Spectra of Graphons: some spectral results from L´ aszl´ o Lov´ asz’s textbook Large Networks and Graph Limits Alexander W. N. Riasanovsky [email protected]Spectral Graph Theory (MATH 595) at Iowa State University April 19, 2017 Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 1 / 22
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Spectra of Graphons:some spectral results from
Laszlo Lovasz’s textbook Large Networks and Graph Limits
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 11 / 22
Revisiting Convergence
Table: An Erdos-Renyi random graph (p = 1/2) and a uniform attachmentrandom graph, both on 21 vertices.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 12 / 22
Revisiting Convergence
Table: An Erdos-Renyi random graph (p = 1/2) and a uniform attachmentrandom graph, both on 22 vertices.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 12 / 22
Revisiting Convergence
Table: An Erdos-Renyi random graph (p = 1/2) and a uniform attachmentrandom graph, both on 23 vertices.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 12 / 22
Revisiting Convergence
Table: An Erdos-Renyi random graph (p = 1/2) and a uniform attachmentrandom graph, both on 24 vertices.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 12 / 22
Revisiting Convergence
Table: An Erdos-Renyi random graph (p = 1/2) and a uniform attachmentrandom graph, both on 25 vertices.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 12 / 22
Revisiting Convergence
Table: An Erdos-Renyi random graph (p = 1/2) and a uniform attachmentrandom graph, both on 26 vertices.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 12 / 22
Revisiting Convergence
Table: An Erdos-Renyi random graph (p = 1/2) and a uniform attachmentrandom graph, both on 27 vertices.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 12 / 22
Revisiting Convergence
Table: An Erdos-Renyi random graph (p = 1/2) and a uniform attachmentrandom graph, both on 28 vertices.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 12 / 22
Revisiting Convergence
Table: The only probable limits in growing sequences of Erdos-Renyi randomgraphs (p = 1/2) and uniform attachment graphs.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 12 / 22
Density and Completeness
TheoremLet W be the metric space ofgraphons modded out by δ� = 0.Then W is compact and the subsetG ⊆ W of graph graphons (those ofthe form WG ) is dense.
In other words...
1 Convergent sequences ofgraphs are graphons
2 Graphons are convergentsequences of graphs
3 Any sequence of graph(on)shas a limit graphon
4 extremal constructions haveone or more limts
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 13 / 22
Density and Completeness
TheoremLet W be the metric space ofgraphons modded out by δ� = 0.Then W is compact and the subsetG ⊆ W of graph graphons (those ofthe form WG ) is dense.
In other words...
1 Convergent sequences ofgraphs are graphons
2 Graphons are convergentsequences of graphs
3 Any sequence of graph(on)shas a limit graphon
4 extremal constructions haveone or more limts
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 13 / 22
Density and Completeness
TheoremLet W be the metric space ofgraphons modded out by δ� = 0.Then W is compact and the subsetG ⊆ W of graph graphons (those ofthe form WG ) is dense.
In other words...
1 Convergent sequences ofgraphs are graphons
2 Graphons are convergentsequences of graphs
3 Any sequence of graph(on)shas a limit graphon
4 extremal constructions haveone or more limts
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 13 / 22
Density and Completeness
TheoremLet W be the metric space ofgraphons modded out by δ� = 0.Then W is compact and the subsetG ⊆ W of graph graphons (those ofthe form WG ) is dense.
In other words...
1 Convergent sequences ofgraphs are graphons
2 Graphons are convergentsequences of graphs
3 Any sequence of graph(on)shas a limit graphon
4 extremal constructions haveone or more limts
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 13 / 22
Density and Completeness
TheoremLet W be the metric space ofgraphons modded out by δ� = 0.Then W is compact and the subsetG ⊆ W of graph graphons (those ofthe form WG ) is dense.
In other words...
1 Convergent sequences ofgraphs are graphons
2 Graphons are convergentsequences of graphs
3 Any sequence of graph(on)shas a limit graphon
4 extremal constructions haveone or more limts
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 13 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn
0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W
and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally,
TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
AG versus TW
AG
Given |V (G )| = n and v ∈ Rn
(AGv)k =n∑
i=1
akivi
AG : Rn → Rn0 1 1 11 0 1 01 1 0 01 0 0 0
πξχρ
=
π + ξ
TW
Given a graphon W and f a“nice” f : [0, 1]→ R
(TW f )(x) =
∫ 1
0W (x , y)f (y)dy
Formally, TW : nice→ nice
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 14 / 22
Spectral Decomposition of TW
Theorem
If W is a graphon, then its eigenvalues can be ordered as1 ≥ |λ1| ≥ |λ2| ≥ · · · so that λk → 0 as k →∞ and
TW ∼∞∑k=1
λk fk(x)fk(y) treating TW : L2[0, 1]→ L2[0, 1]
where the inner product∫ 10 fk(x)fl(y) = 1 if k = l and 0 otherwise.
Example
If W (x , y) = xy , λ1 = 1/4 andf1(x) = 2x and all other λk = 0.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 15 / 22
Spectral Decomposition of TW
Theorem
If W is a graphon, then its eigenvalues can be ordered as1 ≥ |λ1| ≥ |λ2| ≥ · · · so that λk → 0 as k →∞ and
TW ∼∞∑k=1
λk fk(x)fk(y) treating TW : L2[0, 1]→ L2[0, 1]
where the inner product∫ 10 fk(x)fl(y) = 1 if k = l and 0 otherwise.
Example
If W (x , y) = xy ,
λ1 = 1/4 andf1(x) = 2x and all other λk = 0.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 15 / 22
Spectral Decomposition of TW
Theorem
If W is a graphon, then its eigenvalues can be ordered as1 ≥ |λ1| ≥ |λ2| ≥ · · · so that λk → 0 as k →∞ and
TW ∼∞∑k=1
λk fk(x)fk(y) treating TW : L2[0, 1]→ L2[0, 1]
where the inner product∫ 10 fk(x)fl(y) = 1 if k = l and 0 otherwise.
Example
If W (x , y) = xy , λ1 = 1/4 andf1(x) = 2x
and all other λk = 0.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 15 / 22
Spectral Decomposition of TW
Theorem
If W is a graphon, then its eigenvalues can be ordered as1 ≥ |λ1| ≥ |λ2| ≥ · · · so that λk → 0 as k →∞ and
TW ∼∞∑k=1
λk fk(x)fk(y) treating TW : L2[0, 1]→ L2[0, 1]
where the inner product∫ 10 fk(x)fl(y) = 1 if k = l and 0 otherwise.
Example
If W (x , y) = xy , λ1 = 1/4 andf1(x) = 2x and all other λk = 0.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 15 / 22
Exercise: From spec(G ) to spec(W )
To turn a spectral decomposition of AG into one of TW ...
1 Write AG out spectrally.
2 Replace the standard basis of Rn with weighted inticator functions.(Hint: consider indicator functions of intervals. Remember thatdouble integration over line segments produces 0. )
3 Justify that there are no more eigenvalues. (Hint: the “all zeros”graphon 0 · J has all eigenvalues 0 and hence spectral decompositionT0·J = 0.)
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 16 / 22
Exercise: From spec(G ) to spec(W )
To turn a spectral decomposition of AG into one of TW ...
1 Write AG out spectrally.
2 Replace the standard basis of Rn with weighted inticator functions.(Hint: consider indicator functions of intervals. Remember thatdouble integration over line segments produces 0. )
3 Justify that there are no more eigenvalues. (Hint: the “all zeros”graphon 0 · J has all eigenvalues 0 and hence spectral decompositionT0·J = 0.)
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 16 / 22
Exercise: From spec(G ) to spec(W )
To turn a spectral decomposition of AG into one of TW ...
1 Write AG out spectrally.
2 Replace the standard basis of Rn with weighted inticator functions.
(Hint: consider indicator functions of intervals. Remember thatdouble integration over line segments produces 0. )
3 Justify that there are no more eigenvalues. (Hint: the “all zeros”graphon 0 · J has all eigenvalues 0 and hence spectral decompositionT0·J = 0.)
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 16 / 22
Exercise: From spec(G ) to spec(W )
To turn a spectral decomposition of AG into one of TW ...
1 Write AG out spectrally.
2 Replace the standard basis of Rn with weighted inticator functions.(Hint: consider indicator functions of intervals. Remember thatdouble integration over line segments produces 0. )
3 Justify that there are no more eigenvalues. (Hint: the “all zeros”graphon 0 · J has all eigenvalues 0 and hence spectral decompositionT0·J = 0.)
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 16 / 22
Exercise: From spec(G ) to spec(W )
To turn a spectral decomposition of AG into one of TW ...
1 Write AG out spectrally.
2 Replace the standard basis of Rn with weighted inticator functions.(Hint: consider indicator functions of intervals. Remember thatdouble integration over line segments produces 0. )
3 Justify that there are no more eigenvalues.
(Hint: the “all zeros”graphon 0 · J has all eigenvalues 0 and hence spectral decompositionT0·J = 0.)
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 16 / 22
Exercise: From spec(G ) to spec(W )
To turn a spectral decomposition of AG into one of TW ...
1 Write AG out spectrally.
2 Replace the standard basis of Rn with weighted inticator functions.(Hint: consider indicator functions of intervals. Remember thatdouble integration over line segments produces 0. )
3 Justify that there are no more eigenvalues. (Hint: the “all zeros”graphon 0 · J has all eigenvalues 0 and hence spectral decompositionT0·J = 0.)
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 16 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and
spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Spectral Convergence
Table: WKn → J asn→∞.
Table:WKdn/2e,bn/2c →WK2 asn→∞.
Spectra?
spec(Kn) = {n − 1,−1, . . . ,−1} and spec(J) = {1, 0, . . . , 0} and
spec(Kdn/2e,bn/2c) = {n/2, 0, . . . , 0,−n/2} and
spec(WK2) = {1/2, 0, . . . , 0,−1/2}.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 17 / 22
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 18 / 22
The End
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 19 / 22
Applications
Triangle Removal Lemma
For all ε > 0, there exists some ε′ > 0 so that if G a graph on n verticeshas at most ε′n3 triangles, then there exists some triangle-free G ′ ⊆ Gwith e(G )− e(G ′) ≤ εn2.
Quasirandomness [Chung, Graham, Wilson ’89]
If G1,G2, . . . are graphs where Gn is εn-quasirandom (εn as small aspossible) and |Gn| → ∞, then∑
k
λk(Gn)2 → 1/2 and∑k
λk(Gn)4 → 1/16.
if and only if εn →∞.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 20 / 22
More Applications
Many Strong Szemeredi Regularity Lemmas for Graphs and Graphons
All large graphs (all graphons) may be approxmated in cut distance witharbitrary pre-specified precision by a random graph (graphon) which isgiven by an equipartition.
“Disguises” of Graphons
The following models are cryptomorphic (i.e., the same information):
1 a graphon, up to weak isomorphism
2 a multiplicative, normalized simple graph parameter that isnonnegative on signed graphs
3 a consistent and local graph model
4 a local random countable graph model
5 a point in the completion of the space of finite graphs with the cutdistance
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 21 / 22
References
A. Frieze, R. Kannan (1999)
Quick approximation to matrices and applications
Combinatorica 19(3), 175 – 220.
Lovasz (2012)
Large networks and graph limits
Providence: American Mathematical Society 60.
Alexander W. N. Riasanovsky (ISU) Spectra of Graphons April 19, 2017 22 / 22