arXiv math/06100892006-10-02math.MGmath.PR

随机正则图中的测地线和几乎测地线圈

Geodesics and almost geodesic cycles in random regular graphs

Itai Benjamini, Carlos Hoppen, Eran ofek, Pawel Pralat, Nick Wormald

摘要：

图G中的测地线是图G的两个顶点之间的最短路。对于n中的一个特定函数e（n），我们定义G中的一个几乎测地圈C为这样的圈：对于C中的每两个顶点u和v，距离dG（u，v）至少为dC（u，v）-e（n）.设f（n）是任意一个趋于无穷大的函数，且n为零。考虑n阶随机d-正则图。我们证明了几乎所有的顶点对都属于几乎测地圈C，使得e（n）= \log_{d-1} \log_{d-1} n +f（n）且|C| =2 \log_{d-1} n+O（f（n））.一路上，我们得到的结果近测地线路径。我们还给出了这类随机图中两个随机顶点间测地线数目的极限分布。

英文摘要：

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance d_G(u,v) is at least d_C(u,v)-e(n). Let f(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of vertices belong to an almost geodesic cycle C with e(n)= \log_{d-1} \log_{d-1} n +f(n) and |C|=2\log_{d-1}n+O(f(n)). Along the way, we obtain results on near-geodesic paths. We also give the limiting distribution of the number of geodesics between two random vertices in this random graph.

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