完全赋权图的哈密顿圈中的一个图论参数--外因子和
Extra-factorial sum: a graph-theoretic parameter in Hamiltonian cycles of complete weighted graphs
摘要：
一个图论参数，在一个函数的形式，称为额外的阶乘和进行了讨论。主要结果见参考文献[1]（Nastou等人，Optim Lett，10，1203-1220，2016），强烈建议读者研究上述论文。目前的工作提出了主题事项的教程形式与证明和一些较新的未发表的结果接近尾声（引理6扩展和引理7）。额外的阶乘和是有关的哈密顿圈的完全加权图$WH_n$与$n$顶点，并获得了$WH_n$的每一条边。如果这个和乘以$1 / (n - 2)$，那么它直接给出穿过所选边$e_q$的所有哈密顿圈的长度之和$l_i$的算术平均值。该和中的项数是被证明为$(n - 2)!$的阶乘，这表示其值取决于$n$。利用阶乘外和，还可以求出$WH_{n}$的$(n - 1)! / 2$哈密顿圈长度平方和的算术平均值.
英文摘要：
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 The current work presents subject matter in a tutorial form with proofs and some newer unpublished results towards the end (lemma six extension and lemma seven).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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