Trees

                                             TREES 

Trees is a special type of graph,which is acyclic and connected or a connected acyclic  graph is called a tree.

                             eg;paths and stars.

This is a tree.It is also called as connected cycle free graph.

A graph is said to be a tree ,if and only if there exists a unique path between every pair of vertices.

                                                       Coloring of graphs

In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices share the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges share the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

Vertex coloring is the starting point of the subject, and other coloring problems can be transformed into a vertex version. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as is. That is partly for perspective, and partly because some problems are best studied in non-vertex form, as for instance is edge coloring.

The convention of using colors originates from coloring the countries of a map, where each face is literally colored. This was generalized to coloring the faces of a graph embedded in the plane. By planar duality it became coloring the vertices, and in this form it generalizes to all graphs. In mathematical and computer representations, it is typical to use the first few positive or non negative integers as the "colors". In general, one can use any finite set as the "color set". The nature of the coloring problem depends on the number of colors but not on what they are.

Graph coloring enjoys many practical applications as well as theoretical challenges. Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. It has even reached popularity with the general public in the form of the popular number puzzle Sudoku. Graph coloring is still a very active field of research.

This is an example of  vertex coloring.This graph is having proper coloring,because no two adjacent vertices are having same color.The minimum color used is 3.So 3 is the chromatic number of the graph.Chromatic number of the graph is the minimum number of colors needed to have a proper coloring of the graph.

Applications

                    

                                                                                        This is the typical application of coloring of the graph.Here in the map no two adjacent states are having same color.So this technique is used effectively in map creation.

                                                      Rooted trees

A tree with a vertex designated as root and drawn in such a way that every other vertex is coming out of it is called a rooted tree.

In a rooted tree every vertex other than the root with degree 1 is called a leaf.In a rooted tree every vertex other than a half is called an internal vertex.

                                            If m+1 is the maximum degree among all the degree of internal vertices ,then the tree is called an m-ary tree.If every internal vertex has got exactly one degree then that tree is called full m-ary tree.

                                              Minimal spanning tree

Given a connected, undirected graph, a spanning tree of that graph is a subgraph that is a tree and connects all the vertices together. A single graph can have many different spanning trees. We can also assign a weight to each edge, which is a number representing how unfavorable it is, and use this to assign a weight to a spanning tree by computing the sum of the weights of the edges in that spanning tree. A minimum spanning tree (MST) or minimum weight spanning tree is then a spanning tree with weight less than or equal to the weight of every other spanning tree. More generally, any undirected graph (not necessarily connected) has a minimum spanning forest, which is a union of minimum spanning trees for its connected components.
Here the parrt drawn in dark lines denotes the minimum spanning tree of the graph