A345246 Number of 2-connected unlabeled simple graphs with no triangles.
1, 2, 6, 16, 78, 415, 3374, 35860, 524386, 10193061, 263036202, 8948113645, 400280198048
Offset: 4
Examples
For n=4 the only example is a 4-cycle.
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For n=4 the only example is a 4-cycle.
\\ See links in A339645 for combinatorial species functions. edges(v) = {2*sum(i=2, #v, sum(j=1, i-1, gcd(v[i], v[j]))) + sum(i=1, #v, v[i]-1)} graphsCycleIndex(n)={my(s=0); forpart(p=n, s+=permcount(p) * 2^edges(p) * sMonomial(p)); s/n!} graphsSeries(n)={sum(k=0, n, graphsCycleIndex(k)*x^k) + O(x*x^n)} cycleIndexSeries(n)={my(g=graphsSeries(n), gc=sLog(g), gcr=sPoint(gc)); intformal(x*sSolve( sLog( gcr/(x*sv(1)) ), gcr ), sv(1)) + sSolve(subst(gc, sv(1), 0), gcr)} { my(N=15); Vec(-2*x^2 + OgfSeries(cycleIndexSeries(N))) } \\ Andrew Howroyd, Mar 09 2023
For n = 5, all but two of the A002218(5) = 10 2-connected graphs are 1-tough, so a(5) = 8. The exceptions are the complete bipartite graph K_{2,3} and the complete tripartite graph K_{1,1,3}. To see that these graphs are not 1-tough, note that, in both cases, two vertices can be removed resulting in a graph with three components (isolated vertices).
The 3 circle graphs with n = 4 vertices which are 2-connected are K_4, the square and the square with one diagonal.