This document discusses graphs and Euler paths/circuits. It defines what a graph is - a collection of vertices connected by edges. An Euler path uses every edge once, starting and ending at different vertices. An Euler circuit is a closed path that uses every edge once, starting and ending at the same vertex. The Euler theorem states that a connected graph has an Euler circuit if every vertex has an even degree. The document provides examples of Euler paths and circuits and applications to problems like planning a bike route that uses every trail once.

1. Graph: Euler Path and Euler
Circuit
By Prof. Liwayway Memije-Cruz

2. Konigsberg Seven Bridges

3. A graph is a diagram displaying data; showing
the relationship between two or more
quantities, measurements or indicative numbers
that may or may not have a specific
mathematical formula relating them to each
other.

4. What is a Graph?
acollection of pointscalled verticesor nodes, and
connecting segmentscalled edges.
sometimestheedgesaregiven orientationsand
arerepresented by arrowsor aregiven values
(weights).
theposition of thevertices, thelengthsof the
edges, and theshapeof theedgesdo not matter in
agraph. What welook at isthenumber of
verticesand which of them arejoined by edges.

5. EXAMPLESOF GRAPHS

6. The graph
indicates the
social network
of neurological
patients.

7. Molecular Structure

8. Computer Network

9. Judicial
System

10. CONSTRUCTING A GRAPH

11. Thefollowing tablelistsfivestudentsat auniversity.
An “X” indicatesthat thetwo studentsparticipatein
thesamestudy group thissemester.
Joshua Diego Butch Lianne Aby
Joshua --- X X X
Diego X --- X X
Butch X X --- X
Lianne X ---
Aby X X ---

12. Draw a graph that represents this
information where each vertex
represents a student and an edge
connects two vertices if the
corresponding students study together.

14. Simple Graph
also called astrict graph (Tutte1998, p. 2), isan unweighted,
undirected graph containing no graph loopsor multipleedges
(Gibbons1985, p. 2; West 2000, p. 2; Bronshtein and
Semendyayev 2004, p. 346). A simplegraph may beeither
connected or disconnected.
theunqualified term "graph" usually refersto asimplegraph. A
simplegraph with multipleedgesissometimescalled a
multigraph (Skiena1990, p. 89).

15. A graph isconnected if thereisapath
connecting all thevertices.
Two verticesareadjacent if thereisan edge
joining them.
If every pair of verticesof agraph are
adjacent, thegraph iscomplete. A
completegraph with n verticesisdenoted by
Kn.
Thedegreeof avertex isthenumber of edges
attached to it.

16. CompleteGraph
A completegraph isagraph in which each pair of graph
verticesisconnected by an edge. Thecompletegraph with
graph verticesisdenoted and has(thetriangular numbers)
undirected edges, whereisabinomial coefficient.
sometimescalled universal graphs.

17. https://www.youtube.com/watch?
v=Kwj5sdF8b18

18. Euler Path
 apath that uses
every edgeof a
graph exactly once.
 If apath beginsand
endswith thesame
vertex, it isaclosed
path or a
circuit/cycle.
 An Euler path starts
and endsat different
vertices.

19. Leonard Euler
A Swissmathematician and
physicist, oneof thefoundersof
puremathematics.
Henot only madedecisiveand
formativecontributionsto the
subjectsof geometry, calculus,
mechanics, and number theory
but also developed methodsfor
solving problemsin
observational astronomy and
demonstrated useful applications
of mathematicsin technology
and public affairs.

20. Examples of Euler Circuits
isacircuit that
usesevery edgeof
agraph exactly
once
aEuler circuit
startsand endsat
thedifferent
vertices.

21. Thenumber of edgesthat meet at avertex iscalled
thedegreeof avertex.

22. Eulerian Graph Theorem
A connected graph isEulerian if and
only if each vertex of thegraph isof
even degree.
Eulerian Graph Theorem only
guaranteesthat if thedegreesof all the
verticesin agraph areeven,
an Euler circuit exists, but it doesnot
tell ushow to find one.

23. Determinewhether thegraph below isEulerian. If it is, find
an Euler circuit. If it isnot, explain why.

24. Determinewhether thefollowing graph isEulerian. If it
is, find an Eulerian circuit. If it isnot, can you find an
Euler path?

25. Euler Path Theorem
• A connected graph containsan Euler path
if and only if thegraph hastwo verticesof
odd degreewith all other verticesof even
degree.
• Every Euler path must start at oneof the
verticesof odd degreeand end at theother.

26. https://www.youtube.com/watch?v=AwsMTEl79wI

27. Application of Euler Path Theorem
Below isthemap of all the
trailsin anational park.
A biker would liketo
traverseall thetrailsexactly
once.
Isit possiblefor thebiker to
plan atrip that traversesall
thetrailsexactly once?
Isit possiblefor him to
traverseall thetrailsand
return to
thestarting point without
repeating any trail in the
trip?

28. References:
http://mathworld.wolfram.com/Graph.html
https://study.com/academy/lesson/mathematical-models-of-euler
https://www.britannica.com/biography/Leonhard-Euler
https://www.maa.org/press/periodicals/convergence/leonard-eule
Mathematicsof Graphsby Ethel CecilleM. Baltazar