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The paper on the Seven Bridges of Königsberg, written by Leonhard Euler, was published in 1736 and is considered the first paper in the history of graph theory. 6 This article and Vandermonde’s article on the Cavaliers issue continues the analysis point initiated by Leibniz. The Euler formula for the number of edges, vertices and faces of convex polyhedra is studied and generalized by Cauchy and L’Huillier and is the origin of topology. In more than a century after Euler’s paper on the Königsberg bridge and the topologies described above, Cayley studied on the basis of studying specific forms of analysis from differential calculus, which was used to study specific types of graphs, ie trees. . This research has many implications in theoretical chemistry. The techniques involved are primarily concerned with the enumeration of graphs with specific properties. Then, the counting graph theory was consistent with Cayley’s results and Pólya’s basic results published between 1935 and 1937, and De Bruijn’s promotion of them in 1959. Cayley linked his results in the tree to contemporary chemical composition research. stand up. The fusion of ideas from mathematics and ideas from chemistry is the origin of a part of the standard terminology of graph theory.

Objectives :

• Learning what a graph is and how it is used.
• To implement the graph abstract data type using several internal representations.
• See how graphs can be used to solve a variety of problems

What is a Graph?

Graphics is a picturesque representation of a set of objects in which the objects are linked by links. The interrelated objects are represented by points called peaks, and the links that link the peaks are called edges.
Formal graph is a pair of sets (V, E), where V is a set of vertices, and E is a set of edges connecting pairs of peaks. See the chart below:

Definition of 1 (simple graph) A simple graph G is a pair G = (V, E) where V is the final set called G vertices and E is a subset of P2 (V) E of two subsets V) the vertices of the computer and the computer pair are in E if and only when they are connected. Example 2 (Intercity Routes) Imagine four cities with characteristic mathematical spells A, B, C and D. Between these cities there are different routes called a, b, c, d, e, f and g. Here is a picture of this situation:

A tree is a graph without cycles and a spanning tree of a set of points is a tree consisting all of the points. In Figure 2.2, we show three spanning trees of the set of points in Figure 2.1.

Figure 2.2: Three Spanning Trees of the Set of Points in Figure 2.1

Among the three exciting trees, the tree in Figure 2.2 (a) is the shortest and we are interested. Therefore, the least exciting tree problem is defined as follows: We offer a set of points and we want to find the tree with the shortest total length.
How can we find a minimal exciting tree? A very simple algorithm is to quantify all possible trees and one of them should be what we are looking for. Figure 2.3 shows all the possible trees for three points. As you can see, there are only three.

Figure 2.3: All Possible Spanning Trees for Three Points

Tree
An asilic graph is called a tree. In other words, a connected graph is called a cycle of a tree.
The edge of the tree is known as the branch. The elements of the tree are called their nodes. Knots are called knot knot knot knots.
A tree with an “n” has the edge “n-1”. If you have an edge other than n-1, the extra edge must obviously have a pair with two vertices, which leads to the formation of a loop. Then it is a cyclic graph that violates the tree graph.
Example 1
The graph shown here is a tree because it has no cycles and is connected. It has four vertices and three edges, that is, for n-vertices n-1, as defined in the definition.

Every tree has at least two vertices of degree one.

Example 2

In the preceding example, the said “A” and “D” characters have one degree. And in two other angles there are two degrees ‘B’ and ‘C’. This is possible because there is no need to create a circle; the graph should have at least two individual edges anywhere. This is nothing but two edges with nothing but a degree

Forest
A separate axillary diagram is called a forest. In other words, a collection of unique trees is called forests.

Example
The following graph looks like two sub-graphs; but it is a single disconnected graph. There are no cycles in this graph. Hence, clearly it is a forest.

Spanning Trees

Let G be a connected graph, then the sub-graph H of G is called a spanning tree of G if ?
• H is a tree
• H contains all vertices of G.
A spanning tree T of an undirected graph G is a subgraph that includes all of the vertices of G.
Example

The next two are the most commonly used graphic representations.
1. The adjacency matrix
2. List of crosses
There are other representations, for example: the incident matrix and the list of morbidity. The choice of graphic representation is a specific situation. This depends completely on the type of operations performed and the ease of use.

Adjacency matrix:
Adjacency Matrix is a 2D matrix of size V x V, where V is the number of vertices in the graph. Let the 2D matrix be adj , the time interval i j = 1 indicates that there is an edge from the vertex i to the vertex j. The adjacency matrix for an untargeted graph is always symmetric. The adjacency matrix is also used to represent weighted graphs. If adj i j = w, there is an edge from the vertex i to the vertex j with weight w.

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