The detour polynomial of a graph is the characteristic
polynomial of the detour matrix of .

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 has all off-diagonal matrix elements equal to , the detour polynomial of such a graph is given by .

The following table summarizes detour polynomials for some common classes of graphs. Here,
is a Chebyshev polynomial of the
first kind and
is a Chebyshev polynomial of the
second kind .

The following table summarizes the recurrence relations for detour polynomials for some simple classes of graphs.

See also Characteristic Polynomial ,

Detour Index ,

Detour
Matrix
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References Nikolić, S.; Trinajstić, N.; and Mihalić, A. "The Detour Matrix and the Detour Index." Ch. 6 in Topological
Indices and Related Descriptors in QSAR and QSPR (Ed. J. Devillers A. T.
and Balaban). Amsterdam, Netherlands: Gordon and Breach, pp. 279-306, 2000. Referenced
on Wolfram|Alpha Detour Polynomial
Cite this as:
Weisstein, Eric W. "Detour Polynomial."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/DetourPolynomial.html

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