论文标题
图形的边缘加权平均偏心率
On edge-weighted mean eccentricity of graphs
论文作者
论文摘要
令$ g $为订单$ n $和大小$ m $的连接边缘加权图。令$ w：e（g）\ rightarrow \ mathbb {r}^{\ geq 0} $为加权函数。我们假设$ w $是归一化的，即$ \ sum_ {e \ in E（g）} w（e）= m $。任何两个顶点$ u $和$ v $之间的加权距离$ d_w（u，v）$是它们之间的权重，而偏心$ e_w（v）a vertex $ v $的偏心是从$ v $到$ g $的顶点的加权距离。 $ g $，$ avec（g，w）$的平均（平均）偏心率是$ g $中所有偏心率的（加权）平均值。我们在$ n $，$ m $或边缘连接性$λ$的两种情况下获得$ avec（g，w）$的上限和下限：$ g $是一棵树，$ g $是连接的，而不是一棵树。此外，我们获得了边缘加权平均偏心率的Nordhaus-Gaddum型结果。
Let $G$ be a connected edge-weighted graph of order $n$ and size $m$. Let $w:E(G)\rightarrow \mathbb{R}^{\geq 0}$ be the weighting function. We assume that $w$ is normalised, that is, $\sum_{e\in E(G)} w(e)=m$. The weighted distance $d_w(u,v)$ between any two vertices $u$ and $v$ is the least weight between them and the eccentricity $e_w(v)$ of a vertex $v$ is the weighted distance from $v$ to a vertex farthest from it in $G$. The mean(average) eccentricity of $G$, $avec(G,w)$, is the (weighted) mean of all eccentricities in $G$. We obtain upper and lower bounds on $avec(G,w)$ in terms of $n$, $m$ or edge-connectivity $λ$ for two cases: $G$ is a tree and $G$ is connected but not a tree. In addition, we obtain the Nordhaus-Gaddum-type results for edge-weighted average eccentricity.
