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

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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.

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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  edges.

Let us consider an arbitrary edge  such that with endpoints u and v.

Since T is a tree, there is a unique path between u and V.

The union of and is a cycle.

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.

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