6(3)(4:T is a connected graph with e=n-l, we prove"T is a graph with no simple circuit, and if x and y are nonadjacent vertices of T then T+x, y) contains exactly a simple circuit go 1)We first prove that t doesn't contain simple circuit. &o Let us apply induction on the number of vertices of T. .T is connected with n=2 and e=l Thus T doesn't contain any simple circuit. The result holds when n=2 ande=l❖ (3)→(4): T is a connected graph with e=n-1, we prove “T is a graph with no simple circuit, and if x and y are nonadjacent vertices of T then T+{x,y} contains exactly a simple circuit”. ❖ 1)We first prove that T doesn't contain simple circuit. ❖ Let us apply induction on the number of vertices of T. ❖ T is connected with n=2 and e=1, Thus T doesn't contain any simple circuit. The result holds when n=2 and e=1