OFFSET
4,1
COMMENTS
Here, a star graph is a tree on n nodes (n>=4) with one specially designated (center) vertex, v of degree n-1. We are allowed to add edges so that the degree of any node (excepting v) is at most 3 and so that every cycle includes the vertex v with the possible exception of a single cycle of length n-1 through each non-center vertex. We note that anytime edges are added the original "center" node remains specially designated. a(n) is the number of such connected simple labeled graphs with a specially designated node.
If we don't add any edges we have a star graph and there are n labelings.
If we add exactly one edge then we produce a cycle of length 3 and we no longer have a tree.
If we add the maximum number of edges we get a wheel graph A171005.
FORMULA
Ignoring the first 4 terms the e.g.f. is: x*exp(A(x))+ x*(log(1/(1-x))/2 + x^2/4 + x/2) where A(x) = x/(1-x)/2 + x/2.
EXAMPLE
a(4) = 32. There are 4 labelings for the star graph on 4 nodes. There are 12 labelings after we add one edge. There are 12 labelings after we add two edges. There are 4 labelings after we add 3 edges. 4+12+12+4=32.
MATHEMATICA
nn=20; a=x/(1-x)/2+x/2; Drop[Range[0, nn]! CoefficientList[Series[x Exp[a]+x (Log[1/(1-x)]/2+x^2/4+x/2), {x, 0, nn}], x], 4]
CROSSREFS
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Feb 02 2014
STATUS
approved