A quasi-regular graph is a graph such that the degree of every vertex is the same 
except for a single vertex whose degree is 
(Bozóki et al. 2022). By the handshaking lemma, such a graph necessarily has
odd order and odd minimum vertex degree:
if it has 
vertices, the sum of its vertex degrees is 
and must be even, so 
is odd and hence both 
and 
are odd.
Quasi-regular graphs with 
, 5, ..., may be called quasi-cubic,
quasi-quintic, etc.
For odd 
,
the possible values of 
are 1, 3, ..., 
. Distinct values of 
give distinct degree sequences,
so the total numbers of quasi-regular graphs and connected quasi-regular graphs are
obtained by summing the fixed-
counts over these values. There are no quasi-regular graphs
of even order.
Let 
and 
denote the numbers of quasi-
graphs and connected quasi-
graphs on 
vertices, respectively, and let 
denote the number of 
-regular graphs on 
vertices, with 
and 
. Every quasi-
graph has a unique component containing its exceptional
vertex, while all remaining components are 
-regular. Consequently,
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where the last identity uses the corresponding ordinary generating functions. The total counts are 
and 
, with the sums over odd 
.
For every odd 
,
the 
class consists of the single graph disjoint union 
, where 
and 
are path graphs. It is connected
only for 
.
Another disconnected example is the graph disjoint
union 
of wheel graph 
and tetrahedral graph 
.
For 
,
5, 7, 9, ..., the numbers of quasi-regular graphs are 1, 2, 6, 50, 4085, 3289853,
... (OEIS A398642). The corresponding numbers
of connected quasi-regular graphs are 1, 1, 5, 48, 4078, 3289810, ... (OEIS A398643).
At 
,
the fixed-
contributions for 
,
3, 5, 7, 9, and 11 are 1, 1999, 1877695, 1409285, 872, and 1. The corresponding connected
contributions are 0, 1958, 1877694, 1409285, 872, and 1. Hence 43 quasi-regular graphs
on 13 vertices are disconnected, with fixed-
decomposition 
for 
, 3, and 5.
At 
,
the fixed-
contributions for 
,
3, 5, 7, 9, 11, and 13 are 1, 22186, 1505997049, 48750377244, 903496103, 7815, and
1. The corresponding connected contributions are 0, 21879, 1505997026, 48750377244,
903496103, 7815, and 1. For 
, all these graphs are connected.
Indeed, a disconnected graph would require
a quasi-
component on at least 
vertices and a nonempty 
-regular component on at least 
vertices, for a total of at least 
vertices. Hence 331 quasi-regular graphs on 15
vertices are disconnected, with fixed-
decomposition 
for 
, 3, and 5.
