A385693 Number of prime graphs, G, on n vertices which do not contain a degree-1 vertex in G nor in co-G.

0, 0, 0, 0, 1, 6, 76, 1990, 84040, 5749698

Offset

1,6

Comments

Here, "prime" means with respect to modular decomposition (see link).

Links

Table of n, a(n) for n=1..10.

GraphClasses, List of Small Graphs.

Wikipedia, Modular decomposition.

Example

The smallest such graph is the cycle on 5 vertices. The 6 graphs on 6 vertices are the C6, domino, X37 (as named on GraphClasses) and their three respective complements.

Prog

(SageMath)
for n in range(3, 11):
    count = 0
    for g in graphs.nauty_geng(f"{n} -c -d2"):
        degrees = g.degree()
        if max(degrees) < n-2 and g.is_prime():
            count += 1
    print(f"n = {n}: {count} prime graphs")

Crossrefs

Cf. A079473.

Sequence in context: A132613 A009763 A340886 * A305999 A274464 A028979

Adjacent sequences: A385690 A385691 A385692 * A385694 A385695 A385696

Keyword

nonn,more

Author

Jim Nastos and Clara Elliott, Jul 07 2025

Status

approved