[PDF] Graph theory - EPFL PDF solutions2.pdf

Prove the triangle-inequality in graphs: for any three vertices u, v, w in a graph G, there will be a u-v path in the tree, hence in the graph, as well Prove that a forest on n vertices with c connected components has exactly n − c edges Let T be a tree and let u and v be two non-adjacent vertices of T Prove that T +uv

A tree has exactly one path between any pair of vertices Proof: Let x,y be any two distinct vertices Sometimes, vertices of degree 0 are also counted as leaves

[PDF] The University of Sydney MATH2009 GRAPH THEORY Tutorial 4

(ii) Prove that a tree with at least 2 vertices has at least 2 vertices of degree 1 is exactly one path between any pair of vertices, and exactly one cycle created

[PDF] 2 Trees

2 leaf vertices Further, if T has exactly two leaf vertices, then every other vertex of T has degree 2, and it follows that T is a path path from u to v Proof: We proceed by induction on V (T) only one component, and it is a spanning tree D

[PDF] Discussion 4A - EECS: www-insteecsberkeleyedu

with degree 1 (a) Prove that every tree on n ≥ 2 vertices has at least two leaves the contrary that x is not a leaf, so it has degree at least two The case when a tree has only two leaves is called the path graph, which is the graph on V =

[PDF] Tree : A tree is a connected, acyclic, undirected graph

G, we have to show that there is at most one path between any two vertices in G Suppose T is a tree of order n that contains only vertices of degree 1 and 3

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Observe that all the trees on six vertices (figure 2 1) have five edges In 2 1 4 Show that every tree with exactly two vertices of degree one is a path 2 1 5