Minimum spanning trees are important tools in the analysis and design of networks. Many practical applications require their computation, ranging from biology and linguistics to economy and telecommunications. The set of cycles of a network has a vector space structure. Given a spanning tree, the set of non-tree edges defines cycles that determine a basis. The intersection of two such cycles is the number of edges they have in common and the intersection number -- denoted $\cap(G)$ -- is the number of non-empty pairwise intersections of the cycles of the basis. The Minimum Spanning Tree Cycle Intersection problem consists in finding a spanning tree such that the intersection number is minimum. This problem is relevant in order to integrate discrete differential forms. In this paper, we present two lower bounds of the intersection number of an arbitrary connected graph $G=(V,E)$. In the first part, we prove the following statement: $$\frac{1}{2}\left(\frac{\nu^2}{n-1} - \nu\right) \leq \cap(G),$$ where $n = |V|$ and $\nu$ is the \emph{cyclomatic number} of $G$. In the second part, based on some experimental results and a new observation, we conjecture the following improved tight lower bound: $$(n-1) \binom{q}{2} + q \ r\leq \cap(G),$$ where $2 \nu = q (n-1) + r$ is the integer division of $2 \nu$ and $n-1$. This is the first result in a general context, that is for an arbitrary connected graph.

翻译：最小生成树是网络分析与设计中的重要工具。从生物学、语言学到经济学和电信领域，许多实际应用需要计算最小生成树。网络的环集具有向量空间结构。给定一棵生成树，非树边集合定义了构成基的环。两个此类环的交集是它们共有的边数，交集数（记作$\cap(G)$）是基中环之间非空成对交集的数量。最小生成树环交问题旨在寻找一棵交集数最小的生成树。该问题对于积分离散微分形式具有重要意义。本文给出了任意连通图$G=(V,E)$交集数的两个下界。第一部分中，我们证明以下结论：$$\frac{1}{2}\left(\frac{\nu^2}{n-1} - \nu\right) \leq \cap(G),$$其中$n = |V|$，$\nu$为$G$的环数。第二部分中，基于实验结果与新观察，我们猜想以下改进的紧下界：$$(n-1) \binom{q}{2} + q \ r\leq \cap(G),$$其中$2 \nu = q (n-1) + r$是$2 \nu$与$n-1$的整数除法。这是针对一般情形（即任意连通图）的首个结果。