A quasi-cubic graph is a quasi-regular graph, i.e., a graph such that the vertex
degree of every vertex is the same 
except for a single vertex
whose vertex degree is 
(Bozóki et al. 2022), with 
. By the handshaking
lemma, such a graph necessarily has odd graph order: if it has 
vertices, the sum
of its vertex degrees is 
and must be even, so 
is odd.
For 
,
7, 9, ..., the numbers of quasi-cubic graphs are 1, 4, 28, 214, 1999, 22186, ...
(OEIS A398638). The corresponding numbers of
connected quasi-cubic graphs are 1, 4, 27, 208,
1958, 21879, ... (OEIS A398639). Of the quasi-cubic
graphs on 19 vertices, 28526 are disconnected.
The sole disconnected quasi-cubic graph through nine vertices
is the graph disjoint union 
, whose components
are the wheel graph 
and the tetrahedral graph 
. Examples are illustrated above and
are summarized in the table below.
 | quasi-cubic graphs |
| 5 | wheel graph  |
| 7 | Harary graph  , Moser spindle |
| 9 | Harary
graph  ,
 -unit-distance-forbidden
graph,  |
