The detour index ω(G) of a graph G is a graph invariant defined as half the sum of all off-diagonal matrix elements of the detour matrix of G. Unless otherwise stated, hydrogen atoms are usually ignored in the computation of such indices as organic chemists usually do when they write a benzene ring as a hexagon. Precomputed detour indices for many named graphs are available in the Wolfram Language as GraphData[graph, DetourIndex]. Since a Hamilton-connected graph with vertex count n has all off-diagonal matrix elements equal to n - 1, the detour index of such a graph is given by n(n - 1)^2/2.
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