Petersen graph
Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.
As many points as two steps allow
In a graph where every point has d neighbours and every point is within two steps of every other, there can be at most d² + 1 points — one, its d neighbours, and d(d − 1) more reached through them. Graphs that meet the bound exactly are rare to the point of absurdity. The pentagon does it for d = 2, the Petersen graph for d = 3, a fifty-point graph found in 1960 for d = 7, and an eigenvalue argument proves there is nothing else — except possibly one graph with 3,250 points and 57 neighbours each, which nobody has found or ruled out.
Seven graphs that must link
Six points joined in every way cannot be placed in space without two disjoint triangles hooking together. Trade any triangle for a three-pointed star, or a star back for a triangle, and the property survives; doing it in every possible way from K₆ reaches exactly seven graphs, the Petersen graph among them, and stops. Every placement of every one of them links, by the same parity count. Remove any edge from any of them, and placements that link nothing appear at once. The seven are the whole answer: a graph must link exactly when it contains one of them.
The symmetric graphs no tour can close
Take the words of five places with two ones and join two when they share no one. The ten words look exactly alike to the graph, and no closed tour runs through them — because every way of leaving out one edge at each word leaves two pentagons. Only four connected graphs this symmetric are known to fail like that, and nobody knows whether a fifth exists.
Named alongside it
The objects these essays reach for when they reach for this one.
Delta y exchangeEigenvalueExhaustive searchExtremal graphFano planeGirthGraph minorHamiltonian cycleIntrinsically linkedKneser graphLinking numberMoore graph