The detour polynomial of a graph G is the characteristic polynomial of the detour matrix of G. Precomputed detour polynomials for many named graphs are available in the Wolfram Language as GraphData[graph, DetourPolynomial]. Since a Hamilton-connected graph with vertex count n has all off-diagonal matrix elements equal to n - 1, the detour polynomial of such a graph is given by (x - (n - 1)^2)(x + n - 1)^(n - 1).
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