are almost all graphs determined by their...
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![Page 1: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/1.jpg)
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Graphs and MatricesMany Problems
Are Almost All Graphs Determined by theirSpectrum?
Chris Godsil
16 January, 2014
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 2: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/2.jpg)
AgainstFor
Graphs and MatricesMany Problems
Outline
1 AgainstGraphs and MatricesMany Problems
2 ForWalk MatricesWang & Xu
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 3: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/3.jpg)
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Graphs and MatricesMany Problems
Two graphs, two matrices
1 2 3 1 230 1 01 0 10 1 0
0 0 1
0 0 11 1 0
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 4: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/4.jpg)
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Graphs and MatricesMany Problems
Isomorphic implies cospectral
If graphs X and Y are isomorphic, there is a permutation matrixP such that
A(Y ) = PA(X)PT
Since permutation matrices are orthogonal, this means that A(X)and A(Y ) are similar.
DefinitionGraphs X and Y are cospectral if the characteristic polynomials oftheir adjacency matrices are equal.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 5: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/5.jpg)
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Graphs and MatricesMany Problems
Willem Haemers proposes:
ConjectureThe proportion of graphs on n vertices that are determined bytheir spectrum goes to 1 as n →∞.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 6: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/6.jpg)
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The Origin of our Difficulty
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 7: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/7.jpg)
AgainstFor
Graphs and MatricesMany Problems
Outline
1 AgainstGraphs and MatricesMany Problems
2 ForWalk MatricesWang & Xu
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 8: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/8.jpg)
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There’s a problem. . .
φ(X , t) = t5 − 4t3.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 9: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/9.jpg)
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Graphs and MatricesMany Problems
. . . and that’s only the start
If m,n ≥ 1 then the graphs
K1,mn , Km,n ∪ (m − 1)(n − 1)K1
are cospectral.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 10: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/10.jpg)
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Did we forget to say connected?
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 11: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/11.jpg)
AgainstFor
Graphs and MatricesMany Problems
Did we forget to say connected?
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 12: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/12.jpg)
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Graphs and MatricesMany Problems
We have forgotten complements, but. . .
The two graphs are cospectral, and so are their complements.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 13: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/13.jpg)
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Graphs and MatricesMany Problems
We have forgotten complements, but. . .
The two graphs are cospectral, and so are their complements.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 14: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/14.jpg)
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Latin square graphs: regularity is no help
16 squares as vertices; adjacent if in same row, same column, or
same color. Cospectral.
(And there are lots of Latin squares.)
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 15: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/15.jpg)
AgainstFor
Graphs and MatricesMany Problems
Latin square graphs: regularity is no help
16 squares as vertices; adjacent if in same row, same column, or
same color. Cospectral. (And there are lots of Latin squares.)Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 16: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/16.jpg)
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Schwenk’s Particular Tree
u v
If we denote this tree by S, then S \u and S \v are cospectral. Wesay they are cospectral vertices in S .
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 17: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/17.jpg)
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Schwenk’s Pairs of Cospectral Trees
S
S
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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The bad news for trees
Theorem (Schwenk)The proportion of trees on n vertices that are determined by theirspectrum goes to 0 as n →∞.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Well, what does the computer tell us?
n # graphs A A & A5 34 0.059 06 156 0.064 07 1044 0.105 0.0388 12346 0.139 0.0949 274668 0.186 0.16010 12005158 0.213 0.201
Source: Godsil and McKay, Haemers and Spence
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Local Switching: regular and all, half, none
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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We’ve seen it before
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 22: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/22.jpg)
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Graphs and MatricesMany Problems
Features of local switching
Graphs related by local switching are cospectral, and theircomplements are too.The proportion of graphs on n vertices that we can usefullyapply local switching to goes to 0 as n →∞.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Asymptotically useless, but good in the short run
n # graphs A A & A ls5 34 0.059 0 06 156 0.064 0 07 1044 0.105 0.038 0.0388 12346 0.139 0.094 0.0859 274668 0.186 0.160 0.13910 12005158 0.213 0.201 0.171
Source: Godsil and McKay, Haemers and Spence, Brouwer andSpence
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 24: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/24.jpg)
AgainstFor
Walk MatricesWang & Xu
Outline
1 AgainstGraphs and MatricesMany Problems
2 ForWalk MatricesWang & Xu
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 25: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/25.jpg)
AgainstFor
Walk MatricesWang & Xu
And if we look at graphs on 11 and 12 vertices. . .
n # graphs A A & A ls5 34 0.059 0 06 156 0.064 0 07 1044 0.105 0.038 0.0388 12346 0.139 0.094 0.0859 274668 0.186 0.160 0.13910 12005158 0.213 0.201 0.17111 1018997864 0.211 0.208 0.17412 165091172592 0.188 ? ?
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Walk matrices
DefinitionIf X is a graph on n vertices and 1 is the all-ones vector, the walkmatrix W of X is the matrix with columns Ar1 forr = 0, . . . ,n − 1.
DefinitionA graph is controllable if its walk matrix is invertible.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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My Conjecture
ConjectureThe proportion of graphs on n vertices that are controllable goesto 1 as n →∞.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 28: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/28.jpg)
AgainstFor
Walk MatricesWang & Xu
Outline
1 AgainstGraphs and MatricesMany Problems
2 ForWalk MatricesWang & Xu
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
![Page 29: Are Almost All Graphs Determined by their Spectrum?faculty.nps.edu/rgera/Conjectures/JointMeetings-2014/godsil-cospec.pdfAgainst For Walk Matrices Wang & Xu WangandXu Theorem Suppose](https://reader035.vdocuments.net/reader035/viewer/2022071414/610d42bb177c201ca7738b1c/html5/thumbnails/29.jpg)
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Wang and Xu
TheoremSuppose X and Y are cospectral graphs with cospectralcomplements. If X is controllable there is a unique orthogonalmatrix Q such that QT A(X)Q = A(Y ). The matrix Q is rationaland Q1 = 1.
TheoremSuppose X is controllable. If Q is an orthogonal matrix such thatQ1 = 1 and QT AQ is an integer matrix, then Q is rational.
Further, if `W−1 is an integer matrix, so is `Q.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Our level best
DefinitionThe level of a rational matrix M is the least integer ` such that`M is an integer matrix.
Wang and Xu provide considerable evidence that if X iscontrollable and QT A(X)Q is an integer matrix then the level ofQ is two for a positive fraction of the graphs on n vertices. Theyprove that orthogonal matrices of index two are essentially localswitchings and that asymptotically these are negligible.
This provides evidence that a positive fraction of the graphs on nvertices are determined by their spectrum.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Related Questions
We know very little about what happens when we use theLaplacian in place of the adjacency matrix.What proportion of trees on n vertices contain a pair ofcospectral vertices?
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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The End(s)
Chris Godsil Are Almost All Graphs Determined by their Spectrum?