In this chapter, we will cover a few standard examples to demonstrate the concepts we already discussed in the earlier chapters.
Find the number of spanning trees in the following graph.
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.
How many simple non-isomorphic graphs are possible with 3 vertices?
There are 4 non-isomorphic graphs possible with 3 vertices. They are shown below.
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.
By the sum of degrees theorem,
20 Σ i=1 deg(Vi) = 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.
What is the chromatic number of complete graph Kn?
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 Kn = n.
What is the matching number for the following graph?
Number of vertices = 9
We can match only 8 vertices.
Matching number is 4.
What is the line covering number of for the following graph?
Number of vertices = |V| = n = 7
Line covering number = (α1) ≥ [n/2] = 3
α1 ≥ 3
By using 3 edges, we can cover all the vertices.
Hence, the line covering number is 3.