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 A233387 Number of labeled star graphs with added edges. 0

%I #38 Aug 01 2021 01:56:12

%S 32,185,1308,10822,102176,1081908,12681640,162880256,2273437392,

%T 34249286656,553698389888,9558929197560,175471796530816,

%U 3412297318315472,70064350595106336,1514554957975079008,34377185731361631680

%N Number of labeled star graphs with added edges.

%C 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.

%C If we don't add any edges we have a star graph and there are n labelings.

%C If we add exactly one edge then we produce a cycle of length 3 and we no longer have a tree.

%C If we add the maximum number of edges we get a wheel graph A171005.

%F 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.

%e 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.

%t 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]

%Y Cf. A013989 (with appropriate offset) enumerates such graphs where the maximum degree of non-center vertices is restricted to 2.

%K nonn

%O 4,1

%A _Geoffrey Critzer_, Feb 02 2014

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Last modified August 11 07:51 EDT 2024. Contains 375059 sequences. (Running on oeis4.)