  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matter in a tutorial form with proofs and some newer unpublished results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.
