Found 466 Articles for Mathematics

Tree is a discrete structure that represents hierarchical relationships between individual elements or nodes. A tree in which a parent has no more than two children is called a binary tree.Tree and its PropertiesDefinition − A Tree is a connected acyclic undirected graph. There is a unique path between every pair of vertices in G. A tree with N number of vertices contains (N-1) number of edges. The vertex which is of 0 degree is called root of the tree. The vertex which is of 1 degree is called leaf node of the tree and the degree of an internal ... Read More

Hamiltonian graph - A connected graph G is called Hamiltonian graph if there is a cycle which includes every vertex of G and the cycle is called Hamiltonian cycle. Hamiltonian walk in graph G is a walk that passes through each vertex exactly once.Dirac's Theorem - If G is a simple graph with n vertices, where n ≥ 3 If deg(v) ≥ {n}/{2} for each vertex v, then the graph G is Hamiltonian graph.Ore's Theorem - If G is a simple graph with n vertices, where n ≥ 2 if deg(x) + deg(y) ≥ n for each pair of non-adjacent ... Read More

There are various types of graphs depending upon the number of vertices, number of edges, interconnectivity, and their overall structure. We will discuss only a certain few important types of graphs in this chapter.Null GraphA graph having no edges is called a Null Graph.ExampleIn the above graph, there are three vertices named 'a', 'b', and 'c', but there are no edges among them. Hence it is a Null Graph.Trivial GraphA graph with only one vertex is called a Trivial Graph.ExampleIn the above shown graph, there is only one vertex 'a' with no other edges. Hence it is a Trivial graph.Non-Directed ... Read More

A graph can exist in different forms having the same number of vertices, edges, and also the same edge connectivity. Such graphs are called isomorphic graphs. Note that we label the graphs in this chapter mainly for the purpose of referring to them and recognizing them from one another.Isomorphic GraphsTwo graphs G1 and G2 are said to be isomorphic if −Their number of components (vertices and edges) are same.Their edge connectivity is retained.Note − In short, out of the two isomorphic graphs, one is a tweaked version of the other. An unlabelled graph also can be thought of as an ... Read More

Two graphs G1 and G2 are said to be homomorphic, if each of these graphs can be obtained from the same graph 'G' by dividing some edges of G with more vertices. Take a look at the following example −Divide the edge 'rs' into two edges by adding one vertex.The graphs shown below are homomorphic to the first graph.If G1 is isomorphic to G2, then G is homeomorphic to G2 but the converse need not be true.Any graph with 4 or less vertices is planar.Any graph with 8 or less edges is planar.A complete graph Kn is planar if and ... Read More

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