We study undirected multiple graphs of any natural multiplicity \(k>1\). There are edges of three types: ordinary edges, multiple edges and multi-edges. Each edge of the last two types is a union of k linked edges, which connect 2 or \((k+1)\) vertices, correspondingly. The linked edges should be used simultaneously. The multiple tree is a multiple graph with no multiple cycles. The number of edges may be different for multiple trees with the same number of vertices. We prove lower and upper bounds on the number of edges in an arbitrary multiple tree. Also we consider spanning trees of an arbitrary multiple graph. Special interest is paid to the case of complete spanning trees. Their peculiarity is that a multiple path joining any two selected vertices exists in the tree if and only if such a path exists in the initial graph. We study the properties of complete spanning trees and the problem of finding the minimum complete spanning tree of a weighted multiple graph.

我们研究任何自然多重性\(k>1\) 的无向多重图。边有三种类型：普通边、多边和多边。后两种类型的每条边都是k 个连接边的并集，它们连接 2 或\((k+1)\)顶点，相应地。应同时使用链接的边。多重树是没有多重循环的多重图。对于具有相同顶点数的多棵树，边数可能不同。我们证明了任意多重树中边数的下限和上限。我们还考虑了任意多重图的生成树。对完全生成树的情况给予特别关注。它们的特点是当且仅当初始图中存在连接任意两个选定顶点的多条路径时，树中才存在该路径。我们研究完全生成树的性质和求加权多重图的最小完全生成树的问题。