A290778 Number of connected undirected unlabeled loopless multigraphs with 4 vertices and n edges.

0, 0, 0, 2, 5, 11, 22, 37, 61, 95, 141, 203, 288, 393, 531, 704, 918, 1180, 1504, 1887, 2351, 2900, 3546, 4301, 5187, 6202, 7379, 8726, 10262, 12005, 13987, 16209, 18716, 21521, 24652, 28135, 32013, 36291, 41028, 46244, 51977, 58262, 65155, 72667, 80872, 89798

Offset: 0

Author

R. J. Mathar, Aug 10 2017

Keywords

• nonn
• easy

Comments

There are 6 basic underlying simple graphs on 4 vertices: the linear chain with 3 edges (a tree), the star graph with 3 edges (a tree), the 4-cycle (quadrangle) with 4 edges, the triangle extended with one edge protruding to a vertex of degree 1 (4 edges), the complete graph on 4 vertices with 6 edges, a graph with 5 edges (removing one from the complete graph).

Examples

			There are a(3) = 2 connected graphs of 3 edges and 4 vertices, the A000055(4) = 2 trees on 4 vertices.
There are a(4)=5 connected graphs of 4 edges and 4 vertices: duplicate either the middle or a sided edge of the linear chain, duplicate an edge of the star graph, or take any of the two underlying simple graphs with 4 edges.
		

Links

• R. J. Mathar, Connected multigraphs with 4 vertices (A290778)
• R. J. Mathar,Statistics on Small Graphs, arXiv:1709.09000 (2017) Eq. (20).
• Index entries for linear recurrences with constant coefficients, signature (2, 0, 0, -2, -2, 3, 0, 3, -2, -2, 0, 0, 2, -1).

Crossrefs

Column 4 of A191646.

Formula

G.f.: -x^3*(x^10-x^9-2*x^7+x^6-x^5+3*x^4-x^2-x-2)/( (x-1)^6 *(1+x)^2 *(1+x^2) *(1+x+x^2)^2 ). - R. J. Mathar, Aug 11 2017