Ss Theorem 5.14: Let T be a graph with n vertices. The following assertions are equivalent &(1T is a connected graph with no simple circuit '(2) is a graph with no simple circuit and e=n-l where e is number of edges of t 63)T is a connected graph with e=n-l where e is number of edges of t. 4(4T 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. T+x, y is a new graph which is obtained from T by joining x to y. .(5T is connected and if x, yEE(T) then T-x, y is disconnected. Where T-ix, y is a new graph which is obtained from T by removing edge x, y .(6There is a unique simple path between any two of vertices ofT.❖ Theorem 5.14: Let T be a graph with n vertices. The following assertions are equivalent. ❖ (1)T is a connected graph with no simple circuit. ❖ (2)T is a graph with no simple circuit and e=n-1 where e is number of edges of T. ❖ (3)T is a connected graph with e=n-1 where e is number of edges of T. ❖ (4)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. T+{{x,y}} is a new graph which is obtained from T by joining x to y. ❖ (5)T is connected and if {x,y}E(T) then T-{x,y} is disconnected. Where T-{x,y} is a new graph which is obtained from T by removing edge {x,y}. ❖ (6)There is a unique simple path between any two of vertices of T