The number of distinct simple graphs with up to three nodes is

Concept: By convention, when a graph-counting question does not explicitly say "labeled," "distinct simple graphs" means non-isomorphic graphs — two graphs count as the same if relabeling the vertices of one produces the other. So the count is obtained by enumerating distinct edge-arrangements (shapes) after removing relabelings that produce the same shape, not by treating each vertex-labeling as a separate graph.

Application: Count the non-isomorphic simple graphs for n = 1, 2, and 3 nodes, then sum them:

1. n = 1 node: only 1 possible graph — a single isolated vertex (no edge is possible).

2. n = 2 nodes: only 1 possible edge exists between the two nodes, so there are 2 non-isomorphic graphs — no edge, or one edge.

3. n = 3 nodes: 3 possible edges exist among the three pairs. Grouping configurations that are the same shape under relabeling leaves 4 non-isomorphic graphs — 0 edges (empty), 1 edge, 2 edges (a path), and 3 edges (a triangle).

4. Summing across all three sizes: 1 + 2 + 4 = 7.

Cross-check: This matches OEIS sequence A000088 (count of unlabeled simple graphs on n nodes), which gives 1, 2, 4 for n = 1, 2, 3. Contrast this with the labeled-vertex convention: with 3 distinguishable nodes there are 2³ = 8 possible labeled graphs on the 3 possible edges, giving 1 + 2 + 8 = 11 overall. However, 11 is not among the four options offered here (15, 10, 7, 9), and the standard convention for "distinct graphs" without an explicit "labeled" qualifier is to count non-isomorphic shapes. So 7 is the correct count among the given options.