  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.
