  Consider a connected graph $G$ and let $T$ be a spanning tree of $G$. Every
edge $e \in G-T$ induces a cycle in $T \cup \{e\}$. The intersection of two
distinct such cycles is the set of edges of $T$ that belong to both cycles. The
MSTCI problem consists in finding a spanning tree that has the least number of
such non-empty intersections and the instersection number is the number of
non-empty intersections of a solution. In this article we consider three
aspects of the problem in a general context (i.e. for arbitrary connected
graphs). The first presents two lower bounds of the intersection number. The
second compares the intersection number of graphs that differ in one edge. The
last is an attempt to generalize a recent result for graphs with a universal
vertex.
