Problem StatementFind the number of spanning trees in the following graph.SolutionThe 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.

Problem StatementLet '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.SolutionBy the sum of degrees theorem, 20 ∑ i=1  deg(Vi) = 2|E|20(3) = 2|E||E| = 30By Euler’s formula,|V| + |R| = |E| + 220+ |R| = 30 + 2|R| = 12Hence, the number of regions is 12.

Problem StatementHow many simple non-isomorphic graphs are possible with 3 vertices?SolutionThere are 4 non-isomorphic graphs possible with 3 vertices. They are shown below.

Problem StatementWhat is the matching number for the following graph?SolutionNumber of vertices = 9We can match only 8 vertices.Matching number is 4.

Problem StatementWhat is the line covering number for the following graph?SolutionNumber of vertices = |V| = n = 7Line covering number = (α1) ≥ ⌈ n / 2 ⌉ = 3α1 ≥ 3By using 3 edges, we can cover all the vertices.Hence, the line covering number is 3.

Problem StatementWhat is the chromatic number of complete graph Kn?SolutionIn 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.

A graph is traversable if you can draw a path between all the vertices without retracing the same path. Based on this path, there are some categories like Euler’s path and Euler’s circuit which are described in this chapter.Euler’s PathAn Euler’s path contains each edge of ‘G’ exactly once and each vertex of ‘G’ at least once. A connected graph G is said to be traversable if it contains an Euler’s path.ExampleEuler’s Path = d-c-a-b-d-e.Euler’s CircuitIn an Euler’s path, if the starting vertex is same as its ending vertex, then it is called an Euler’s circuit.ExampleEuler’s Path = a-b-c-d-a-g-f-e-c-a.Euler’s Circuit ... Read More

A graph is a set of points, called nodes or vertices, which are interconnected by a set of lines called edges. The study of graphs, or graph theory is an important part of a number of disciplines in the fields of mathematics, engineering and computer science.Graph TheoryDefinition − A graph (denoted as G = (V, E)) consists of a non-empty set of vertices or nodes V and a set of edges E. A vertex a  represents an endpoint of an edge. An edge joins two vertices a, b  and is represented by set of vertices it connects.Example − Let us ... Read More

Distance between Two VerticesIt is number of edges in a shortest path between Vertex U and Vertex V. If there are multiple paths connecting two vertices, then the shortest path is considered as the distance between the two vertices.Notation − d(U, V)There can be any number of paths present from one vertex to other. Among those, you need to choose only the shortest one.ExampleTake a look at the following graph −Here, the distance from vertex 'd' to vertex 'e' or simply 'de' is 1 as there is one edge between them. There are many paths from vertex 'd' to vertex ... Read More

It is the number of vertices adjacent to a vertex V. Notation − deg(V). In a simple graph with n number of vertices, the degree of any vertices is − deg(v) = n – 1 ∀ v ∈ G A vertex can form an edge with all other vertices except by itself. So the degree of a vertex will be up to the number of vertices in the graph minus 1. This 1 is for the self-vertex as it cannot form a loop by itself. If there is a loop at any of the vertices, then it is not a ... Read More