Are Almost All Graphs Determined by their...
Transcript of Are Almost All Graphs Determined by their...
AgainstFor
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?
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Graphs and MatricesMany Problems
Outline
1 AgainstGraphs and MatricesMany Problems
2 ForWalk MatricesWang & Xu
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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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?
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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?
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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?
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Graphs and MatricesMany Problems
The Origin of our Difficulty
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Graphs and MatricesMany Problems
Outline
1 AgainstGraphs and MatricesMany Problems
2 ForWalk MatricesWang & Xu
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Graphs and MatricesMany Problems
There’s a problem. . .
φ(X , t) = t5 − 4t3.
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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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?
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Graphs and MatricesMany Problems
Did we forget to say connected?
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
AgainstFor
Graphs and MatricesMany Problems
Did we forget to say connected?
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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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?
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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?
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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?
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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?
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Graphs and MatricesMany Problems
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?
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Graphs and MatricesMany Problems
Schwenk’s Pairs of Cospectral Trees
S
S
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Graphs and MatricesMany Problems
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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Graphs and MatricesMany Problems
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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Graphs and MatricesMany Problems
Local Switching: regular and all, half, none
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Graphs and MatricesMany Problems
We’ve seen it before
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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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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Graphs and MatricesMany Problems
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?
AgainstFor
Walk MatricesWang & Xu
Outline
1 AgainstGraphs and MatricesMany Problems
2 ForWalk MatricesWang & Xu
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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 MatricesWang & Xu
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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Walk MatricesWang & Xu
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?
AgainstFor
Walk MatricesWang & Xu
Outline
1 AgainstGraphs and MatricesMany Problems
2 ForWalk MatricesWang & Xu
Chris Godsil Are Almost All Graphs Determined by their Spectrum?
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Walk MatricesWang & Xu
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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Walk MatricesWang & Xu
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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Walk MatricesWang & Xu
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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Walk MatricesWang & Xu
The End(s)
Chris Godsil Are Almost All Graphs Determined by their Spectrum?