Which graph properties are characterized by the spectrum? Willem H - PowerPoint PPT Presentation

Which graph properties are characterized by the spectrum? Willem H Haemers Tilburg University The Netherlands

Which graph properties are characterized by the spectrum? Willem H Haemers Tilburg University The Netherlands Celebrating 80 years Reza Khosrovshahi

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 ✉ adjacency spectrum √ √ √ √ {− 1 − 3 , − 1 , − 1 , 1 − 3 , − 1 + 3 , 1 , 1 , 1 + 3 }

0 1 0 0 0 0 0 0   1 2 3 4 ✉ ✉ ✉ ✉ 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   5 6 7 8 0 0 1 0 0 0 1 0

0 1 0 0 0 0 0 0   1 2 3 4 ✉ ✉ ✉ ✉ 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   5 6 7 8 0 0 1 0 0 0 1 0 adjacency spectrum √ √ √ √ {− 1 − 3 , − 1 , − 1 , 1 − 3 , − 1 + 3 , 1 , 1 , 1 + 3 }

adjacency spectrum √ √ √ √ {− 1 − 3 , − 1 , − 1 , 1 − 3 , − 1 + 3 , 1 , 1 , 1 + 3 }

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

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

0 0 0 0 1 0 0 0   1 3 5 7 ✉ ✉ ✉ ✉ 0 0 0 0 1 1 0 1 ✑ ✁ ❅ � ✁ ✁   ✑   ✁ ❅ � ✁ ✑ ✁ 0 0 0 0 1 1 1 0   ✑ ✁ � ❅ ✁ ✁   ✑ 0 0 0 0 1 0 1 1   ✁ � ✑ ✁ ❅ ✁   ✑ 1 1 1 1 0 0 0 0 ✁ � ✁ ❅ ✁   ✑   ✁ � ✑ ✁ ✁ ❅ 0 1 1 0 0 0 0 0   ✑ ✉ ✉ ✉ ✉ ✁ � ✁ ✁ ❅   ✑ ✁ � ✁ ✁ ❅ 0 0 1 1 0 0 0 0   2 4 6 8 0 1 0 1 0 0 0 0 adjacency spectrum √ √ √ √ {− 1 − 3 , − 1 , − 1 , 1 − 3 , − 1 + 3 , 1 , 1 , 1 + 3 }

λ 1 ≥ . . . ≥ λ n are the adjacency eigenvalues of G Theorem n n � � G has n vertices, 1 λ 2 i edges and 1 λ 3 i triangles 2 6 i =1 i =1 Theorem G is regular if and only if λ 1 equals the average degree

λ 1 ≥ . . . ≥ λ n are the adjacency eigenvalues of G Theorem n n � � G has n vertices, 1 λ 2 i edges and 1 λ 3 i triangles 2 6 i =1 i =1 Theorem n � G is regular if and only if λ 1 equals 1 λ 2 m i n i =1

λ 1 ≥ . . . ≥ λ n are the adjacency eigenvalues of G Theorem n n � � G has n vertices, 1 λ 2 i edges and 1 λ 3 i triangles 2 6 i =1 i =1 Theorem n � G is regular if and only if λ 1 equals 1 λ 2 m i n i =1 Drawback Spectrum does not tell everything

√ √ √ √ {− 1 − 3 , − 1 , − 1 , 1 − 3 , − 1 + 3 , 1 , 1 , 1 + 3 }

√ √ √ √ {− 1 − 3 , − 1 , − 1 , 1 − 3 , − 1 + 3 , 1 , 1 , 1 + 3 } 8 vertices, 10 edges, bipartite

√ √ √ √ {− 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 0  0 0 0 0 0 ✉ ✉ ✉ ✉ ✉     0 0 0 0 0     0 0 0 0 0     0 0 0 0 0     0 0 0     0 0 0   ✉ ✉ ✉ 0 0 0

√ √ √ √ {− 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 0  0 0 0 0 0 ✉ ✉ ✉ ✉ ✉     0 0 0 0 0     0 0 0 0 0     0 0 0 0 0     0 0 0     0 0 0   ✉ ✉ ✉ 0 0 0

√ √ √ √ {− 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   0 0 0 0 ✉ ✉ ✉ ✉    0 0 0 0      0 0 0 0     0 0 0 0     0 0 0 0     0 0 0 0   ✉ ✉ ✉ ✉ 0 0 0 0

√ √ √ √ {− 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 1   ✉ ✉ ✉ ✉ ◗◗◗◗◗◗◗◗◗◗ ✑ 0 0 0 0 1 0 1 0 ❅ ❆ ✁ ❆ � ❆ �   ✑  0 0 0 0 1 0 0 1  ❆ ❅ ✁ ❆ � ❆ ✑ �   ✑ ❆ ❅ ✁ � ❆ � ❆   ✑ 0 0 0 0 1 1 0 0   ❆ ✁ ❅ � ✑ ❆ � ❆   ✑ 1 1 1 1 0 0 0 0 ✁ ❆ � ❅ ❆ � ❆   ✑   ✁ � ✑ ❆ � ❅ ❆ ❆ 1 0 0 1 0 0 0 0   ✑ ✉ ✁ � ❆ ✉ � ❅ ❆ ✉ ❆ ✉ ✁ � ✑ ❆ � ❅ ❆ ◗ ❆   1 1 0 0 0 0 0 0   1 0 1 0 0 0 0 0 degree sequence (2 , 2 , 2 , 2 , 2 , 2 , 4 , 4)

√ √ √ √ {− 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 0   ✉ ✉ ✉ ✉ ✑ 0 0 0 0 1 1 0 1 ✁ ❅ � ✁ ✁   ✑  0 0 0 0 1 1 1 0  ✁ ❅ � ✁ ✑ ✁   ✑ ✁ � ❅ ✁ ✁   ✑ 0 0 0 0 1 0 1 1   ✁ � ✑ ✁ ❅ ✁   ✑ 1 1 1 1 0 0 0 0 ✁ � ✁ ❅ ✁   ✑   ✁ � ✑ ✁ ✁ ❅ 0 1 1 0 0 0 0 0   ✑ ✉ ✁ � ✉ ✁ ✉ ✁ ❅ ✉ ✁ ✑ � ✁ ✁ ❅   0 0 1 1 0 0 0 0   0 1 0 1 0 0 0 0 degree sequence (1 , 2 , 2 , 2 , 3 , 3 , 3 , 4)

Observation The degree sequence of a graph is not determined by the adjacency spectrum Question Are the sizes of the two parts of a bipartite graph determined by the adjacency spectrum?

Observation The degree sequence of a graph is not determined by the adjacency spectrum Question Are the sizes of the two parts of a bipartite graph determined by the adjacency spectrum? General answer is NO!

✉ ✉ ✉ ✉ ✉ ✉ ✉ ◗ ✑ ◗ ✑ ◗ ✑✑✑✑✑ ◗ ✑✑✑✑✑ ❆ ✁ ❆ ✁ ◗ ❆ ✁ ❆ ◗ ✁ ◗ ◗ ❆ ✁ ❆ ✁ ◗ ◗ ✉ ✉ ✉ ◗ ❆ ✁ ❆ ◗ ✁ both graphs have adjacency spectrum {− 2 , 0 , 0 , 0 , 2 }

Problem (Zwierzy´ nski 2006) Can one determine the size of a bipartition given only the spectrum of a connected bipartite graph?

Problem (Zwierzy´ nski 2006) Can one determine the size of a bipartition given only the spectrum of a connected bipartite graph? Theorem (van Dam, WHH 2008) NO!

✉ ✉ ✉ ✉ ✉ ✉ ✉ ◗ ✑ ◗ ✑ ◗ ✑✑✑✑✑ ◗ ✑✑✑✑✑ ❆ ✁ ❆ ✁ ◗ ❆ ✁ ❆ ◗ ✁ ◗ ◗ ❆ ✁ ❆ ✁ ◗ ◗ ✉ ✉ ✉ ◗ ❆ ✁ ❆ ◗ ✁ NOT determined by the adjacency spectrum are: • being connected • being a tree • the girth

Laplacian (matrix)  3 -1 -1 -1 0 0  ✉ � ❅ -1 2 0 0 0 -1 � ❅   �   ❅ -1 0 2 0 0 -1 ✉ ✉ ❅ ✉ ✉ �     ❅ � -1 0 0 3 -1 -1 �   ❅ �   ❅ 0 0 0 -1 1 0 ✉ ❅�   0 -1 -1 -1 0 3 Laplacian spectrum √ √ { 0 , 3 − 5 , 2 , 3 , 3 , 3+ 5 }

0 = µ 1 ≤ . . . ≤ µ n are the Laplacian eigenvalues of G Theorem n n � � • G has 1 µ i edges, and 1 µ i spanning trees 2 n i =2 i =2 • the number of connected components of G equals the multiplicity of 0 Theorem n n � � µ i ) 2 G is regular if and only if n µ i ( µ i − 1) = ( i =2 i =2