Extra-factorial sum: a graph-theoretic parameter in Hamiltonian cycles of complete weighted graphs
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A graph-theoretic parameter, in a form of a function, called the extra-factorial sum is discussed. The main results are presented in ref. [1] (Nastou et al., Optim Lett, 10, 1203-1220, 2016) and the reader is strongly advised to study the aforementioned paper. The current work presents subject matter in a tutorial form with proofs and some newer unpublished results towards the end (lemma six extension and lemma seven). The extra-factorial sum is relevant to Hamiltonian cycles of complete weighted graphs WH_n with n vertices and is obtained for each edge of WH_n. If this sum is multiplied by 1 / (n - 2) then it gives directly the arithmetic mean of the sum of lengths l_i of all Hamiltonian cycles that traverse a selected edge e_q. The number of terms in this sum is a factorial proven to be (n - 2)! which signifies that its value depends on n. Using the extra-factorial sum, the arithmetic mean of the sum of the squared lengths of (n - 1)! / 2 Hamiltonian cycles of WH_n can be obtained as well.
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