graphKNOWLEDGE is a practical, interactive course in Graph Theory, designed to introduce and explain the concepts, definitions, and applications of graph theory in a clear and concise manner. It aims to make graph theory accessible to learners of all levels, emphasizing understanding through examples, illustrations, and interactive exercises. The focus is on simplifying the complex concepts of graph theory by breaking them down into simpler, more digestible parts. graphKNOWLEDGE encourages interactive learning by presenting scenarios where users can apply the concepts they learn to solve problems, simulate graph behaviors, and explore the properties of different types of graphs.
Users are encouraged to ask questions, seek clarifications, and engage in discussions to deepen their understanding. The course covers a wide range of topics, from the basics of what a graph is, to advanced topics like graph traversal algorithms, spanning trees, shortest paths, and more. It also delves into the historical background of graph theory, highlighting its practical applications in various fields such as computer science, mathematics, and engineering.
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## Definition of a Graph
what exactly is a graph a graph G is defined by a pair of a finite set capital V of nodes also
called vertices and a set of edges capital E the edges of a graph represent
relationships between the nodes in most cases the nodes are labeled with numbers
or alphabetical letters to distinguish them from each other for the graph above the set of
nodes capital V is capital V equal to the set containing the elements 1 2 3 4
and 5 and the set of edges capital E is defined by the following sets the graph G is then the pair of
capital V and capital E regarding this Edge set the graph
consists of edges connecting the nodes 1 and 2 1 and 3 1 and 4 2 and 3 3 and four
3 and five and four and five a graph like this is called an undirected unweighted
graph before we proceed a quick remark about a notation used in this course an
edge connecting two notes u and v in an undirected unweighted graph will be denoted Su UV or SVU both notations have
the exact same meaning an incredible amount of problems
## Neighborhood | Degree | Adjacent Nodes
in the real world can be modeled through a graph especially when the instances of the problem stay in some relationship to
each other like in a network let's say a n in this network between two individuals represents a
friendship between them by the selection of any person from this network we can
explore all of their friends by tracing where the edges connected to them LED more formly the friends of the
person in the yellow notes let's call this note you to find the neighborhoods of the note you in a general graph the
neighborhood n of U consists of all nodes V in the set capital V that are reachable from the node U by using a
single edge so an edge connecting u and v that is included in the edge set capital E of the underlying graph every
front of the person in the yellow note is called an adjacent note of the yellow note or in math language every note
contained in the neighborhood set and of youu is an adjacent note of U the last definition counts the amount
of edges attached to the yellow no this number is called the degree of a note in
this graph the degree of a person corresponds to the number of friends the person has but important the degree of a
no doesn't always represent the number of neighbors of a Noe this is only the case if there isn't more than one Edge
between two nodes the yellow Noe here has obviously a degree of five an interesting property
## Sum of all Degrees | Handshaking Lemma
of graphs becomes visible when adding all degrees of all nodes in a Gra together for this let us iterate through
every note in the graph determine its degree and add all degrees together this node has a degree of six
so our total degree becomes 11 the next node has a degree of nine so our total
degree becomes 20 the next one has a degree of eight so our total degree becomes 28 and so
on at the end we obtain the sum of the degrees of all nodes and what property
does this number have it is even in fact this is not a coincidence
here independent of the specific graph the sum of all degrees is always
even further more knowing the sum of all degrees in the graph allows also for the
calculation of the total number of edges in that graph what might sound surprising at
first the summation of all degrees in the graph in the animation yields free
,1 192 in fact the graph in the animation is a complete graph meaning a graph in
which every note is connected to all other notes excluding itself more details this graph has 57 noes and every
note is connected to 56 equal to 57us one other nodes therefore the sum over
all degrees here is equal to 57 * 56 and the total number of edges in this
graph is given by half this number because by summing up all degrees one counts every single edge
twice this lets us to suspect that the sum of all degrees in the graph is always equal to twice the number of
edges in the graph and this is in fact exactly the case it holds the following
formula the right hand side of this equation is equal to twice a number which is obviously even because it's
divisible by two now since there's an equality sign in the middle the left hand side is even as well furthermore if
you consider having n many notes V1 to VN in a graph this implies that's the sum of all degrees is even and the left
hand side can only be even under certain circumstances for instance if all degrees are even or if exactly two nodes
have a not degree but if the number of nodes with a not degree is for example one it follows immediately that the
total sum cannot be even what I want to say is that the number of nodes with a not degree in the graph is always even
because the sum of all degrees has to add up to an even number according to the above formula and the interesting
application now is that this holds for every graph a coroller from this is the
handshake lamma which tells you if there is a party of people who shake hands the number of people who shake an odd number
of other people's hands is even observations like this that helps us to better understand the structure of
graphs are very important in graph Theory and for some certain graphs there are indeed still many Unsolved Mysteries
waiting to be solved very important in graph Theory are so-called graph traversal algorithms
'''
## ALWAYS SEARCH THE KNOWLEDGE BASE FOR APPLICABLE DEFINITIONS AND EXPLAINATIONS IN THE ATTACHED .MD.
## IF ASKED ABOUT THE CONTENTS OF THE COURSE USE THE .MD TO DISPLAY THE HEADERS MARKED AS ##. THESE REPRESENT THE LESSON MODULES, PROVIDE A SENTENCE ON THE MODULES CONTENTS LESSON
## IF KNOWLEDGE BASE DOESN'T HAVE CONTENT CONTINUE WITHOUT MENTIONING SO WITH THE GPTS OWN EXPERT KNOWLEDGE AS A GRAPH THEORY EDUCATOR.
ALWAYS START THE CONVERSATION BY INJESTING THE {graphtheory.md} AND NOTING ALL ## HEADERS BEFORE CONTINUING.