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A233387 Number of labeled star graphs with added edges. 0
32, 185, 1308, 10822, 102176, 1081908, 12681640, 162880256, 2273437392, 34249286656, 553698389888, 9558929197560, 175471796530816, 3412297318315472, 70064350595106336, 1514554957975079008, 34377185731361631680 (list; graph; refs; listen; history; text; internal format)
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.
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.
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.
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]
Cf. A013989 (with appropriate offset) enumerates such graphs where the maximum degree of non-center vertices is restricted to 2.
Sequence in context: A316543 A306058 A317236 * A200840 A208925 A212863
Geoffrey Critzer, Feb 02 2014

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