network theory ( Graph theory components )

The network graph theory/network topology is a graphical representation of electric circuits. The main concept of graph theory is to make an easy representation of circuits which are so much complex. Using network graph theory we can find KVL  and KCL equations. Some important terms of graph theory:- 1.Graph:- The graphical representation of an electrical network circuit in terms of node and branches is known as the Graph of a network. 2. Node: A node is a junction point or inter-connection point of two or more network elements. 3.Branch or Edge: An element is a line segment representing one network element or a combination of network elements connected between two nodes. 4. Degree of node:-It is an integer number that represents incoming branches to one node. To analyze network circuits, first, learn types of a graph. By knowing the type of graph it is easy to find out many more elements of the graph, which we find by solving a lot of equations. Basic types of graphs:- Connected Graph: When there exits at least one path between every pair of nodes, then the graph is called a connected graph.connected graph

unconnected graph

Unconnected Graph

If there exists at least one node in the graph that remains unconnected by even a single branch, then it is called an unconnected graph. So, there will be one or more isolated nodes in an unconnected graph. Undirected Graph:- If the branches of a graph are not represented with arrows, then that graph is called as an undirected graph. Since there are no directions of current flow, this graph is also called as an unoriented graph. The above-given graphs are unoriented /undirected graphs.

current directions are given

Complete graph :-

• Every node pair has one line segment . There should be not more or less than one line segment between two nodes. number of trees= n(n-1)/2
• where n is number of nodes.
• Sub Graph:-sub graph is said to be sub graph of A graph, if every node of graph A is present in that sub graph.

Tie-set:

 Tie-set is a set of branches which forms a closed path or loop.

A basic tie-set or a fundamental tie-set is a tie-set having one and only one link branch, the other elements being tree branches

Cut-set:

 A fundamental cut-set of a graph w.r.t a tree is a cut-set formed by one and onlyone twig and a set of links, which must be cut to divide the network graph into two parts.The conventional direction for the fundamental cut-set is taken to be the same as the direction of the tree branch defines the particular cut-set.

Tree:-

tree is a subgraph having all nodes and no closed loop.

Twigs:-

The branches of the tree are known as twigs.

Co-Tree:-

the subgraph that is not present in the tree is known as co tree.

Links:-

the branches of the co tree are known as links.

Example Let us consider the following electric circuit.

In the above circuit, there are four principal nodes and those are labeled with 1, 2, 3, and 4. There are seven branches in the above circuit, among which one branch contains a 20 V voltage source, another branch contains a 4 A current source and the remaining five branches contain resistors having resistances of 30 Ω, 5 Ω, 10 Ω, 10 Ω and 20 Ω respectively. An equivalent graph corresponding to the above electric circuit is shown in the following figure.

There are four nodes and those are labeled with 1, 2, 3 & 4 respectively. These are same as that of principal nodes in the electric circuit. There are six branches in the above graph and those are labeled with a, b, c, d, e & f respectively.

In this case, we got one branchless in the graph because the 4 A current source is made as open circuit while converting the electric circuit into its equivalent graph.

From this example, we can conclude the following points − The number of nodes present in a graph will be equal to the number of principal nodes present in an electric circuit.

• The number of branches present in a graph will be less than or equal to the number of branches present in an electric circuit.

• 

  Add caption

  This is an example of connected and complete graph.

The solid line segments (twigs) represent a tree (containing  all nodes of the original graph and no closed loop)

the dashed line segments (links) represent the co tree ( subgraph exclude from the tree or can say not present in a tree).

Conclusion

Original graph=tree+ cotree

number of total branches = twigs+links

twigs=n-1

n- number of nodes

b=n-1+l

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Six Mysterious cases in India

1. Burari Mass Suicide case

The Chundawat family (also known as Bhatia family by neighbours) had been living in the double-storey house in Burari's Sant Nagar neighbourhood for around twenty years, after moving from their native town in Rajasthan. The family ran a grocery shop and plywood business in the area. The family included Narayani Devi - 77, her two sons Bhavnesh - 50, and Lalit - 45, daughters in law Savita - 48 and Teena - 42, Narayani's daughter Pratibha Bhatia (née Chundawat) - 57, and grandchildren Priyanka - 33 (daughter of Pratibha), Nitu - 25 (elder daughter of Bhavnesh), Monu - 23 (younger daughter of Bhavnesh), Dhruv - 15 (youngest son of Bhavnesh) and Shivam - 15 (only child of Lalit).
 On 1st july , 2018 Ten family members were found hanged, while the oldest family member, the grandmother, was strangled. 
Police found eleven diaries in the house, all of them maintained for the period of eleven years. Joint Commissioner of Police (Crime) Alok Kumar stated: "We have found handwritten notes detailing how hands and legs are to be tied and are quite similar to the manner in which the bodies of 10 persons were found. They are exhaustive notes and we are studying them."^[10] Details/directions provided in the diaries match how the bodies were found with their faces covered, mouths taped, and cotton balls in ears. The diary also mentions: "everyone will tie their own hands and when the kriya (ritual) is done then everyone will help each other untie their hands." Signifying the family was not expecting to die.
Psychologists have commented that these deaths are a result of ‘shared psychotic disorder’ in which members of a group blindly trust one among them and follow instructions without questioning. Psychologists feel that Lalit suffered from a ‘delusional disorder’.For more information visit link1 and link2.

2. Double murder case

Aarushi-Hemraj murder case
Location	Noida, India
Date	15–16 May 2008
Attack type	Murders
Weapons	Undetermined
Deaths	2
Victims	Aarushi Talwar Hemraj Banjade
Perpetrators	Undetermined

The Noida double murder case refers to the unsolved murders of 13-year-old girl Aarushi Talwar and 45-year-old Hemraj Banjade, a male live-in domestic worker employed by her family. The two were killed on the night of 15–16 May 2008 at Aarushi's home in Noida, India. The case aroused public interest as a whodunit story, and received heavy media coverage. The sensational media coverage, which included salacious allegations against Aarushi and the suspects, was criticized by many as a trial by media.The CBI said that dentist Rajesh Talwar had found his daughter and his domestic help in "objectionable position" and killed them using his golf stick and a sharp-edged weapon.Nobody knows the real truth.

For more information wikipedia.

3.Sheena Bora

Sheena Bora, an executive working for Mumbai Metro One based in Mumbai,went missing on 24 April 2012. In August 2015 Mumbai Police arrested her mother Indrani Mukerjea, her stepfather Sanjeev Khanna, and her mother's driver, Shyamvar Pinturam Rai, for allegedly abducting and killing her and subsequently burning her corpse. Khanna and Rai confessed to the crime, and Mukerjea has said that Sheena Bora is living in the United States.

On 24 April 2012, Sheena took a leave of absence and "sent in her written resignation".^[18] On the same day, Rahul Mukerjea (Sheena's step-brother who she was dating) received a breakup SMS from Sheena's phone.^[19] Her mother, Indrani, said that Sheena had gone to the United States for higher studies and hence a missing First Information Report was never filed.^[6][20][21] Sheena was never seen after 24 April 2012.

On Rahul's insistence, police visited Indrani's Worli residence where they were informed by the staff that Indrani was out of India. Upon her return, Indrani visited the Worli police station and informed the police that Rahul was trying to stalk Sheena and that was why Sheena had moved to the United States without informing him.^[22]

On 23 May 2012, local police in Pen tehsil, Raigad, found a body after villagers complained of a foul odor. No identification was made and the remains were sent to JJ Hospital in Mumbai. No link was made with the Sheena Bora case.

4. Sundana Pushkar

Sunanda Pushkar

Sunanda Pushkar Nath Dass (27 June 1962 – 17 January 2014) was an Indian businesswoman and the wife of Indian former diplomat and politician Shashi Tharoor. She was a sales director in the Dubai-based TECOM Investments, and a co-owner of the India-based Rendezvous Sports World (RSW), a cricketfranchise in the Indian Premier League.

On 17 January 2014, a day after the Twitter controversy, Sunanda was found dead in room number 345 of the Leela Palace hotel in Chanakyapuri, New Delhi, where the couple had moved to, as their house was being renovated and painted. Shashi Tharoor discovered her body, when she did not wake up from her sleep in the evening. He informed the Delhi Police, who recovered the body from the hotel and sent it for postmortem. According to initial reports, Sunanda was suspected to have committed suicide. Later reports stated that the cause of death was unnatural; the doctors at the All India Institute of Medical Sciences gave a preliminary autopsy report that revealed injury marks on her body. 

Doctors at KIMS Hospital Trivandrum, who had examined her a few days earlier said that Sunanda did not have serious health problems. However, Sunanda had hinted about her death, hours before her body was recovered from the hotel.

On 1 July 2014, controversy over her death deepened when AIIMS doctor Sudhir Gupta claimed that he was pressured to give a false report in the case.^[31]

On 10 October 2014, the medical team probing her death concluded that she died of poisoning.^[32] On 6 January 2015, Delhi Police reported that Sunanda was murdered and filed FIR in the regard. ^[3]Pushkar's domestic help alleged that Pushkar often fought with Shashi Tharoor and days before her murder, she had threatened Tharoor that she would "disclose everything" and as such, he was "finished".

Subramanian Swamy, a BJP politician, tried to fight this case against Shashi Tharoor in January 2015. On 20 May 2015, a trial court allowed Delhi Police to conduct lie detector test on 3 suspects related to her death case.This case is subjudice.

5.Arun Tikku and Karan Kakkad 

The sensational murders of two Delhi residents at the hands of a money-crazy criminal and his female accomplice are no short of a thriller. Former gangster and prime accused Vijay Palande used his model wife Simran Sood as a 'honeytrap' to commit the crimes for usurping properties of the victims, police said. In April 2012, Palande along with his aides Dhananjay Shinde and Manoj Gajkosh, murdered Delhi-based businessman Arunkumar Tikku, 67. 

Palande had befriended the victim's actor son Anuj Tikku with a motive to acquire his plush apartment in upscale in Mumbai’s upscale Lokhandwala Complex, police had said. The trio were nabbed by the Mumbai crime branch. Simran Sood was also arrested in connection with the abduction and murder of Delhi-based aspiring producer Karankumar Kakkad, 28. Palande confessed that he killed Kakkad because he suspected the latter of being an underworld mole, who would eventually kill him. 

Simran introduced Palande as her "brother" to Anuj and Karankumar, police said. 

Palande was earlier convicted of a double murder in 1998, and again in 2002, he jumped parole in 2003, went to Bangkok for a cosmetic surgery to change his features, and returned to Mumbai in 2005. 

6. Neeraj Grover

Neeraj Grover was a television executive working for Synergy Adlabs, a Mumbai based production house. He was found dead in May 2008, a crime for which actress Maria Susairaj and her boyfriend Lieutenant Emile Jerome Mathew were arrested. Mathew was eventually found guilty of culpable homicide not amounting to murder and for destroying evidence. Susairaj was acquitted of the killing itself, but found guilty of destroying evidence.The killing and subsequent trial received significant press coverage in India.

Grover visited Susairaj's Malad apartment Dhiraj Solitaire at 22:00 on 6 May to help her settle in. At the same time, Mathew called her up and got upset to hear a male voice in background. When Susairaj told Mathew that Grover had come over to help her, Jerome asked Maria not to allow Neeraj to stay overnight. Mathew at that time was living in Kochi. However, he took a flight and came to Mumbai the next day without telling Susairaj. He reached her apartment at 07:30. Susairaj opened the door and Mathew went straight into the bedroom to find Grover in the room. As per media and police reports, the two men had argument and then engaged in a physical fight. Mathew grabbed a kitchen knife and stabbed Grover, who died on the spot.

On 25 May, the crime branch recovered a nine-inch knife from Susairaj's apartment complex. Police believed that the knife was used to kill Grover.

Around 11:00 am the same day, Susairaj went to a nearby mall and purchased bags, air freshener, curtains, bed sheets & knife. She later told police that she did this on Jerome's instructions.

They both then took the dead body to the kitchen and chopped it into several pieces,stuffing them into the bags Susairaj had earlier bought. While many media outlets reported that Grover's body had been chopped into 300 pieces, the judge in the case would later state that this was "...far from the facts on record", and that such a description had caused "confusion" and had "outraged the general public".^[8]

Susairaj borrowed a car from her friend and dumped the bags in the vehicle's boot whilst the building security guard watched them load the car. At 16:30 the same day, they drove towards Manor after buying two bottles of Petrol at Bhayander. They soaked the bags in petrol and set it on fire in a jungle near Amgaon village in Manor. After burning the dead body, they returned to Mumbai the same night at 21:30.

for more cases visit rediff.