Figure $2$: Graph for Problem $1$

Problem $2.$ Consider Figure $2$ again. $(1)$ Write down the adjacent nodes for a and $f.$

$(2)$ write down the adjacency matrix in the order of $v_1-v_5.$

Problem $3.$ Given $6$ nodes, construct a binary tree with the following requirement:

$(1)$ Draw a tree with minimum height.

$(2)$ Draw a tree with maximum height.

$(3)$ Prove that your solution for $(2)$ by mathematical induction.