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 (Szádoczki 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,