## Problem 1076

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

Solution:-

A tree is a graph that is connected and has no circuit. All trees have the properties listed below.

a. There is no 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 given graph is connected. All vertices A, B, C, and D can be reached from all other vertices, so the graph is connected.

Next, determine whether the graph has circuits. The graph contains the circuit A, B, C, D.

There is more than one path connecting each vertex to the other vertices.

The graph is connected, but it has a circuit. Therefore, the graph is not a tree. Note that is any one of the edges were removed, the circuit would be broken and the graph would be a tree.

## Graph lines

Graph lines

The graph of an equation is the set of points corresponding to all ordered pairs that satisfy the equation. It gives a “picture” of the equation. Most equations in two variables are satisfied by an infinite number of ordered pairs, so their graphs include an infinite number of points.

To graph an equation, we plot a number of ordered pairs that satisfy the equation until we have enough points to suggest the shape of the graph. For example, to graph 2x  + 3y =  6, we plot all the ordered pairs found in Objective 2 and Example 1 on the previous page. These points, shown in a table of values and plotted in Figure 4(a), appear to lie on a straight line. If all the ordered pairs that satisfy the equation 2x  + 3y =  6 were graphed, they would form the straight line shown in Figure 4(b).

The equation 2x + 3y = 6 is called a first-degree equation because it has no term with a variable to a power greater than one.

The graph of any first-degree equation in two variables is a straight line.

## Use number lines

Use number lines

A good way to get a picture of a set

of numbers is by using a number line. To construct a number line, choose

any point on a horizontal line and label it 0. Next, choose a point to the right

of 0 and label it 1. The distance from 0 to 1 establishes a scale that can be

used to locate more points, with positive numbers to the right of 0 and negative

numbers to the left of 0. The number 0 is neither positive nor negative.

A number line is shown in Figure 1.

The set of numbers identified on the number line in Figure 1, including

positive and negative numbers and 0, is part of the set of integers, written

I 5 {. . . , 23, 22, 21, 0, 1, 2, 3, . . . }.

Each number on a number line is called the coordinate of the point that

it labels, while the point is the graph of the number. Figure 2 shows a number

line with several selected points graphed on it.