Graph Theory

 

Hi, Im leonard Schulz your graph theory teacher, check me out at Leonardschulz29@hotmail.com

Graph theory is an esciting course, hope you enjoi it very much

Topic 1

Assignment 1

Assignment due: 7/jan/2022

iIntroduction to graph theory, submit by Jan/7/2022

Graph theory, branch of mathematics concerned with networks of points connected by lines. The subject of graph theory had its beginnings in recreational math problems (see number game), but it has grown into a significant area of mathematical research, with applications in chemistry, operations research, social sciences, and computer science.

Topic 2

Assignment 2

Assignment due:17/jan/2022

The history of graph theory may be specifically traced to 1735, when the Swiss mathematician Leonhard Euler solved the Königsberg bridge problem. The Königsberg bridge problem was an old puzzle concerning the possibility of finding a path over every one of seven bridges that span a forked river flowing past an island—but without crossing any bridge twice. Euler argued that no such path exists. His proof involved only references to the physical arrangement of the bridges, but essentially he proved the first theorem in graph theory.

Topic 3

Assignment 3

Assignment due: 26/jan/2022

As used in graph theory, the term graph does not refer to data charts, such as line graphs or bar graphs. Instead, it refers to a set of vertices (that is, points or nodes) and of edges (or lines) that connect the vertices. When any two vertices are joined by more than one edge, the graph is called a multigraph. A graph without loops and with at most one edge between any two vertices is called a simple graph. Unless stated otherwise, graph is assumed to refer to a simple graph. When each vertex is connected by an edge to every other vertex, the graph is called a complete graph. When appropriate, a direction may be assigned to each edge to produce what is known as a directed graph, or digraph.

Topic 4

Assignment 4

Assignment due: 7/jan/2022

Talk about basic types of graphs in a research paper, and give me the answer by 31/jan/2022


An important number associated with each vertex is its degree, which is defined as the number of edges that enter or exit from it. Thus, a loop contributes 2 to the degree of its vertex. For instance, the vertices of the simple graph shown in the diagram all have a degree of 2, whereas the vertices of the complete graph shown are all of degree 3. Knowing the number of vertices in a complete graph characterizes its essential nature. For this reason, complete graphs are commonly designated Kn, where n refers to the number of vertices, and all vertices of Kn have degree n − 1. (Translated into the terminology of modern graph theory, Euler’s theorem about the Königsberg bridge problem could be restated as follows: If there is a path along edges of a multigraph that traverses each edge once and only once, then there exist at most two vertices of odd degree; furthermore, if the path begins and ends at the same vertex, then no vertices will have odd degree.)


Another important concept in graph theory is the path, which is any route along the edges of a graph. A path may follow a single edge directly between two vertices, or it may follow multiple edges through multiple vertices. If there is a path linking any two vertices in a graph, that graph is said to be connected. A path that begins and ends at the same vertex without traversing any edge more than once is called a circuit, or a closed path. A circuit that follows each edge exactly once while visiting every vertex is known as an Eulerian circuit, and the graph is called an Eulerian graph. An Eulerian graph is connected and, in addition, all its vertices have even degree.

Topic 5

Assignment 5

Assignment due: Jan/31/2022

Describe double traversal nodes in a research paper, submit by 31/jan/2022





In 1857 the Irish mathematician William Rowan Hamilton invented a puzzle (the Icosian Game) that he later sold to a game manufacturer for £25. The puzzle involved finding a special type of path, later known as a Hamiltonian circuit, along the edges of a dodecahedron (a Platonic solid consisting of 12 pentagonal faces) that begins and ends at the same corner while passing through each corner exactly once. The knight’s tour (see number game: Chessboard problems) is another example of a recreational problem involving a Hamiltonian circuit. Hamiltonian graphs have been more challenging to characterize than Eulerian graphs, since the necessary and sufficient conditions for the existence of a Hamiltonian circuit in a connected graph are still unknown.

Topic 6

Assignment 6

Assignment due: Feb/2/2022 at 11:00 pm

describe all types of sequence queues in a research paper, submit by Feb/03/2022 at 11 pm