Graphs come with various properties which are used for characterization of graphs depending on their structures. These properties are defined in specific terms pertaining to the domain of graph theory. In this chapter, we will discuss a few basic properties that are common in all graphs.Radius of a Connected GraphThe minimum eccentricity from all the vertices is considered as the radius of the Graph G. The minimum among all the maximum distances between a vertex to all other vertices is considered as the radius of the Graph G.Notation − r(G)From all the eccentricities of the vertices in a graph, the ... Read More

A graph is a diagram of points and lines connected to the points. It has at least one line joining a set of two vertices with no vertex connecting itself. The concept of graphs in graph theory stands up on some basic terms such as point, line, vertex, edge, degree of vertices, properties of graphs, etc. Here, in this chapter, we will cover these fundamentals of graph theory.PointA point is a particular position in a one-dimensional, two-dimensional, or three-dimensional space. For better understanding, a point can be denoted by an alphabet. It can be represented with a dot.ExampleHere, the dot ... Read More

Euler Graph - A connected graph G is called an Euler graph, if there is a closed trail which includes every edge of the graph G.Euler Path - An Euler path is a path that uses every edge of a graph exactly once. An Euler path starts and ends at different vertices.Euler Circuit - An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler circuit always starts and ends at the same vertex. A connected graph G is an Euler graph if and only if all vertices of G are of even degree, ... Read More

Cardinality of a set S, denoted by |S|, is the number of elements of the set. The number is also referred as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞.Example − |{1, 4, 3, 5}| = 4, |{1, 2, 3, 4, 5, ....}| = ∞If there are two sets X and Y, |X| = |Y| denotes two sets X and Y having same cardinality. It occurs when the number of elements in X is exactly equal to the number of elements in Y. In this case, there exists a bijective function ‘f’ ... Read More

Graph coloring is the procedure of assignment of colors to each vertex of a graph G such that no adjacent vertices get same color. The objective is to minimize the number of colors while coloring a graph. The smallest number of colors required to color a graph G is called its chromatic number of that graph. Graph coloring problem is a NP Complete problem.Method to Color a GraphThe steps required to color a graph G with n number of vertices are as follows −Step 1 − Arrange the vertices of the graph in some order.Step 2 − Choose the first ... Read More

A Function assigns to each element of a set, exactly one element of a related set. Functions find their application in various fields like representation of the computational complexity of algorithms, counting objects, study of sequences and strings, to name a few. The third and final chapter of this part highlights the important aspects of functions.Function - DefinitionA function or mapping (Defined as f: X → Y) is a relationship from elements of one set X to elements of another set Y (X and Y are non-empty sets). X is called Domain and Y is called Codomain of function ‘f’.Function ... Read More

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.