## Problem 1077

Determine whether the graph is a tree. If the graph is not a tree, given the reason why.

**Solution:-**

Definition and Properties of a Tree

A tree is a graph that is connected and has no circuits. All trees have the following properties:

a. There is one and only one path joining any two vertices.

b. Every edge is a bridge.

c. A tree with n vertices must have n -1 edges.

First, determine whether the graph is connected. Since each of the five vertices, A, B, C, D, and E, can be reached from all other vertices, the given graph is connected.

Next, determine whether there are any circuits. Since there is one and only one path joining any two of the graph’s vertices, there are no circuits.

The graph is connected and has no circuits. Therefore, the graph is a tree. There is one and only one path joining any two vertices. Every edge is a bridge. It has 5 vertices and 5 -1, or 4, edges.