In this chapter, we will cover a few standard examples to demonstrate the concepts we already discussed in the earlier chapters.

**Example 1**

**Find the number of spanning trees in the following graph.**

**Solution**

The number of spanning trees obtained from the above graph is 3. They are as follows −

These three are the spanning trees for the given graphs. Here the graphs I and II are isomorphic to each other. Clearly, the number of non-isomorphic spanning trees is two.

**Example 2**

**How many simple non-isomorphic graphs are possible with 3 vertices?**

**Solution**

There are 4 non-isomorphic graphs possible with 3 vertices. They are shown below.

**Example 3**

**Let ‘G’ be a connected planar graph with 20 vertices and the degree of each vertex is 3. Find the number of regions in the graph.**

**Solution**

By the sum of degrees theorem,

20 ∑ i=1 deg(V_{i}) = 2|E|

20(3) = 2|E|

|E| = 30

By Euler’s formula,

|V| + |R| = |E| + 2

20+ |R| = 30 + 2

|R| = 12

Hence, the number of regions is 12.

**Example 4**

**What is the chromatic number of complete graph K _{n}?**

**Solution**

In a complete graph, each vertex is adjacent to is remaining (n–1) vertices. Hence, each vertex requires a new color. Hence the chromatic number *K _{n} = n*.

**Example 5**

**What is the matching number for the following graph?**

**Solution**

Number of vertices = 9

We can match only 8 vertices.

Matching number is 4.

**Example 6**

**What is the line covering number of for the following graph?**

**Solution**

Number of vertices = |V| = n = 7

Line covering number = (α_{1}) ≥ ⌈*n*2⌉ = 3

α_{1} ≥ 3

By using 3 edges, we can cover all the vertices.

Hence, the line covering number is 3.