A quasi-cubic graph is a quasi-regular graph, i.e., a graph such that degree of every vertex is the same except for a single vertex whose degree is (Bozóki et al. 2020), with .

The numbers of connected quasi-quintic graphs on , 2, ... nodes are 0, 0, 0, 0, 1, 0, 4, 0, 27, 0, ... and
the numbers of not-necessarily connected quasi-quintic graphs on , 2, ... nodes are 0, 0, 0, 0, 1, 0, 4, 0, 28, 0, .... The
sole disconnected quasi-cubic graph on 10 nodes or fewer is the graph
union
of the 5-wheel graph and the tetrahedral
graph. Examples are illustrated above and are summarized in the table below.

Bozóki S.; Szadoczki, Z.; and Tekile, H. A. "Filling in Pattern Designs for Incomplete Pairwise Comparison Matrices: (Quasi-)Regular
Graphs With Minimal Diameter." 13 May 2020. https://arxiv.org/abs/2006.01127.