We investigate a graph theoretic analog of geodesic geometry. In a graph $G=(V,E)$ we consider a system of paths $\mathcal{P}=\{P_{u,v}|u,v\in V\}$ where $P_{u,v}$ connects vertices $u$ and $v$. This system is consistent in that if vertices $y, z$ are in $P_{u,v}$, then the sub-path of $P_{u,v}$ between them coincides with $P_{y,z}$. A map $w: E\to(0,\infty)$ is said to induce $\mathcal{P}$ if for every $u, v\in V$ the path $P_{u,v}$ is $w$-geodesic. We say that $G$ is metrizable if every consistent path system is induced by some such $w$. As we show, metrizable graphs are very rare, whereas there exist infinitely many $2$-connected metrizable graphs.
Slovak University of Technology（斯洛伐克技术大学）；University of Split（斯普利特大学）；Faculty of Information Studies, Novo Mesto（诺沃梅斯托信息研究学院）；University of Ljubljana（卢布尔雅那大学）；Rudolfovo - Science and Technology Centre（鲁多尔福夫科学与技术中心）；Shanghai Jiao Tong University（上海交通大学）
On the number of geodesics of Petersen graph $GP(n,2)$
在任何网络中，节点之间通过测地线的互连以及节点之间存在的测地线的数量是重要的。存在一类基于通过顶点的测地线数目的中心性度量。介数中心性指示顶点的介数或顶点在测地线上出现在其他顶点之间的频率。它在网络分析中有着广泛的应用。考虑$GP(n,k)$。对于每个$n$和$k \,(n > 2k)$，广义Petersen图$GP ( n , k )$是具有点集$\{ u_ i ,\, v_ i \,|\, 0 \leq i \leq n - 1 \}$和边集$\{ u_ i u_ {i + 1} , u_ i v _i , v_ i v_{ i + k}\, |\, 0\leq i \leq n - 1, \hbox{ subscripts reduced modulo } n \}$的三价图。有三种边缘，即外边缘，辐条和内边缘。外部顶点生成一个称为外部循环的$n$循环，内部顶点生成一个或多个内部循环。在本文中，我们考虑了$GP(n,2)$，并找到表达式的测地线和介数中心。
