The term 
spoke graph and notation 
are used in this work for the graph obtained from a star graph 
by placing 
vertices along each of its
spokes, in addition to the shared central
vertex. While this graph
has been considered by various authors, it apparently has not been given a standard
name. This terminology avoids confusion with star graphs
and permutation star graphs.
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.
