A Hamiltonian path is a path that visits each vertex of the graph exactly once.
A Hamiltonian circuit is a path that uses each vertex of a graph exactly once and returns to the starting vertex.
A graph is a diagram displaying data which show 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.
what is Hamilton path and Euler path?
History of Euler path and Hamilton path
Vertex(node) and edge
Hamilton path and Hamilton circuit
Euler path and Euler circuit
Degree of vertex and comparison of Euler and Hamilton path
Solving a problem
A graph is a diagram displaying data which show 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.
what is Hamilton path and Euler path?
History of Euler path and Hamilton path
Vertex(node) and edge
Hamilton path and Hamilton circuit
Euler path and Euler circuit
Degree of vertex and comparison of Euler and Hamilton path
Solving a problem
One of the main reasons for the popularity of Dijkstra's Algorithm is that it is one of the most important and useful algorithms available for generating (exact) optimal solutions to a large class of shortest path problems. The point being that this class of problems is extremely important theoretically, practically, as well as educationally.
The solution to the single-source shortest-path tree problem in graph theory. This slide was prepared for Design and Analysis of Algorithm Lab for B.Tech CSE 2nd Year 4th Semester.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
One of the main reasons for the popularity of Dijkstra's Algorithm is that it is one of the most important and useful algorithms available for generating (exact) optimal solutions to a large class of shortest path problems. The point being that this class of problems is extremely important theoretically, practically, as well as educationally.
The solution to the single-source shortest-path tree problem in graph theory. This slide was prepared for Design and Analysis of Algorithm Lab for B.Tech CSE 2nd Year 4th Semester.
In mathematics, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects.Graph theory is also important in real life.
Analysis & Design of Algorithms
Backtracking
N-Queens Problem
Hamiltonian circuit
Graph coloring
A presentation on unit Backtracking from the ADA subject of Engineering.
In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph concepts from the field of graph theory within mathematics.
Ant Colony (-based) Optimisation – a way to solve optimisation problems based on the way that ants indirectly communicate directions to each other we call Stigmergy.
Second-level Digital Divide and experiences of Schools and TeachersLiwayway Memije-Cruz
The second-level digital divide, is referred to as the production gap, and it describes the gap that separates the consumers of content on the Internet from the producers of content.
Science and technology studies, or science, technology and society studies (STS) is the study of how society, politics, and culture affect scientific research and technological innovation, and how these, in turn, affect society, politics and culture.
A hydrocarbon is a molecule whose structure includes only hydrogen and carbon atoms. Hydrocarbons form bonds with other atoms in order to create organic compounds.
Hydrocarbon derivatives are based on simple hydrocarbon compounds that contain only hydrogens and carbons. Hydrocarbon derivatives contain at least one element other than hydrogen or carbon, such as oxygen, nitrogen or one of the halogen atoms (elements in column 7A of the Periodic Table.
Organic reactions are chemical reactions involving organic compounds. Organic reactions are used in the construction of new organic molecules. The production of many man-made chemicals such as drugs, plastics, food additives, fabrics depend on organic reactions.
Organic chemistry involves the study of the structure, properties, composition, reactions, and preparation of carbon-containing compounds, which include not only hydrocarbons but also compounds with any number of other elements, including hydrogen (most compounds contain at least one carbon–hydrogen bond), nitrogen, oxygen, halogens, phosphorus, silicon, and sulfur.
This branch of chemistry was originally limited to compounds produced by living organisms but has been broadened to include human-made substances such as plastics. The range of application of organic compounds is enormous and also includes, but is not limited to, pharmaceuticals, petrochemicals, food, explosives, paints, and cosmetics.
Organic chemistry is the study of the structure, properties, composition, reactions, and preparation of carbon-containing compounds, which include not only hydrocarbons but also compounds with any number of other elements, including hydrogen (most compounds contain at least one carbon–hydrogen bond), nitrogen, oxygen,
Science and technology studies, or science, technology and society studies (STS) is the study of how society, politics, and culture affect scientific research and technological innovation, and how these, in turn, affect society, politics and culture.
Isomers are molecules with the same molecular formula, but different structural or spatial arrangements of the atoms within the molecule. The reason there are such a colossal number of organic compounds which is more than 10 million is partly due to isomerism.
Apportionment is Apportionment involves dividing something up, just like fair division.
Voting is a method for a group, such as, a meeting or an electorate to make a collective decision or express an opinion, usually following discussions, debates or election campaigns.
Lipid metabolism entails the oxidation of fatty acids to either generate energy or synthesize new lipids from smaller constituent molecules. Lipid metabolism is associated with carbohydrate metabolism, as products of glucose (such as acetyl CoA) can be converted into lipids.
Carbohydrate metabolism involves the different biochemical processes responsible for the formation, breakdown, and interconversion of carbohydrates in living organisms.
Every organism is composed of several different types of human body tissue. The human body tissue is another way of describing how our cells are grouped together in a highly organized manner according to specific structure and function. These groupings of cells form tissues, which then make up organs and various parts of the body.
Reproduction means producing offspring that may or may not be exact copies of their parents. It is a part of a life cycle, which is a series of events wherein individuals grow, develop, and reproduce according to a program of instructions encoded in DNA, which they inherit from their parents. When cells divide, each daughter cell receives a complete copy of DNA and enough cytoplasmic machinery to start up its own operation. DNA contains the blueprints for making different proteins.
.Enzymes are proteins that catalyze or speed up chemical reactions. They also help digest the foods we eat food and heal our wounds. They play major roles in respiration, making proteins, and DNA replication..
Reproduction means producing offspring that may or may not be exact copies of their parents. It is a part of a life cycle, which is a series of events wherein individuals grow, develop, and reproduce according to a program of instructions encoded in DNA, which they inherit from their parents. When cells divide, each daughter cell receives a complete copy of DNA and enough cytoplasmic machinery to start up its own operation. DNA contains the blueprints for making different proteins.
Problem solving is the process of finding solutions to difficult or complex issues ,w hile reasoning is the action of thinking about something in a logical, sensible way.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
2. Irish mathematician
who contributed to the
development of optics,
dynamics, and algebra
—in particular,
discovering thealgebra
of quaternions. His
work proved significant
for thedevelopment of
quantum mechanics.
William Rowan Hamilton
4. Hamiltonian Path and Circuit
A Hamiltonian path isapath that visits
each vertex of thegraph exactly once.
A Hamiltonian circuit isapath that uses
each vertex of agraph exactly onceand
returnsto thestarting vertex. A graph that
containsaHamiltonian circuit iscalled
Hamiltonian.
5. In Euler circuits, welooked at closed pathsthat use
every edgeexactly once, possibly visiting avertex
morethan once.
In Hamiltonian circuits, welook at pathsthat visit
each vertex exactly once, possibly not passing
through someof theedges.
But unliketheEuler circuit, wheretheEulerian
Graph Theorem isused to determinewhether it
containsan Euler circuit or not, wedo not havea
straightforward criterion to determinewhether or not
aHamiltonian circuit existsin agraph.
7. Determine whether the graph below is Hamiltonian
or not. If it is, find a Hamiltonian circuit. If it is not,
explain why?
Answer: A – B – C – E – D – F – G – A.
9. Dirac’s Theorem
Consider aconnected graph with at least threeverticesand no
multipleedges. Let n bethenumber of verticesin thegraph. If
every vertex hasdegreeof at least n/2, then thegraph must be
Hamiltonian.
10. Application of Hamiltonian Circuit
Thegraph below shows
theavailableflightsof a
popular
airline. An edgebetween
two verticesindicatesthat
there
isadirect flight between
thetwo cities. Determine
whether
thegraph isHamiltonian.
If it is, find aHamiltonian
circuit.
11. Solution
Thereareten verticesin thegraph, and n/2 =5 . Now,
vertex Manilahas9 edges, Tokyo 5, Seoul 5, Taipei 6,
Hongkong 7, Macau 9, Ho Chi Minh 5, KualaLumpur 5,
and Singapore5. Using Dirac’stheorem, thegraph is
Hamiltonian.
Thismeansthat thegraph containsacircuit that visitseach
vertex and return to itsstarting point without visiting a
vertex morethan once.
By trial and error, oneHamiltonian circuit isManila–
Tokyo – Seoul – Taipei – Hongkong – Macau – Bangkok –
Ho Chi Minh – KualaLumpur – Singapore– Manila.
14. Weighted Graph
A weighted graph isagraph in which each
edgeisassociated with avalue, called a
weight.
15. Travelling Salesman Problem
The travelling salesman problem (TSP) asksthe
following question: "Given alist of citiesand the
distancesbetween each pair of cities, what isthe
shortest possibleroutethat visitseach city exactly
onceand returnsto theorigin city?“
Thetravelling salesman problem consistsof a
salesman and aset of cities. Thesalesman hasto
visit each oneof thecitiesstarting from acertain
one(e.g. thehometown) and returning to thesame
city. Thechallengeof theproblem isthat the
travelling salesman wantsto minimizethetotal
length of thetrip.
16. Example: Travelling Salesman Problem
Thetablebelow listsdown thedistances(miles) between
thecitieshaving direct routesaswell asthecorresponding
distancesbetween them.
Draw agraph therepresentsthisinformation and find two
different routesthat visit each of theplacesand return to its
starting point without visiting any city twice.
19. TheGreedy Algorithm
A method of finding aHamiltonian circuit in acomplete
weighted graph isgiven by thefollowing greedy algorithm.
1.Chooseavertex to start at, then travel along theconnected
edgethat hasthesmallest weight.
2.After arriving at thenext vertex, travel along theedgeof
smallest weight that connectsto avertex not yet visited.
Continuethisprocessuntil you havevisited all vertices.
3.Return to thestarting vertex.
Take Note:
Thegreedy algorithm attemptsto giveacircuit of
minimal total weight, although it doesnot always
succeed.
20. Example
Aaron, Belle, Carol, Donna, Eric, and Fearebest of
friends. Thefigurebelow showsthedistances(km)
from afriend’splaceto another. If Aaron wantsto
visit each of hisfriends’ housesexactly once, what is
theshortest routethat hemust take?
22. TheEdge-Picking Algorithm
Another method of finding aHamiltonian circuit in
acompleteweighted graph isgiven by the
following edge-picking algorithm.
1.Mark theedgeof smallest weight in thegraph.
2.Mark theedgeof thenext smallest weight in the
graph, aslong asit doesnot completeacircuit and
doesnot add athird marked edgeto asinglevertex.
3.Continuetheprocessuntil you can no longer mark
any edges. Then mark thefinal edgethat completes
theHamiltonian circuit.
23. TheEdge-Picking Algorithm
Aaron, Belle, Carol, Donna, Eric, and Fearebest of friends.
Thefigurebelow showsthedistances(km) from afriend’s
placeto another. If Aaron wantsto visit each of hisfriends’
housesexactly once, what istheshortest routethat hemust
take?
24. Solution
First wemark thelinesegment from Aaron’shouseto Belle’s
house, of weight 1.
Next wemark thesegment from Belle’sto Carol’shouse, of
weight 2, followed by Carol’sto Donna’shouse, of weight 3,
followed by Eric’sto Fe’shouse, of weight 6.
Takenotethat wecannot mark thesegment from Eric’shouseto
Aaron’sbecauseit can completeacircuit. Also, wecannot mark
thesegment from Carol’sto Fe’shousebecauseit can makethe
third marked edgeon avertex.
Finally to completethecircuit, wemark thelinesegment from
Fe’shouseback to Aaron’s.
Thefinal Hamiltonian circuit, of total weight
1+2+3+6+9+12=33, isAaron’shouse– Belle’shouse– Carol’s
house– Donna’shouse– Eric’shouse– Fe’shouseand back to
Aaron’s.
26. Application
Thetableshowsthelengthsof cablesneeded to
connect computersto createanetwork. Find the
minimum length of cablematerial needed using the
edge-picking algorithm.
A B C D E F
A -- 10 22 9 15 8
B 10 -- 12 14 16 5
C 22 12 -- 14 9 15
D 9 14 14 -- 7 16
E 15 16 9 7 -- 13
F 8 5 16 15 13 --
27. Planar Graphs
A planar graph isagraph that can bedrawn so
that no edgesintersect each other (except at
vertices).
29. Subgraphs
A part of agraph G iscalled asubgraph of G.
Subgraph Theorem
“If agraph G hasasubgraph that isnot planar, theG isalso not
planar. In particular, if containstheUtilitiesGraph or K5 asa
subgraph, G isnot planar.”
Nonplanar Graph Theorem
A graph isnonplanar if and only if it hastheUtilitiesGraph or
K5 asasubgraph, or it hasasubgraph that can becontracted to
theUtilitiesGraph or K5.
31. Euler’sFormula
In aconnected planar graph drawn with no
intersecting edges, let v bethenumber of vertices, e
thenumber of edges, and f thenumber of faces.
Then v + f = e+ 2.
32. Graph Coloring
If themap isdivided into regionsin somemanner, what is
theminimum number of colorsrequired if theneighboring
regionsareto becolored differently?
Thereisaconnection between map coloring and graph
theory. Mapscan bemodeled by graphsusing the
countriesastheverticesand two vertices(countries) are
adjacent if they shareacommon boundary.
In graph coloring, each vertex of agraph will beassigned
onecolor in such away that no two adjacent verticeshave
thesamecolor. Theinteresting ideahereisto determine
theminimum number of (distinct) colorsto beused so that
wecan color each vertex of agraph with no two adjacent
verticeshavethesamecolor
33. Four-Color Theorem
Theminimum number of colorsneeded to color a
graph so that no edgeconnectsverticesof thesame
color iscalled thechromatic number.
Four-Color Theorem
Thechromatic number of aplanar graph isutmost 4.
34. 2-ColorableGraph Theorem
A graph is2-colorableif and only if it hasno circuits
that consist of an odd number of vertices
Determinewhether thegraph is2-colorable
35. Scheduling Problem
Six collegeaccreditation committeesneed to hold
meetingson thesameday, but someteachers
belong to morethan onecommittee. In order to
avoid membersmissing meetings, themeetings
need to bescheduled during different timeslots.
An “X” in thetableindicatesthat thetwo
corresponding committeesshareat least one
member. Usegraph coloring to determinethe
minimum number of timeslotsnecessary to
ensurethat all faculty memberscan attend all
meetings.
37. Solution
First wedraw agraph representing thesix committeesusing six
verticesor nodesin any configuration. An edgeconnectstwo
committeesthat shareat least onemember.
Then assign each vertex of thegraph with onecolor in such a
way that no two adjacent verticeshavethesamecolor.
38. Conclusion
Obviously, thegraph isnot 2-colorablebecausewe
can find circuitsof odd length but thegraph is3-
colorable. Hence, theminimum number of timeslots
necessary to ensurethat all faculty memberscan
attend all meetingsis3.
Scheduleof Meetings
First timeslot: Faculty Instruction, Student
Welfare
Second timeslot: Faculty Development,
Outreach
Program
Third timeslot: Library Facility, Physical Facility