MCA 520: Graph Theory Instructor Neelima Gupta

Table of Contents Tree: Connected Acyclic Graph

Properties Every tree with at least two vertices has at least two leaves (leaf is a vertex of degree 1). Deleting a leaf from an n-vertex tree produces an (n-1)-vertex tree……it allows many things to be done…many facts to be proved by induction for tree.

Characterization of Tree For an n-vertex graph G (n > 1), following are equivalent: A.G is connected and has no cycles. B.G is connected and has n-1 edges. C.G has n-1 edges and no cycles. A => B : Proof by induction. B => C: Construct G’ from G making G’ acyclic. C => A: Let G1, G2….Gk be the components, each being acyclic….

Another characterization of Tree D. G has no loops and has exactly one u-v path for all u and v in V(G). A => D D => A