which graph properties are characterized by the...
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![Page 1: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/1.jpg)
Which graph properties arecharacterized by the spectrum?
Willem H Haemers
Tilburg University
The Netherlands
![Page 2: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/2.jpg)
Which graph properties arecharacterized by the spectrum?
Willem H Haemers
Tilburg University
The Netherlands
Celebrating 80 years
Reza Khosrovshahi
![Page 3: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/3.jpg)
![Page 4: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/4.jpg)
![Page 5: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/5.jpg)
![Page 6: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/6.jpg)
![Page 7: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/7.jpg)
0 1 0 0 0 0 0 0
1 0 1 0 1 0 1 0
0 1 0 1 0 0 0 1
0 0 1 0 1 0 0 0
0 1 0 1 0 1 0 0
0 0 0 0 1 0 1 0
0 1 0 0 0 1 0 1
0 0 1 0 0 0 1 0
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adjacency spectrum
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
![Page 8: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/8.jpg)
0 1 0 0 0 0 0 0
1 0 1 0 1 0 1 0
0 1 0 1 0 0 0 1
0 0 1 0 1 0 0 0
0 1 0 1 0 1 0 0
0 0 0 0 1 0 1 0
0 1 0 0 0 1 0 1
0 0 1 0 0 0 1 0
21 3 4
65 7 8
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![Page 9: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/9.jpg)
0 1 0 0 0 0 0 0
1 0 1 0 1 0 1 0
0 1 0 1 0 0 0 1
0 0 1 0 1 0 0 0
0 1 0 1 0 1 0 0
0 0 0 0 1 0 1 0
0 1 0 0 0 1 0 1
0 0 1 0 0 0 1 0
21 3 4
65 7 8
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adjacency spectrum
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
![Page 10: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/10.jpg)
adjacency spectrum
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
![Page 11: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/11.jpg)
adjacency spectrum
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
Theorem
The adjacency spectrum is symmetric around 0
if and only if the graph is bipartite
![Page 12: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/12.jpg)
adjacency spectrum
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
Theorem (Coulson, Rushbrooke 1940, Sachs 1966)
The adjacency spectrum is symmetric around 0
if and only if the graph is bipartite
![Page 13: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/13.jpg)
0 0 0 0 1 0 0 00 0 0 0 1 1 0 10 0 0 0 1 1 1 00 0 0 0 1 0 1 11 1 1 1 0 0 0 00 1 1 0 0 0 0 00 0 1 1 0 0 0 00 1 0 1 0 0 0 0
31 5 7
42 6 8
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uu u uadjacency spectrum
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
![Page 14: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/14.jpg)
λ1 ≥ . . . ≥ λn are the adjacency eigenvalues of G
Theorem
G has n vertices, 12
n∑i=1
λ2i edges and 16
n∑i=1
λ3i triangles
Theorem
G is regular if and only if λ1 equals the average degree
![Page 15: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/15.jpg)
λ1 ≥ . . . ≥ λn are the adjacency eigenvalues of G
Theorem
G has n vertices, 12
n∑i=1
λ2i edges and 16
n∑i=1
λ3i triangles
Theorem
G is regular if and only if λ1 equals 1n
n∑i=1
λ2i m
![Page 16: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/16.jpg)
λ1 ≥ . . . ≥ λn are the adjacency eigenvalues of G
Theorem
G has n vertices, 12
n∑i=1
λ2i edges and 16
n∑i=1
λ3i triangles
Theorem
G is regular if and only if λ1 equals 1n
n∑i=1
λ2i m
Drawback
Spectrum does not tell everything
![Page 17: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/17.jpg)
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
![Page 18: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/18.jpg)
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
8 vertices, 10 edges, bipartite
![Page 19: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/19.jpg)
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
8 vertices, 10 edges, bipartite
Can the bipartition have parts of unequal size?
0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0
0 0 00 0 00 0 0
uu u u u
u u u
![Page 20: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/20.jpg)
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
8 vertices, 10 edges, bipartite
Can the bipartition have parts of unequal size? NO!
0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0
0 0 00 0 00 0 0
uu u u u
u u u
![Page 21: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/21.jpg)
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
8 vertices, 10 edges, bipartite with parts of size 4
0 0 0 00 0 0 00 0 0 00 0 0 0
0 0 0 00 0 0 00 0 0 00 0 0 0
uu u u
uu u u
![Page 22: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/22.jpg)
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
8 vertices, 10 edges, bipartite with parts of size 4
0 0 0 0 1 1 1 10 0 0 0 1 0 1 00 0 0 0 1 0 0 10 0 0 0 1 1 0 01 1 1 1 0 0 0 01 0 0 1 0 0 0 01 1 0 0 0 0 0 01 0 1 0 0 0 0 0
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![Page 23: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/23.jpg)
{−1−√3 , −1 , −1 , 1−
√3 , −1 +
√3 , 1 , 1 , 1 +
√3}
8 vertices, 10 edges, bipartite with parts of size 4
0 0 0 0 1 0 0 00 0 0 0 1 1 0 10 0 0 0 1 1 1 00 0 0 0 1 0 1 11 1 1 1 0 0 0 00 1 1 0 0 0 0 00 0 1 1 0 0 0 00 1 0 1 0 0 0 0
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![Page 24: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/24.jpg)
Observation
The degree sequence of a graph is not determined bythe adjacency spectrum
Question
Are the sizes of the two parts of a bipartite graphdetermined by the adjacency spectrum?
![Page 25: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/25.jpg)
Observation
The degree sequence of a graph is not determined bythe adjacency spectrum
Question
Are the sizes of the two parts of a bipartite graphdetermined by the adjacency spectrum?
General answer is NO!
![Page 26: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/26.jpg)
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both graphs have adjacency spectrum
{−2, 0, 0, 0, 2}
![Page 27: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/27.jpg)
Problem (Zwierzynski 2006)
Can one determine the size of a bipartition given onlythe spectrum of a connected bipartite graph?
![Page 28: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/28.jpg)
Problem (Zwierzynski 2006)
Can one determine the size of a bipartition given onlythe spectrum of a connected bipartite graph?
Theorem (van Dam, WHH 2008)
NO!
![Page 29: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/29.jpg)
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NOT determined by the adjacency spectrum are:
• being connected
• being a tree
• the girth
![Page 30: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/30.jpg)
Laplacian (matrix)
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3 -1 -1 -1 0 0-1 2 0 0 0 -1-1 0 2 0 0 -1-1 0 0 3 -1 -10 0 0 -1 1 00 -1 -1 -1 0 3
Laplacian spectrum
{0, 3−√
5, 2, 3, 3, 3+√
5}
![Page 31: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/31.jpg)
0 = µ1 ≤ . . . ≤ µn are the Laplacian eigenvalues of G
Theorem
• G has 12
n∑i=2
µi edges, and 1n
n∏i=2
µi spanning trees
• the number of connected components of G equalsthe multiplicity of 0
Theorem
G is regular if and only if nn∑
i=2
µi(µi − 1) = (n∑
i=2
µi)2
![Page 32: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/32.jpg)
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uLaplacian spectrum
{0, 3−√
5, 2, 3, 3, 3+√
5}
![Page 33: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/33.jpg)
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uNOT determined by the Laplacian spectrum are:
• number of triangles
• bipartite
• degree sequence
• girth
![Page 34: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/34.jpg)
If G is regular of degree k , then L = kI − Ahence µi = k − λi for i = 1 . . . n
Properties determined by one spectrum are alsodetermined by the other spectrum
For regular graphs the following are determined bythe spectrum:
• number of vertices, edges, triangles; bipartite
• number of spanning trees, connected components
![Page 35: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/35.jpg)
If G is regular of degree k , then L = kI − Ahence µi = k − λi for i = 1 . . . n
Properties determined by one spectrum are alsodetermined by the other spectrum
For regular graphs the following are determined bythe spectrum:
• number of vertices, edges, triangles; bipartite
• number of spanning trees, connected components
• degree sequence
• girth
![Page 36: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/36.jpg)
Strongly regular graph SRG(n, k , λ, µ)
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A2 = kI + λA + µ(J − I − A)
(A− rI )(A− sI ) = µJ , r + s = λ− µ, rs = µ− k
Every adjacency eigenvalue is equal to k , r , or s
![Page 37: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/37.jpg)
Example SRG(16, 9, 2, 4); Latin square graph
A C B D
D A C B
B D A C
C B D A
vertices: entries of the Latin square
adjacent: same row, column, or letter
adjacency spectrum {(−3)6, 19, 9}
![Page 38: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/38.jpg)
Example SRG(16, 9, 2, 4); Latin square graph
A C B D
D A C B
B D A C
C B D A
A C B D
C A D B
B D A C
D B C A
vertices: entries of the Latin square
adjacent: same row, column, or letter
adjacency spectrum {(−3)6, 19, 9}
![Page 39: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/39.jpg)
Theorem (Shrikhande, Bhagwandas 1965)
G is strongly regular
if and only if
G is regular and connected and has exactly threedistinct eigenvalues, or G is regular and disconnectedwith exactly two distinct eigenvalues∗
∗ i.e. G is the disjoint union of m > 1 complete graphs of order ` > 1
![Page 40: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/40.jpg)
Incidence graph of a symmetric (v , k , λ)-design
bipartite
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Adjacency spectrum
{−k ,−√k − λ(v−1),
√k − λ(v−1), k}
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Example Heawood graph, the incidence graph of theunique symmetric (7, 3, 1)-design (Fano plane)
A =
[O NN> O
]where N =
1 1 0 1 0 0 0
0 1 1 0 1 0 0
0 0 1 1 0 1 0
0 0 0 1 1 0 1
1 0 0 0 1 1 0
0 1 0 0 0 1 1
1 0 1 0 0 0 1
Spectrum {−3, −
√26,√
26, 3}
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Theorem (Cvetkovic, Doob, Sachs 1984)
G is incidence graph of a symmetric (v , k , λ)-design
if and only if G has adjacency spectrum
{−k , −√k − λ(v−1),
√k − λ(v−1), k}
Corollary There exists a projective plane of order mif and only if there exists a graph with adjacencyspectrum
{−m − 1, −√m
m(m+1),√m
m(m+1), m + 1}
![Page 43: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/43.jpg)
For the following properties there exist a pair ofcospectral regular graphs where one graph has theproperty and the other one not
![Page 44: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/44.jpg)
For the following properties there exist a pair ofcospectral regular graphs where one graph has theproperty and the other one not
• being distance-regular of diameter d ≥ 3(d ≥ 4 Hoffman 1963, d = 3 WHH 1992)
(A distance-regular graphs of diameter 2 is strongly regular)
![Page 45: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/45.jpg)
For the following properties there exist a pair ofcospectral regular graphs where one graph has theproperty and the other one not
• being distance-regular of diameter d ≥ 3(d ≥ 4 Hoffman 1963, d = 3 WHH 1992)• having diameter d ≥ 2 (WHH, Spence 1995)
![Page 46: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/46.jpg)
For the following properties there exist a pair ofcospectral regular graphs where one graph has theproperty and the other one not
• being distance-regular of diameter d ≥ 3(d ≥ 4 Hoffman 1963, d = 3 WHH 1992)• having diameter d ≥ 2 (WHH, Spence 1995)• having a perfect matching (n2 disjoint edges)
(Blazsik, Cummings, WHH 2015)
![Page 47: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/47.jpg)
For the following properties there exist a pair ofcospectral regular graphs where one graph has theproperty and the other one not
• being distance-regular of diameter d ≥ 3(d ≥ 4 Hoffman 1963, d = 3 WHH 1992)• having diameter d ≥ 2 (WHH, Spence 1995)• having a perfect matching (n2 disjoint edges)
(Blazsik, Cummings, WHH 2015)• having vertex connectivity ≥ 3 (WHH 2019)
![Page 48: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/48.jpg)
For the following properties there exist a pair ofcospectral regular graphs where one graph has theproperty and the other one not
• being distance-regular of diameter d ≥ 3(d ≥ 4 Hoffman 1963, d = 3 WHH 1992)• having diameter d ≥ 2 (WHH, Spence 1995)• having a perfect matching (n2 disjoint edges)
(Blazsik, Cummings, WHH 2015)• having vertex connectivity ≥ 3 (WHH 2019)
• having edge connectivity ≥ 6 (WHH 2019)
![Page 49: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/49.jpg)
For most NP-hard properties (chromatic number,clique number etc.) it is not hard to find a pair ofcospectral regular graphs, where one has the property,and the other one not.
![Page 50: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/50.jpg)
For most NP-hard properties (chromatic number,clique number etc.) it is not hard to find a pair ofcospectral regular graphs, where one has the property,and the other one not.
Problem Does there exist a pair of cospectral regulargraphs of degree k , where one has chromatic index(edge chromatic number) k , and the other k + 1?
![Page 51: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/51.jpg)
Characterizations from the spectral point of view
Proposition G has two distinct adjacency eigenvaluesif and only if G is the disjoint union of completegraphs having the same order m > 1
Proposition G has two distinct Laplacian eigenvaluesif and only if G is the disjoint union of completegraphs having the same order m > 1, possiblyextended with some isolated vertices
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Can we characterize the graphs with three distinctadjacency eigenvalues?
![Page 53: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/53.jpg)
Can we characterize the graphs with three distinctadjacency eigenvalues?
If the graphs are regular and connected, then they areprecisely the connected strongly regular graphs
If regularity is not assumed, then there exist otherexamples, but no characterization is known
![Page 54: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/54.jpg)
Theorem (van Dam, WHH 1998)
A connected graph G has three distinct Laplacianeigenvalues if and only if µ and µ are constant
![Page 55: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/55.jpg)
Theorem (van Dam, WHH 1998)
A connected graph G has three distinct Laplacianeigenvalues if and only if µ and µ are constant
µ����uu �
��
@@@
µ����uu
![Page 56: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/56.jpg)
Theorem (van Dam, WHH 1998)
A connected graph G has three distinct Laplacianeigenvalues if and only if µ and µ are constant
µ����uu �
��
@@@
µ����uu
If G is regular of degree k , then µ = n − 2k + λ, andG is an SRG(n, k , λ, µ)
![Page 57: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/57.jpg)
Example n = 7, µ = 1, µ = 2
������������
��������
QQQQ
TTTTTTTTTTTTu
uu
uu u
uLaplacian spectrum {0, 3−
√23, 3 +
√23}
![Page 58: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/58.jpg)
Theorem (Cameron, Goethals, Seidel, Shult 1976)
A graph G has least adjacency eigenvalue ≥ −2 ifand only if G is a generalized line graph, or G belongsto a finite set of exceptional graphs (n ≤ 36)
Book: Spectral generalisations of line graphs,Cvetkovic, Rowlinson, Simic 2004
![Page 59: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/59.jpg)
Proposition G has least adjacency eigenvalue ≥ −1if and only if G is the disjoint union of completegraphs
![Page 60: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/60.jpg)
Proposition G has least adjacency eigenvalue ≥ −1if and only if G is the disjoint union of completegraphs
Proof 1 A + I is positive semi-definite, so it is theGram matrix of a set of unit vectors with innerproduct 0 or 1
![Page 61: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/61.jpg)
Proposition G has least adjacency eigenvalue ≥ −1if and only if G is the disjoint union of completegraphs
Proof 1 A + I is positive semi-definite, so it is theGram matrix of a set of unit vectors with innerproduct 0 or 1
Proof 2 The path P3 = u u u has spectrum{−√
2, 0,√
2}, and by interlacing it can not be aninduced subgraph of G
![Page 62: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/62.jpg)
a2 + b2 = c2 Pythagoras!
![Page 63: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/63.jpg)
a2 + b2 = c2 Pythagoras!
![Page 64: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/64.jpg)
a2 + b2 = c2 Pythagoras!
eπi + 1 = 0 Euler!
![Page 65: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/65.jpg)
a2 + b2 = c2 Pythagoras!
eπi + 1 = 0 Euler!
![Page 66: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/66.jpg)
a2 + b2 = c2 Pythagoras!
eπi + 1 = 0 Euler!
1 = 0 !
![Page 67: Which graph properties are characterized by the spectrum?math.ipm.ac.ir/IPMCCC2019/slides/Haemers.pdfadjacency spectrum f 1 p 3 ; 1 ; 1 ; 1 p 3 ; 1+ p 3 ; 1 ; 1 ; 1+ p 3g Theorem The](https://reader034.vdocuments.site/reader034/viewer/2022042212/5eb5a54b786bd40c7744e082/html5/thumbnails/67.jpg)
a2 + b2 = c2 Pythagoras!
eπi + 1 = 0 Euler!
1 = 0 !