Consider a complete graph $K_n$ with $n$ vertices ( $n>4$ ). Note that multiple spanning trees can be constructed over $K_n$. Each of these spanning trees is represented as a set of edges. The Jaccard coefficient between any two sets is defined as the ratio of the size of the intersection of the two sets to the size of the union of the two sets. Which one of the following options gives the lowest possible value for the Jaccard coefficient between any two spanning trees of $K_n$ ?

Let $G(V, E)$ be a simple, undirected graph. A vertex cover of $G$ is a subset $V^{\prime} \subseteq V$ such that for every $(u, v) \in E, u \in V^{\prime \prime}$ or $v \in V^{\prime}$. Let the size of the smallest vertex cover in $G$ be $k$. Let $S$ be any vertex cover of size $k$.

For a vertex $v \in V$, which of the following constraints will always ensure that $v \in S$ ?

Let $G$ be an undirected graph, which is a path on 8 vertices. The number of matchings in $G$ is $\_\_\_\_$ (answer in integer)

The chromatic number of a graph is the minimum number of colours used in a proper colouring of the graph. The chromatic number of the following graph is ________