A spider graph, spider tree, or simply "spider," is a tree with one vertex of vertex
degree at least 3 and all others with vertex degree
at most 2. The numbers of spiders up to graph isomorphism
on 
, 2, ... nodes
are 0, 0, 0, 1, 2, 4, 7, 11, 17, 25, 36, 50, 70, 94, ... (OEIS A004250).
The count 
of spider trees with 
nodes is the same as the number of integer
partitions of 
into three or more parts. For 
, it also has closed form
 |
(1)
|
where 
is the partition function P and 
is the floor
function. A generating function for 
is given by
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
where 
is a q-Pochhammer symbol.
Not all spiders are caterpillar graphs, nor are all spiders lobster graphs.
It is conjectured that every spider is a graceful graph (Saengsura and Poomsa-ard 2025, pp. 1-2).
