Betweenness centrality is defined as follows. Let
$\sigma_{x,y}$ number of shortest paths between $x$ and $y$ and for
$\sigma_{x,y}(v)$ number of shortest paths between $x$ and $y$ passing through $v$, then
$$C_B(v) = \sum_{\substack{x,y \in V(G) \setminus v\\x \neq y}} \frac{\sigma_{x,y}(v)}{\sigma_{x,y}}$$Global betweenness centrality $$\bar{C}_B(G) = \frac{1}{n} \sum_{v \in V(G)} C_B(v)$$
Average shortest path length $$L(G) = \frac{1}{n(n-1)} \sum_{\substack{u,v \in V(G)\\ u \neq v}} d(u,v)$$
Theorem Let $G$ be unoriented graph of size $|V(G)| = n$, $L(G)$ its average path length and $\overline{C}_B(G)$ its global betweeness centrality. Then it holds that $\overline{C}_B(G) = (n - 1) (L(G) - 1)$.
Lemma Let $G$ be a graph of size $n$ and let $G'$ be a graph obtained from $G$ by adding an edge between random vertices $u,v$ in distance $d = d(u,v) > 1$. Then it holds that $$\overline{C}_B(G') \leq \overline{C}_B(G) - \frac{2(d - 1)}{n}.$$
A vertex is called a leaf if $\deg(v) = 1$ and a k-neighbourhood of a vertex is defined as $$\Gamma_k(v) = \{v\in V : d(u,v) = k\}.$$
Theorem Let $G$ be a graph of size $n$, $w \in V$ a vertex with eccentricity $\epsilon(w)$ and let $G'$ be a graph obtained from $G$ by adding a vertex $u$ which is connected to $G$ by the edge $\{u,w\}$. Then $$\overline{C}_B(G') = \frac{n}{n - 1}\overline{C}_B(G) + \frac{2}{n - 1}\sum_{k=1}^{\epsilon(w)} k|\Gamma_k(w)|.$$
Task 1: Express the betweenness centrality of vertices in
Use $n$ as number of vertices in these graphs. Use some additional parameter to express betweenness if necessary.
Task 2: Bound the change of betweenness after