%N Number of simple graphs with n unlabeled vertices with the degree of each vertex a prime number.
%C A simple graph, also called a strict graph, is an unweighted, undirected graph containing no graph loops or multiple edges. Given an undirected graph, a degree sequence is a monotonic nonincreasing sequence of the vertex degrees (valencies) of its graph vertices.
%H Darryn Bryant and Tom McCourt, <a href="http://wiki.smp.uq.edu.au/G-designs/index.php/Graphs_with_five_vertices">Graphs with five vertices</a>, Figure 1.
%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/SimpleGraph.html">MathWorld: Simple Graph</a>
%e a(3) = 1 because there is a unique graph with 3 vertices each having prime degree, the triangle, with degree sequence (2,2,2).
%e a(4) = 3 because there are 3 graphs with 4 vertices each having prime degree: the 4-cycle (2,2,2,2); the complete graph K_4 with degree sequence (3,3,3,3); and two triangle graphs sharing a common edge, with degree sequence (3,3,2,2).
%e a(5) = 4 because there are 4 graphs with 5 vertices each having prime degree: the 5-cycle with degree sequence (2,2,2,2,2); a square graph sharing an edge with a triangle graph (G_13 in the linked-to illustration) with degree sequence (3,3,2,2,2); G_14 in the linked-to illustration with degree sequence (3,3,2,2,2); G_18 in the linked-to illustration with degree sequence (3,3,3,3,2).
%t a[n_Integer] := Count[And @@ PrimeQ /@ GraphData[#, "Degrees"] & /@ GraphData[n], True] (* _Charles R Greathouse IV_, Apr 11 2012 *)
%t show[n_Integer] := Map[Graph, GraphData[#, "EdgeRules"] & /@
%t Select[GraphData[n], And @@ PrimeQ /@ GraphData[#, "Degrees"] &]] (* _Charles R Greathouse IV_, Apr 12 2012 *)
%Y Cf. A000040, A000088, A004251.
%A _Jonathan Vos Post_, Apr 11 2012
%E a(5)-a(7) from _Charles R Greathouse IV_, Apr 11 2012
%E a(8)-a(15) from _Andrew Howroyd_, Mar 08 2020
%E a(16)-a(20) from _Andrew Howroyd_, May 03 2020