论文标题
订单树及其拓扑指数的极端图
The extremal graphs of order trees and their topological indices
论文作者
论文摘要
最近，D。Vuki $ \ check {c} $ evi $ \ acute {c} $和J. sedlar in \ cite {vuki}在$ \ mathcal {t} _n $上引入了“ $ \ prepeq $”订单$ \ langle \ Mathcal {t} _n，\ prepeq \ rangle $。它提供了一种新方法来确定拓扑指数$ f $的极端图。通过使用该方法，他们确定了$ \ Mathcal {t} _n $的公共最大值和/或最小值，相对于Wiener类型和反向维纳类型的拓扑指数。在他们的研究中，我们进一步研究了订单集合$ \ langle \ Mathcal {t} _n，\ \ \ \ preceq \ rangle $，并给出了确定其顺序的标准，这使我们能够在四个规定的子类别中获得$ \ langle \ langle \ nangle \ natercal \ nathcal {t} t} _n，\ \ \ \ \ \ \ \ creang $ \ f \ creang con。所有这些极端图被确认为Wiener类型和抗媒体类型的拓扑指数的常见最大值和/或最小图。此外，我们以订单设置$ \ langle \ Mathcal {c}（n，k），\ prepeq \ rangle $，$ \ langle \ langle \ mathcal \ Mathcal {t} _ {n}（n}（q）（q），\ preceq \ preceq \ rangle $ $ $ $ $ $ $ $ $ $ $ $ $ \ langle $， $ \ langle \ mathcal {t} _ {n}^δ，\ preceq \ rangle $。
Recently, D. Vuki$\check{c}$evi$\acute{c}$ and J. Sedlar in \cite{Vuki} introduced an order "$\preceq$" on $\mathcal{T}_n$, the set of trees on $n$ vertices, such that the topological index $F$ of a graph is a function defined on the order set $\langle\mathcal{T}_n,\preceq\rangle$. It provides a new approach to determine the extremal graphs with respect to topological index $F$. By using the method they determined the common maximum and/or minimum graphs of $\mathcal{T}_n$ with respect to topological indices of Wiener type and anti-Wiener type. Motivated by their researches we further study the order set $\langle\mathcal{T}_n,\preceq\rangle$ and give a criterion to determine its order, which enable us to get the common extremal graphs in four prescribed subclasses of $\langle\mathcal{T}_n,\preceq\rangle$. All these extremal graphs are confirmed to be the common maximum and/or minimum graphs with respect to the topological indices of Wiener type and anti-Wiener type. Additionally, we calculate the exact values of Wiener index for the extremal graphs in the order sets $\langle\mathcal{C}(n,k),\preceq\rangle$, $\langle\mathcal{T}_{n}(q),\preceq\rangle$ and $\langle\mathcal{T}_{n}^Δ,\preceq\rangle$.
