On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block...
Transcript of On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block...
![Page 1: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/1.jpg)
On the Proof of the Friendship Theorem
Hussein Awala
Temple University
Philadelphia, PA
Hussein Awala (Temple University) 1 / 18
![Page 2: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/2.jpg)
Preliminaries
A graph in the plane is an ordered pair G = (V ,E) where:V = V (G) is called the vertex set, and is the set of vertices ornodes in the graph G.E = E(G) is called the edge set, and is the set of edges orconnections between the vertices.
For u, v ∈ V (G).The undirected edges are represented by {u, v}.The directed edges are represented by (u, v).
Finally let G = (V ,E) be an undirected graph.Two nodes that are connected by an edge are adjacent.The neighbourhood of a node v , denoted N(v), is the set of all thenodes that are adjacent to it, and |N(v)| is it’s cardinality.
Hussein Awala (Temple University) 2 / 18
![Page 3: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/3.jpg)
Preliminaries
A graph in the plane is an ordered pair G = (V ,E) where:V = V (G) is called the vertex set, and is the set of vertices ornodes in the graph G.E = E(G) is called the edge set, and is the set of edges orconnections between the vertices.
For u, v ∈ V (G).The undirected edges are represented by {u, v}.The directed edges are represented by (u, v).
Finally let G = (V ,E) be an undirected graph.Two nodes that are connected by an edge are adjacent.The neighbourhood of a node v , denoted N(v), is the set of all thenodes that are adjacent to it, and |N(v)| is it’s cardinality.
Hussein Awala (Temple University) 2 / 18
![Page 4: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/4.jpg)
Preliminaries
A graph in the plane is an ordered pair G = (V ,E) where:V = V (G) is called the vertex set, and is the set of vertices ornodes in the graph G.E = E(G) is called the edge set, and is the set of edges orconnections between the vertices.
For u, v ∈ V (G).The undirected edges are represented by {u, v}.The directed edges are represented by (u, v).
Finally let G = (V ,E) be an undirected graph.Two nodes that are connected by an edge are adjacent.The neighbourhood of a node v , denoted N(v), is the set of all thenodes that are adjacent to it, and |N(v)| is it’s cardinality.
Hussein Awala (Temple University) 2 / 18
![Page 5: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/5.jpg)
Main Theorem
TheoremSuppose that G = (V ,E) is a finite graph in which
∀ u, v ∈ V (G) |N(u) ∩ N(v)| = 1.
Then there exists a vertex z ∈ V (G) that is adjacent to all othervertices.
Informally, one can say
TheoremSuppose a group of at least three people has the property that everypair of people has exactly one friend in common, then there must beone person who is a friend to all the others.
Hussein Awala (Temple University) 3 / 18
![Page 6: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/6.jpg)
Main Theorem
TheoremSuppose that G = (V ,E) is a finite graph in which
∀ u, v ∈ V (G) |N(u) ∩ N(v)| = 1.
Then there exists a vertex z ∈ V (G) that is adjacent to all othervertices.
Informally, one can say
TheoremSuppose a group of at least three people has the property that everypair of people has exactly one friend in common, then there must beone person who is a friend to all the others.
Hussein Awala (Temple University) 3 / 18
![Page 7: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/7.jpg)
History and Application
It is not yet known who formulated this problem, or who gave it thehuman touch that it has.
In 1966, Paul Erdos, Alfred Renyi and Vera Sos Published the firstproof of this theorem.This Theorem has a lot of applications, it is mainly used in thefields of
Block DesignsCoding TheorySet Theory
The Friendship Theorem is listed among Paul and Jack Abad’s"100 Greatest Theorems".In 2001, Aigner and Ziegler mentioned the Friendship Theorem inthere book titled "Proofs from the Book "as one of the greatesttheorems of Erdos of all time.
Hussein Awala (Temple University) 4 / 18
![Page 8: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/8.jpg)
History and Application
It is not yet known who formulated this problem, or who gave it thehuman touch that it has.In 1966, Paul Erdos, Alfred Renyi and Vera Sos Published the firstproof of this theorem.
This Theorem has a lot of applications, it is mainly used in thefields of
Block DesignsCoding TheorySet Theory
The Friendship Theorem is listed among Paul and Jack Abad’s"100 Greatest Theorems".In 2001, Aigner and Ziegler mentioned the Friendship Theorem inthere book titled "Proofs from the Book "as one of the greatesttheorems of Erdos of all time.
Hussein Awala (Temple University) 4 / 18
![Page 9: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/9.jpg)
History and Application
It is not yet known who formulated this problem, or who gave it thehuman touch that it has.In 1966, Paul Erdos, Alfred Renyi and Vera Sos Published the firstproof of this theorem.This Theorem has a lot of applications, it is mainly used in thefields of
Block DesignsCoding TheorySet Theory
The Friendship Theorem is listed among Paul and Jack Abad’s"100 Greatest Theorems".In 2001, Aigner and Ziegler mentioned the Friendship Theorem inthere book titled "Proofs from the Book "as one of the greatesttheorems of Erdos of all time.
Hussein Awala (Temple University) 4 / 18
![Page 10: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/10.jpg)
History and Application
It is not yet known who formulated this problem, or who gave it thehuman touch that it has.In 1966, Paul Erdos, Alfred Renyi and Vera Sos Published the firstproof of this theorem.This Theorem has a lot of applications, it is mainly used in thefields of
Block DesignsCoding TheorySet Theory
The Friendship Theorem is listed among Paul and Jack Abad’s"100 Greatest Theorems".
In 2001, Aigner and Ziegler mentioned the Friendship Theorem inthere book titled "Proofs from the Book "as one of the greatesttheorems of Erdos of all time.
Hussein Awala (Temple University) 4 / 18
![Page 11: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/11.jpg)
History and Application
It is not yet known who formulated this problem, or who gave it thehuman touch that it has.In 1966, Paul Erdos, Alfred Renyi and Vera Sos Published the firstproof of this theorem.This Theorem has a lot of applications, it is mainly used in thefields of
Block DesignsCoding TheorySet Theory
The Friendship Theorem is listed among Paul and Jack Abad’s"100 Greatest Theorems".In 2001, Aigner and Ziegler mentioned the Friendship Theorem inthere book titled "Proofs from the Book "as one of the greatesttheorems of Erdos of all time.
Hussein Awala (Temple University) 4 / 18
![Page 12: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/12.jpg)
More History
Several different approaches have been used to prove this theorem:In 1971, Herbert Wilf provided a geometric proof.
In 1972, Judith Longyear and Torrence Parsons gave a proof bycounting neighbors, walks and cycles in regular graphs.In 1983, Hammersley provided a proof using numericaltechniques. He later extended this theorem to the so called "LoveProblem".In 2001, Douglus West gave a proof similar to the one byLongyear and Parsons, counting common neighbors and cycles.Finally, in 2002 Craig Huneke gave two proofs, one being morecombinatorial and one that combines combinatorics and linearalgebra.
Hussein Awala (Temple University) 5 / 18
![Page 13: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/13.jpg)
More History
Several different approaches have been used to prove this theorem:In 1971, Herbert Wilf provided a geometric proof.In 1972, Judith Longyear and Torrence Parsons gave a proof bycounting neighbors, walks and cycles in regular graphs.
In 1983, Hammersley provided a proof using numericaltechniques. He later extended this theorem to the so called "LoveProblem".In 2001, Douglus West gave a proof similar to the one byLongyear and Parsons, counting common neighbors and cycles.Finally, in 2002 Craig Huneke gave two proofs, one being morecombinatorial and one that combines combinatorics and linearalgebra.
Hussein Awala (Temple University) 5 / 18
![Page 14: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/14.jpg)
More History
Several different approaches have been used to prove this theorem:In 1971, Herbert Wilf provided a geometric proof.In 1972, Judith Longyear and Torrence Parsons gave a proof bycounting neighbors, walks and cycles in regular graphs.In 1983, Hammersley provided a proof using numericaltechniques. He later extended this theorem to the so called "LoveProblem".
In 2001, Douglus West gave a proof similar to the one byLongyear and Parsons, counting common neighbors and cycles.Finally, in 2002 Craig Huneke gave two proofs, one being morecombinatorial and one that combines combinatorics and linearalgebra.
Hussein Awala (Temple University) 5 / 18
![Page 15: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/15.jpg)
More History
Several different approaches have been used to prove this theorem:In 1971, Herbert Wilf provided a geometric proof.In 1972, Judith Longyear and Torrence Parsons gave a proof bycounting neighbors, walks and cycles in regular graphs.In 1983, Hammersley provided a proof using numericaltechniques. He later extended this theorem to the so called "LoveProblem".In 2001, Douglus West gave a proof similar to the one byLongyear and Parsons, counting common neighbors and cycles.
Finally, in 2002 Craig Huneke gave two proofs, one being morecombinatorial and one that combines combinatorics and linearalgebra.
Hussein Awala (Temple University) 5 / 18
![Page 16: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/16.jpg)
More History
Several different approaches have been used to prove this theorem:In 1971, Herbert Wilf provided a geometric proof.In 1972, Judith Longyear and Torrence Parsons gave a proof bycounting neighbors, walks and cycles in regular graphs.In 1983, Hammersley provided a proof using numericaltechniques. He later extended this theorem to the so called "LoveProblem".In 2001, Douglus West gave a proof similar to the one byLongyear and Parsons, counting common neighbors and cycles.Finally, in 2002 Craig Huneke gave two proofs, one being morecombinatorial and one that combines combinatorics and linearalgebra.
Hussein Awala (Temple University) 5 / 18
![Page 17: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/17.jpg)
Proof
TheoremSuppose that G = (V ,E) is a finite graph in which
∀ u, v ∈ V (G) |N(u) ∩ N(v)| = 1. (?)
Then there exists a vertex z ∈ V (G) that is adjacent to all othervertices.
Proof :
We will proceed by contradiction. Suppose that G is a finite graphsatisfying (?). furthermore suppose that G has no vertex adjacent to allother vertices. Our proof today will be divided into two parts.
Hussein Awala (Temple University) 6 / 18
![Page 18: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/18.jpg)
Proof
TheoremSuppose that G = (V ,E) is a finite graph in which
∀ u, v ∈ V (G) |N(u) ∩ N(v)| = 1. (?)
Then there exists a vertex z ∈ V (G) that is adjacent to all othervertices.
Proof :
We will proceed by contradiction. Suppose that G is a finite graphsatisfying (?). furthermore suppose that G has no vertex adjacent to allother vertices. Our proof today will be divided into two parts.
Hussein Awala (Temple University) 6 / 18
![Page 19: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/19.jpg)
Proof
TheoremSuppose that G = (V ,E) is a finite graph in which
∀ u, v ∈ V (G) |N(u) ∩ N(v)| = 1. (?)
Then there exists a vertex z ∈ V (G) that is adjacent to all othervertices.
Proof :
We will proceed by contradiction. Suppose that G is a finite graphsatisfying (?). furthermore suppose that G has no vertex adjacent to allother vertices.
Our proof today will be divided into two parts.
Hussein Awala (Temple University) 6 / 18
![Page 20: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/20.jpg)
Proof
TheoremSuppose that G = (V ,E) is a finite graph in which
∀ u, v ∈ V (G) |N(u) ∩ N(v)| = 1. (?)
Then there exists a vertex z ∈ V (G) that is adjacent to all othervertices.
Proof :
We will proceed by contradiction. Suppose that G is a finite graphsatisfying (?). furthermore suppose that G has no vertex adjacent to allother vertices. Our proof today will be divided into two parts.
Hussein Awala (Temple University) 6 / 18
![Page 21: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/21.jpg)
Proof-Part one
We claim that G is a regular graph. Here a graph T =(V (T ),E(T )
)is
called regular whenever
∀ u, v ∈ V (T ) |N(u)| = |N(v)|.
Turning our attention back to proving the Claim. Notice that thecondition (?) ensures that G has no cycles of length 4, indeed considerthe 4-cycle.
where the RED, and the GREEN vertices, have to vertices in commonviolating (?).
Hussein Awala (Temple University) 7 / 18
![Page 22: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/22.jpg)
Proof-Part one
We claim that G is a regular graph. Here a graph T =(V (T ),E(T )
)is
called regular whenever
∀ u, v ∈ V (T ) |N(u)| = |N(v)|.
Turning our attention back to proving the Claim. Notice that thecondition (?) ensures that G has no cycles of length 4, indeed considerthe 4-cycle.
where the RED, and the GREEN vertices, have to vertices in commonviolating (?).
Hussein Awala (Temple University) 7 / 18
![Page 23: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/23.jpg)
Proof-Part one
We claim that G is a regular graph. Here a graph T =(V (T ),E(T )
)is
called regular whenever
∀ u, v ∈ V (T ) |N(u)| = |N(v)|.
Turning our attention back to proving the Claim. Notice that thecondition (?) ensures that G has no cycles of length 4, indeed considerthe 4-cycle.
where the RED, and the GREEN vertices, have to vertices in commonviolating (?).
Hussein Awala (Temple University) 7 / 18
![Page 24: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/24.jpg)
Proof-Part one
We claim that G is a regular graph. Here a graph T =(V (T ),E(T )
)is
called regular whenever
∀ u, v ∈ V (T ) |N(u)| = |N(v)|.
Turning our attention back to proving the Claim. Notice that thecondition (?) ensures that G has no cycles of length 4, indeed considerthe 4-cycle.
where the RED, and the GREEN vertices, have to vertices in commonviolating (?).Hussein Awala (Temple University) 7 / 18
![Page 25: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/25.jpg)
Continuation of Proof-Part one
Now let U and Y be two non-adjacent distinct vertices in G. Consider|N(U)| = k for some k ∈ N. Consider N(U) = {W1,W2, ...,Wk}
W1 W2 W3 .......
U
Wk
Y
Hussein Awala (Temple University) 8 / 18
![Page 26: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/26.jpg)
Continuation of Proof-Part one
U and Y must have exactly one vertex in common. Let us say that isW2. Similarly W2 and U must have exactly one vertex in common. Letus say that is W1.
W1W2
.......W3
U
Wk
Y
Hussein Awala (Temple University) 9 / 18
![Page 27: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/27.jpg)
Continuation of Proof-Part one
Turning our attention to Y , it must have exactly one vertex in commonwith each Wi . Call these common vertices Zi ′s for i ∈ {2, ..., k}.
W1W2
.......
.......Z2
Z3 Zk
.......W3
U
Wk
Y
Hussein Awala (Temple University) 10 / 18
![Page 28: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/28.jpg)
Continuation of Proof-Part one
Turning our attention to Y , it must have exactly one vertex in commonwith each Wi . Call these common vertices Zi ′s for i ∈ {2, ..., k}.
W1W2
.......
Z2Z3 Zk
.......W3
U
Wk
Y
The Zi’s that we just added have to be distinct other wise we get a4-cycle , which we proved can’t exist.Hussein Awala (Temple University) 11 / 18
![Page 29: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/29.jpg)
Continuation of Proof-Part one
Hence this is what our graph looks like now. Moving on, we count theedges of Y .
W1W2
.......
.......Z2
Z3 Zk
.......W3
U
Wk
Y
We get that |N(Y )| ≥ |N(U)|. Finally, by symmetry we conclude that|N(Y )| = |N(U)|. To finish the proof of the claim, Notice that asidefrom W2 any vertex is either non-adjacent to Y or non-adjacent to U.
Hussein Awala (Temple University) 12 / 18
![Page 30: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/30.jpg)
Continuation of Proof-Part one
Hence this is what our graph looks like now. Moving on, we count theedges of Y .
W1W2
.......
.......Z2
Z3 Zk
.......W3
U
Wk
Y
We get that |N(Y )| ≥ |N(U)|. Finally, by symmetry we conclude that|N(Y )| = |N(U)|. To finish the proof of the claim, Notice that asidefrom W2 any vertex is either non-adjacent to Y or non-adjacent to U.Hussein Awala (Temple University) 12 / 18
![Page 31: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/31.jpg)
Continuation of Proof-Part one
To finish the first part of the proof we will count the number of verticeswe have in total in G.
Starting from U we have k adjacent vertices Wi ′s
Each Wi also has k adjacent vertices, so we get k2.
But one of the vertices adjacent to each Wi is U. Hence the totalis k2 − k + 1. Since we counted U k-times.
Notice that each Wi is adjacent to exactly one other Wj other wise ifadjacent to more we run again into a 4-cycle.
W1 W2
W3
.......
U
Wk
Hussein Awala (Temple University) 13 / 18
![Page 32: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/32.jpg)
Continuation of Proof-Part one
To finish the first part of the proof we will count the number of verticeswe have in total in G.
Starting from U we have k adjacent vertices Wi ′s
Each Wi also has k adjacent vertices, so we get k2.
But one of the vertices adjacent to each Wi is U. Hence the totalis k2 − k + 1. Since we counted U k-times.
Notice that each Wi is adjacent to exactly one other Wj other wise ifadjacent to more we run again into a 4-cycle.
W1 W2
W3
.......
U
Wk
Hussein Awala (Temple University) 13 / 18
![Page 33: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/33.jpg)
Continuation of Proof-Part one
To finish the first part of the proof we will count the number of verticeswe have in total in G.
Starting from U we have k adjacent vertices Wi ′s
Each Wi also has k adjacent vertices, so we get k2.
But one of the vertices adjacent to each Wi is U. Hence the totalis k2 − k + 1. Since we counted U k-times.
Notice that each Wi is adjacent to exactly one other Wj other wise ifadjacent to more we run again into a 4-cycle.
W1 W2
W3
.......
U
Wk
Hussein Awala (Temple University) 13 / 18
![Page 34: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/34.jpg)
Continuation of Proof-Part one
To finish the first part of the proof we will count the number of verticeswe have in total in G.
Starting from U we have k adjacent vertices Wi ′s
Each Wi also has k adjacent vertices, so we get k2.
But one of the vertices adjacent to each Wi is U. Hence the totalis k2 − k + 1. Since we counted U k-times.
Notice that each Wi is adjacent to exactly one other Wj other wise ifadjacent to more we run again into a 4-cycle.
W1 W2
W3
.......
U
Wk
Hussein Awala (Temple University) 13 / 18
![Page 35: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/35.jpg)
Continuation of Proof-Part one
To finish the first part of the proof we will count the number of verticeswe have in total in G.
Starting from U we have k adjacent vertices Wi ′s
Each Wi also has k adjacent vertices, so we get k2.
But one of the vertices adjacent to each Wi is U. Hence the totalis k2 − k + 1. Since we counted U k-times.
Notice that each Wi is adjacent to exactly one other Wj other wise ifadjacent to more we run again into a 4-cycle.
W1 W2
W3
.......
U
Wk
Hussein Awala (Temple University) 13 / 18
![Page 36: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/36.jpg)
Proof-Part two
Moving on, we will present the second part of the proof which is abeautiful algebraic argument to reach contradiction. Start byintroducing the adjacency matrix A, which for G =
(V (G),E(G)
)is
defined as follows:
aij :=
{1 if {vi , vj} ∈ E(G)
0 otherwise
Here, V (G) = {v1, ...., vn}, where n = k2 − k + 1. With this in hand,and using the condition (?) we have
A2 =
k 1 · · · 11 k 1...
. . ....
1 · · · 1 k
= (k − 1)In + Jn (??)
Here In is the identity Matrix, and Jn is a matrix of 1′s.
Hussein Awala (Temple University) 14 / 18
![Page 37: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/37.jpg)
Proof-Part two
Moving on, we will present the second part of the proof which is abeautiful algebraic argument to reach contradiction. Start byintroducing the adjacency matrix A, which for G =
(V (G),E(G)
)is
defined as follows:
aij :=
{1 if {vi , vj} ∈ E(G)
0 otherwise
Here, V (G) = {v1, ...., vn}, where n = k2 − k + 1. With this in hand,and using the condition (?) we have
A2 =
k 1 · · · 11 k 1...
. . ....
1 · · · 1 k
= (k − 1)In + Jn (??)
Here In is the identity Matrix, and Jn is a matrix of 1′s.
Hussein Awala (Temple University) 14 / 18
![Page 38: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/38.jpg)
Proof-Part two
Moving on, we will present the second part of the proof which is abeautiful algebraic argument to reach contradiction. Start byintroducing the adjacency matrix A, which for G =
(V (G),E(G)
)is
defined as follows:
aij :=
{1 if {vi , vj} ∈ E(G)
0 otherwise
Here, V (G) = {v1, ...., vn}, where n = k2 − k + 1.
With this in hand,and using the condition (?) we have
A2 =
k 1 · · · 11 k 1...
. . ....
1 · · · 1 k
= (k − 1)In + Jn (??)
Here In is the identity Matrix, and Jn is a matrix of 1′s.
Hussein Awala (Temple University) 14 / 18
![Page 39: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/39.jpg)
Proof-Part two
Moving on, we will present the second part of the proof which is abeautiful algebraic argument to reach contradiction. Start byintroducing the adjacency matrix A, which for G =
(V (G),E(G)
)is
defined as follows:
aij :=
{1 if {vi , vj} ∈ E(G)
0 otherwise
Here, V (G) = {v1, ...., vn}, where n = k2 − k + 1. With this in hand,and using the condition (?) we have
A2 =
k 1 · · · 11 k 1...
. . ....
1 · · · 1 k
= (k − 1)In + Jn (??)
Here In is the identity Matrix, and Jn is a matrix of 1′s.Hussein Awala (Temple University) 14 / 18
![Page 40: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/40.jpg)
Continuation of Proof-Part two
Now Using (??), we deduce that the eigenvalues if A2 are
{k − 1 + n = K 2 of multiplicity 1
K − 1 of multiplicity n − 1
So A has the following eigenvaluesk of multiplicity 1√
K − 1 of multiplicity r
−√
K − 1 of multiplicity s
where r + s = n − 1.
Hussein Awala (Temple University) 15 / 18
![Page 41: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/41.jpg)
Continuation of Proof-Part two
Now Using (??), we deduce that the eigenvalues if A2 are{k − 1 + n = K 2 of multiplicity 1
K − 1 of multiplicity n − 1
So A has the following eigenvaluesk of multiplicity 1√
K − 1 of multiplicity r
−√
K − 1 of multiplicity s
where r + s = n − 1.
Hussein Awala (Temple University) 15 / 18
![Page 42: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/42.jpg)
Continuation of Proof-Part two
Now Using (??), we deduce that the eigenvalues if A2 are{k − 1 + n = K 2 of multiplicity 1
K − 1 of multiplicity n − 1
So A has the following eigenvaluesk of multiplicity 1√
K − 1 of multiplicity r
−√
K − 1 of multiplicity s
where r + s = n − 1.
Hussein Awala (Temple University) 15 / 18
![Page 43: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/43.jpg)
Continuation of Proof-Part two
Finally, we use the fact that the trace of A ( which is 0 )is equal to thesum of its eigenvalues , we have
K + r√
k − 1− s√
k − 1 = 0
This implies that r 6= s, and we can write
√k − 1 =
ks − r
Using the fact that, if the square root of a natural number is rationalthen it must be an integer, we deduce that ∃h ∈ N s.t.
h =√
k − 1 which implies h(s − r) = k = h2 + 1.
Then, h divides h2 + 1, and h2. This forces h = 1 and thusk = 2 =⇒ n = 3 , which satisfies the theorem. Hence acontradiction.
Hussein Awala (Temple University) 16 / 18
![Page 44: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/44.jpg)
Continuation of Proof-Part two
Finally, we use the fact that the trace of A ( which is 0 )is equal to thesum of its eigenvalues , we have
K + r√
k − 1− s√
k − 1 = 0
This implies that r 6= s, and we can write
√k − 1 =
ks − r
Using the fact that, if the square root of a natural number is rationalthen it must be an integer, we deduce that ∃h ∈ N s.t.
h =√
k − 1 which implies h(s − r) = k = h2 + 1.
Then, h divides h2 + 1, and h2. This forces h = 1 and thusk = 2 =⇒ n = 3 , which satisfies the theorem. Hence acontradiction.
Hussein Awala (Temple University) 16 / 18
![Page 45: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/45.jpg)
Continuation of Proof-Part two
Finally, we use the fact that the trace of A ( which is 0 )is equal to thesum of its eigenvalues , we have
K + r√
k − 1− s√
k − 1 = 0
This implies that r 6= s, and we can write
√k − 1 =
ks − r
Using the fact that, if the square root of a natural number is rationalthen it must be an integer, we deduce that ∃h ∈ N s.t.
h =√
k − 1 which implies h(s − r) = k = h2 + 1.
Then, h divides h2 + 1, and h2. This forces h = 1 and thusk = 2 =⇒ n = 3 , which satisfies the theorem. Hence acontradiction.
Hussein Awala (Temple University) 16 / 18
![Page 46: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/46.jpg)
Continuation of Proof-Part two
Finally, we use the fact that the trace of A ( which is 0 )is equal to thesum of its eigenvalues , we have
K + r√
k − 1− s√
k − 1 = 0
This implies that r 6= s, and we can write
√k − 1 =
ks − r
Using the fact that, if the square root of a natural number is rationalthen it must be an integer, we deduce that ∃h ∈ N s.t.
h =√
k − 1 which implies h(s − r) = k = h2 + 1.
Then, h divides h2 + 1, and h2. This forces h = 1 and thusk = 2 =⇒ n = 3 , which satisfies the theorem. Hence acontradiction.
Hussein Awala (Temple University) 16 / 18
![Page 47: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/47.jpg)
Continuation of Proof-Part two
Finally, we use the fact that the trace of A ( which is 0 )is equal to thesum of its eigenvalues , we have
K + r√
k − 1− s√
k − 1 = 0
This implies that r 6= s, and we can write
√k − 1 =
ks − r
Using the fact that, if the square root of a natural number is rationalthen it must be an integer, we deduce that ∃h ∈ N s.t.
h =√
k − 1 which implies h(s − r) = k = h2 + 1.
Then, h divides h2 + 1, and h2. This forces h = 1 and thusk = 2 =⇒ n = 3 , which satisfies the theorem. Hence acontradiction.Hussein Awala (Temple University) 16 / 18
![Page 48: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/48.jpg)
Kotzig’s Conjecture
Let us first rephrase our theorem, the condition (?) can be formulatedas :
Suppose G is a graph with the property that between any two verticesthere is exactly one path of length 2.
This lead to the consideration of the following problem:
Let ` ≥ 2. Is there a finite graphs with the property that
between any two vertices there is precisely one path of length `.
Kotzig conjectured in 1974 that there exists no such graphs. Heproved this conjecture for 3 ≤ ` ≤ 8.Kostochka proved in 1988 that the conjecture is true for ` ≤ 20.Xing and Hu proved the Kotzig’s conjecture in 1994 for ` ≥ 12.
Thus, the Kotzig’s conjecture is valid now as a theorem.
Hussein Awala (Temple University) 17 / 18
![Page 49: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/49.jpg)
Kotzig’s Conjecture
Let us first rephrase our theorem, the condition (?) can be formulatedas :
Suppose G is a graph with the property that between any two verticesthere is exactly one path of length 2.
This lead to the consideration of the following problem:
Let ` ≥ 2. Is there a finite graphs with the property that
between any two vertices there is precisely one path of length `.
Kotzig conjectured in 1974 that there exists no such graphs. Heproved this conjecture for 3 ≤ ` ≤ 8.Kostochka proved in 1988 that the conjecture is true for ` ≤ 20.Xing and Hu proved the Kotzig’s conjecture in 1994 for ` ≥ 12.
Thus, the Kotzig’s conjecture is valid now as a theorem.
Hussein Awala (Temple University) 17 / 18
![Page 50: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/50.jpg)
Kotzig’s Conjecture
Let us first rephrase our theorem, the condition (?) can be formulatedas :
Suppose G is a graph with the property that between any two verticesthere is exactly one path of length 2.
This lead to the consideration of the following problem:
Let ` ≥ 2. Is there a finite graphs with the property that
between any two vertices there is precisely one path of length `.
Kotzig conjectured in 1974 that there exists no such graphs. Heproved this conjecture for 3 ≤ ` ≤ 8.
Kostochka proved in 1988 that the conjecture is true for ` ≤ 20.Xing and Hu proved the Kotzig’s conjecture in 1994 for ` ≥ 12.
Thus, the Kotzig’s conjecture is valid now as a theorem.
Hussein Awala (Temple University) 17 / 18
![Page 51: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/51.jpg)
Kotzig’s Conjecture
Let us first rephrase our theorem, the condition (?) can be formulatedas :
Suppose G is a graph with the property that between any two verticesthere is exactly one path of length 2.
This lead to the consideration of the following problem:
Let ` ≥ 2. Is there a finite graphs with the property that
between any two vertices there is precisely one path of length `.
Kotzig conjectured in 1974 that there exists no such graphs. Heproved this conjecture for 3 ≤ ` ≤ 8.Kostochka proved in 1988 that the conjecture is true for ` ≤ 20.
Xing and Hu proved the Kotzig’s conjecture in 1994 for ` ≥ 12.Thus, the Kotzig’s conjecture is valid now as a theorem.
Hussein Awala (Temple University) 17 / 18
![Page 52: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/52.jpg)
Kotzig’s Conjecture
Let us first rephrase our theorem, the condition (?) can be formulatedas :
Suppose G is a graph with the property that between any two verticesthere is exactly one path of length 2.
This lead to the consideration of the following problem:
Let ` ≥ 2. Is there a finite graphs with the property that
between any two vertices there is precisely one path of length `.
Kotzig conjectured in 1974 that there exists no such graphs. Heproved this conjecture for 3 ≤ ` ≤ 8.Kostochka proved in 1988 that the conjecture is true for ` ≤ 20.Xing and Hu proved the Kotzig’s conjecture in 1994 for ` ≥ 12.
Thus, the Kotzig’s conjecture is valid now as a theorem.
Hussein Awala (Temple University) 17 / 18
![Page 53: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/53.jpg)
Kotzig’s Conjecture
Let us first rephrase our theorem, the condition (?) can be formulatedas :
Suppose G is a graph with the property that between any two verticesthere is exactly one path of length 2.
This lead to the consideration of the following problem:
Let ` ≥ 2. Is there a finite graphs with the property that
between any two vertices there is precisely one path of length `.
Kotzig conjectured in 1974 that there exists no such graphs. Heproved this conjecture for 3 ≤ ` ≤ 8.Kostochka proved in 1988 that the conjecture is true for ` ≤ 20.Xing and Hu proved the Kotzig’s conjecture in 1994 for ` ≥ 12.
Thus, the Kotzig’s conjecture is valid now as a theorem.Hussein Awala (Temple University) 17 / 18
![Page 54: On the Proof of the Friendship Theoremstorage.googleapis.com/wzukusers/user-16668366... · Block Designs Coding Theory Set Theory The Friendship Theorem is listed amongPaulandJack](https://reader031.fdocuments.in/reader031/viewer/2022041111/5f13a064df5eac497d0ed940/html5/thumbnails/54.jpg)
THANK YOU
Hussein Awala (Temple University) 18 / 18