Trees and distance, Graph theory, networks and applications, MAA600 Christopher Engstr¨om Trees and distance, Lecture 5, trees and Graph theory, networks and applications, distance MAA600 Distance in trees and graphs Christopher Engstr¨om September 15, 2014 I If the graph is connected and acyclic it is called a tree I A vertex with degree one (only 1 connecting edge) is called a leaf. I A subgraph of G with the same vertex set V (G) is called a spanning subgraph. (note that it does not need to be connected nor acyclic). I A spanning forest is a spanning subgraph which is also a forest. I A spanning tree is a spanning subgraph which is also a tree. Trees and Definitions distance, Graph theory, networks and applications, Definition MAA600 Christopher Engstr¨om I A graph with no cycle is called acyclic. Such a graph is called a forest. Lecture 5, trees and distance Distance in trees and graphs I A vertex with degree one (only 1 connecting edge) is called a leaf. I A subgraph of G with the same vertex set V (G) is called a spanning subgraph. (note that it does not need to be connected nor acyclic). I A spanning forest is a spanning subgraph which is also a forest. I A spanning tree is a spanning subgraph which is also a tree. Trees and Definitions distance, Graph theory, networks and applications, Definition MAA600 Christopher Engstr¨om I A graph with no cycle is called acyclic. Such a graph is called a forest. Lecture 5, trees and distance I If the graph is connected and acyclic it is called a tree Distance in trees and graphs I A subgraph of G with the same vertex set V (G) is called a spanning subgraph. (note that it does not need to be connected nor acyclic). I A spanning forest is a spanning subgraph which is also a forest. I A spanning tree is a spanning subgraph which is also a tree. Trees and Definitions distance, Graph theory, networks and applications, Definition MAA600 Christopher Engstr¨om I A graph with no cycle is called acyclic. Such a graph is called a forest. Lecture 5, trees and distance I If the graph is connected and acyclic it is called a tree Distance in I A vertex with degree one (only 1 connecting edge) is trees and graphs called a leaf. I A spanning forest is a spanning subgraph which is also a forest. I A spanning tree is a spanning subgraph which is also a tree. Trees and Definitions distance, Graph theory, networks and applications, Definition MAA600 Christopher Engstr¨om I A graph with no cycle is called acyclic. Such a graph is called a forest. Lecture 5, trees and distance I If the graph is connected and acyclic it is called a tree Distance in I A vertex with degree one (only 1 connecting edge) is trees and graphs called a leaf. I A subgraph of G with the same vertex set V (G) is called a spanning subgraph. (note that it does not need to be connected nor acyclic). I A spanning tree is a spanning subgraph which is also a tree. Trees and Definitions distance, Graph theory, networks and applications, Definition MAA600 Christopher Engstr¨om I A graph with no cycle is called acyclic. Such a graph is called a forest. Lecture 5, trees and distance I If the graph is connected and acyclic it is called a tree Distance in I A vertex with degree one (only 1 connecting edge) is trees and graphs called a leaf. I A subgraph of G with the same vertex set V (G) is called a spanning subgraph. (note that it does not need to be connected nor acyclic). I A spanning forest is a spanning subgraph which is also a forest. Trees and Definitions distance, Graph theory, networks and applications, Definition MAA600 Christopher Engstr¨om I A graph with no cycle is called acyclic. Such a graph is called a forest. Lecture 5, trees and distance I If the graph is connected and acyclic it is called a tree Distance in I A vertex with degree one (only 1 connecting edge) is trees and graphs called a leaf. I A subgraph of G with the same vertex set V (G) is called a spanning subgraph. (note that it does not need to be connected nor acyclic). I A spanning forest is a spanning subgraph which is also a forest. I A spanning tree is a spanning subgraph which is also a tree. This essentially means that every tree with more than one vertex can be constructed from a smaller tree by adding a new leaf. Trees and Trees distance, Graph theory, networks and applications, MAA600 Christopher Engstr¨om Lecture 5, trees and Lemma distance Every tree with at least two vertices has at least two leaves. Distance in trees and Deleting a leaf from a tree with n vertices produces a new tree graphs with n − 1 vertices. Trees and Trees distance, Graph theory, networks and applications, MAA600 Christopher Engstr¨om Lecture 5, trees and Lemma distance Every tree with at least two vertices has at least two leaves. Distance in trees and Deleting a leaf from a tree with n vertices produces a new tree graphs with n − 1 vertices. This essentially means that every tree with more than one vertex can be constructed from a smaller tree by adding a new leaf. I G is connected and has n − 1 edges. I G has n − 1 edges and has no cycles. I G has no loops, and for every u; v 2 V (G) there is exactly one path from u to v. We will show the first statement, the others can be proved in a similar manner. Trees and Trees distance, Graph theory, networks and applications, MAA600 Christopher Engstr¨om Theorem For an n-vertex graph G (with n ≥ 1), the following statements Lecture 5, trees and are equivalent. distance Distance in I G is connected and has no cycles (a tree). trees and graphs I G has n − 1 edges and has no cycles. I G has no loops, and for every u; v 2 V (G) there is exactly one path from u to v. We will show the first statement, the others can be proved in a similar manner. Trees and Trees distance, Graph theory, networks and applications, MAA600 Christopher Engstr¨om Theorem For an n-vertex graph G (with n ≥ 1), the following statements Lecture 5, trees and are equivalent. distance Distance in I G is connected and has no cycles (a tree). trees and graphs I G is connected and has n − 1 edges. I G has no loops, and for every u; v 2 V (G) there is exactly one path from u to v. We will show the first statement, the others can be proved in a similar manner. Trees and Trees distance, Graph theory, networks and applications, MAA600 Christopher Engstr¨om Theorem For an n-vertex graph G (with n ≥ 1), the following statements Lecture 5, trees and are equivalent. distance Distance in I G is connected and has no cycles (a tree). trees and graphs I G is connected and has n − 1 edges. I G has n − 1 edges and has no cycles. We will show the first statement, the others can be proved in a similar manner. Trees and Trees distance, Graph theory, networks and applications, MAA600 Christopher Engstr¨om Theorem For an n-vertex graph G (with n ≥ 1), the following statements Lecture 5, trees and are equivalent. distance Distance in I G is connected and has no cycles (a tree). trees and graphs I G is connected and has n − 1 edges. I G has n − 1 edges and has no cycles. I G has no loops, and for every u; v 2 V (G) there is exactly one path from u to v. We will show the first statement, the others can be proved in a similar manner. Trees and Trees distance, Graph theory, networks and applications, MAA600 Christopher Engstr¨om Theorem For an n-vertex graph G (with n ≥ 1), the following statements Lecture 5, trees and are equivalent. distance Distance in I G is connected and has no cycles (a tree). trees and graphs I G is connected and has n − 1 edges. I G has n − 1 edges and has no cycles. I G has no loops, and for every u; v 2 V (G) there is exactly one path from u to v. Trees and Trees distance, Graph theory, networks and applications, MAA600 Christopher Engstr¨om Theorem For an n-vertex graph G (with n ≥ 1), the following statements Lecture 5, trees and are equivalent. distance Distance in I G is connected and has no cycles (a tree). trees and graphs I G is connected and has n − 1 edges. I G has n − 1 edges and has no cycles. I G has no loops, and for every u; v 2 V (G) there is exactly one path from u to v. We will show the first statement, the others can be proved in a similar manner. Trees and Proof distance, Graph theory, networks and applications, MAA600 Christopher Engstr¨om To prove that statement 1; 2; 3 are equivalent it is enough to Lecture 5, show that any two of (connected, acyclic, n − 1 edges) implies trees and distance the third. Distance in (connected, acyclic) ) n − 1 edges: It is easy to see that trees and graphs n − 1 edges are enough and necessary in order for a graph to be connected.