OEIS A000088: Number of nonisomorphic simple graphs on n nodes

Explore A000088, the OEIS entry counting unlabeled simple graphs on n nodes. We’ll trace the first terms (1, 1, 2, 4, 11, 34) and explain why the count grows so rapidly, roughly like 2^{n(n-1)/2} divided by n! as most graphs have no nontrivial automorphisms. We’ll glimpse how to derive connected graphs via the Euler transform and even see how the same sequence appears in counting equivalence classes of sign patterns of certain matrices. Finally, we’ll connect these ideas to real-world fields—from social networks and chemistry to physics and computer science—and discuss the computational challenges of enumerating graphs.

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