Graph Theory Calculator

This calculator estimates structural properties of a graph — average degree, density, likely connectivity, and whether an Eulerian circuit is possible — from just three summary numbers: how many vertices it has, how many edges connect them, and whether the graph is directed. It never receives the actual list of which vertex connects to which; every output here is derived purely from those counts, by the handshaking lemma (average degree equals twice the edge count divided by the vertex count for an undirected graph) and by comparing the edge count to how many edges a complete graph on that many vertices would need. Vertices pulls Average Degree down more sharply than Edges pulls it up, since Vertices sits in the denominator of that ratio and a bigger denominator shrinks a fraction faster than the same proportional change to the numerator grows it.

Because this page never sees real vertex-to-vertex connections, "Possibly Connected" and "Eulerian Circuit Possible" are genuinely just estimates: a graph can satisfy the edge-count condition checked here for "Possibly Connected" while still being disconnected in reality. For "Eulerian Circuit Possible," the check confirms only that the average degree comes out to a whole number (2E divisible by V) — that's weaker than confirming the average degree is even, and far weaker than confirming every individual vertex has an even degree, which a genuine Eulerian circuit requires; a graph with, say, 6 vertices and 9 edges has a whole-number average degree of 3, which is odd, and would still be reported "Possible" here. The calculator also performs no check that your edge count is achievable without duplicate connections between the same pair of vertices.