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In combinatorics there are very simple basic graphs from which a whole lot of theory came. For example the complete graphs K_5 and K_{3,3} which alone provide the ground level for any non-planar graph according to Kuratowski's theorem. Another simple graph that gave rise to a huge amount of theory is Petersen's graph, which I like to think as the graph whose vertices are the eight binary triples (with entries 0 or 1)ten two-element subsets of {1,2,3,4,5}, and for which two such vertices are connected iff the associated triples have Hamming distance 1they are disjoint.
In combinatorics there are very simple basic graphs from which a whole lot of theory came. For example the complete graphs K_5 and K_{3,3} which alone provide the ground level for any non-planar graph according to Kuratowski's theorem. Another simple graph that gave rise to a huge amount of theory is Petersen's graph, which I like to think as the graph whose vertices are the eight binary triples (with entries 0 or 1), and for which two such vertices are connected iff the associated triples have Hamming distance 1.