphs, the term 
spoke graph and notation 
are used.
Special cases are summarized in the following table.
Considering vertices by their distance from the central vertex shows that the bipartition classes of 
have sizes 
and 
for even 
, and sizes 
and 
for odd 
. Therefore the bipartition sizes differ by 1 for even 
and by 
for odd 
. In particular, for odd 
and 
,
is an untraceable
graph; for even 
,
the same bipartition-size obstruction shows that 
is nonhamiltonian.
For 
, 
is also untraceable
since a traceable tree
must be a path graph.
