Solution Found!
a. Show that in a graph with n nodes and n edges, there must be a loop. b. A graph is
Chapter 3, Problem 4(choose chapter or problem)
a. Show that in a graph with n nodes and n edges, there must be a loop. b. A graph is called a tree if it contains no loops. Show that if a graph is a tree with n nodes, then it has at most n 1 edges. (Thus, a tree with n nodes and n 1 edges is called a maximal tree.) c. Give an example of an incidence matrix A with dim N(A) > 1. Draw a picture of the graph corresponding to your matrix and explain.
Questions & Answers
QUESTION:
a. Show that in a graph with n nodes and n edges, there must be a loop. b. A graph is called a tree if it contains no loops. Show that if a graph is a tree with n nodes, then it has at most n 1 edges. (Thus, a tree with n nodes and n 1 edges is called a maximal tree.) c. Give an example of an incidence matrix A with dim N(A) > 1. Draw a picture of the graph corresponding to your matrix and explain.
ANSWER:Step 1 of 3
Let G be a graph with n nodes and n edges.
Since G is connected, it must have a spanning tree T with n vertices and 
Let us consider an arbitrary edge 
Since T is a tree, there is a unique path 
The union of 
Thus, it is proved that in a graph with n nodes and n edges, there must be a loop.
A graph is called a tree if it contains no loops.