The inverse eigenvalue problem of a graph: Multiplicities and minors Jephian C.-H. Lin Department of Mathematics and Statistics, University of Victoria Jan 13, 2018 Joint Mathematics Meetings, San Diego, CA IEPG: Multiplicities and minors 1/14 Math & Stats, University of Victoria
Joint work with Wayne Barrett, Steve Butler, Shaun M. Fallat, H. Tracy Hall, Leslie Hogben, Bryan L. Shader and Michael Young. IEPG: Multiplicities and minors 2/14 Math & Stats, University of Victoria
Inverse eigenvalue problem of a graph Let G be a simple graph on n vertices. The family S ( G ) consists � � of all n × n real symmetric matrix M = M i , j with M i , j = 0 if i � = j and { i , j } is not an edge , M i , j � = 0 if i � = j and { i , j } is an edge , M i , j ∈ R if i = j . 0 1 0 1 − 1 0 2 0 . 1 0 , , , · · · S ( ) ∋ 1 0 1 − 1 2 − 1 0 . 1 1 π 0 1 0 0 − 1 1 0 0 π The inverse eigenvalue problem of a graph (IEPG) asks what are all spectra appeared in S ( G ) for a given graph G . IEPG: Multiplicities and minors 3/14 Math & Stats, University of Victoria
Theorem (Monfared and Shader 2013) Let G be a graph on n vertices. For any n distinct real numbers { λ 1 , . . . , λ n } , there is a matrix A ∈ S ( G ) with spec( A ) = { λ 1 , . . . , λ n } . Key idea: Use Implicit Function Theorem to perturb the diagonal matrix. λ 1 0 0 0 0 ∼ λ 1 ǫ 0 ǫ 0 0 0 0 0 ∼ λ 2 0 λ 2 ǫ ǫ ǫ 0 0 0 0 0 ∼ λ 3 0 λ 3 ǫ ǫ − → ... ... 0 0 0 0 ǫ 0 ǫ ǫ 0 0 0 0 0 0 ∼ λ n λ n ǫ ǫ IEPG: Multiplicities and minors 4/14 Math & Stats, University of Victoria
Theorem (Monfared and Shader 2013) Let G be a graph on n vertices. For any n distinct real numbers { λ 1 , . . . , λ n } , there is a matrix A ∈ S ( G ) with spec( A ) = { λ 1 , . . . , λ n } . Key idea: Use Implicit Function Theorem to perturb the diagonal matrix. λ 1 0 0 0 0 ∼ λ 1 ǫ 0 ǫ 0 0 0 0 0 ∼ λ 2 0 λ 2 ǫ ǫ ǫ 0 0 0 0 0 ∼ λ 3 0 λ 3 ǫ ǫ − → ... ... 0 0 0 0 ǫ 0 ǫ ǫ 0 0 0 0 0 0 ∼ λ n λ n ǫ ǫ IEPG: Multiplicities and minors 4/14 Math & Stats, University of Victoria
Example on P 3 0 x 1 y A ( x 1 , x 2 , x 3 , y ) = y x 2 y 0 y x 3 Goal: Given λ i ’s, find x i ’s and y � = 0 such that spec( A ( x 1 , x 2 , x 3 , y )) = { λ i } 3 i =1 . Note: spec( A ( λ 1 , λ 2 , λ 3 , 0)) = { λ i } 3 i =1 , but y = 0. IEPG: Multiplicities and minors 5/14 Math & Stats, University of Victoria
Example on P 3 x 1 y 0 A ( x 1 , x 2 , x 3 , y ) = y x 2 y 0 y x 3 Consider the function f ( x 1 , x 2 , x 3 , y ) �→ (tr( A ) , 1 2 tr( A 2 ) , 1 3 tr( A 3 )) . The right hand side controls the spectrum. When x i = λ i and y = 0, it has the desired spectrum. IEPG: Multiplicities and minors 5/14 Math & Stats, University of Victoria
Example on P 3 x 1 y 0 A ( x 1 , x 2 , x 3 , y ) = y x 2 y 0 y x 3 independent ) �→ (tr( A ) , 1 2 tr( A 2 ) , 1 � �� � 3 tr( A 3 ) ( ) . x 1 , x 2 , x 3 , y ���� � �� � independent dependent IEPG: Multiplicities and minors 5/14 Math & Stats, University of Victoria
Example on P 3 0 x 1 y A ( x 1 , x 2 , x 3 , y ) = y x 2 y 0 y x 3 dependent ) �→ (tr( A ) , 1 2 tr( A 2 ) , 1 � �� � 3 tr( A 3 ) ( x 1 , x 2 , x 3 , y ) . ���� � �� � independent fixed IEPG: Multiplicities and minors 5/14 Math & Stats, University of Victoria
Example on P 3 0 x 1 y A ( x 1 , x 2 , x 3 , y ) = y x 2 y 0 y x 3 dependent ) �→ (tr( A ) , 1 2 tr( A 2 ) , 1 � �� � 3 tr( A 3 ) ( x 1 , x 2 , x 3 , y ) . ���� � �� � independent fixed tr( A ) = x 1 + x 2 + x 3 1 1 1 � tr( A 2 ) = x 2 1 + x 2 2 + x 2 3 + y (???) = ⇒ Jac = λ 1 λ 2 λ 3 � x i = λ i y =0 λ 2 λ 2 λ 2 tr( A 3 ) = x 3 1 + x 3 2 + x 3 3 + y (???) 1 2 3 IEPG: Multiplicities and minors 5/14 Math & Stats, University of Victoria
Example on P 3 x 1 y 0 A ( x 1 , x 2 , x 3 , y ) = y x 2 y 0 y x 3 dependent ) �→ (tr( A ) , 1 2 tr( A 2 ) , 1 � �� � 3 tr( A 3 ) ( x 1 , x 2 , x 3 , y ) . ���� � �� � independent fixed When λ i ’s are all distinct, the Jacobian matrix is invertible. We may perturb y to ǫ � = 0 and x i ∼ λ i , while preserving the same spectrum. (This proof follows from [Monfared and Khanmohammadi 2018].) IEPG: Multiplicities and minors 5/14 Math & Stats, University of Victoria
Another point of view of the theorem Theorem (Monfared and Shader 2013) Let K n be a spanning subgraph of G. If A ∈ S ( K n ) has some nice property, then there is B ∈ S ( G ) with spec( B ) = spec( A ) . Theorem (BFHHLS 2017) Let H be a spanning subgraph of G. If A ∈ S ( H ) has the Strong Spectral Property, then there is B ∈ S ( G ) with spec( B ) = spec( A ) . IEPG: Multiplicities and minors 6/14 Math & Stats, University of Victoria
Another point of view of the theorem Theorem (Monfared and Shader 2013) Let K n be a spanning subgraph of G. If A ∈ S ( K n ) has some nice property, then there is B ∈ S ( G ) with spec( B ) = spec( A ) . Theorem (BFHHLS 2017) Let H be a spanning subgraph of G. If A ∈ S ( H ) has the Strong Spectral Property, then there is B ∈ S ( G ) with spec( B ) = spec( A ) . IEPG: Multiplicities and minors 6/14 Math & Stats, University of Victoria
Isospectral manifolds and pattern manifolds Let A ∈ S ( H ). The isospectral manifold is E A = { Q ⊤ AQ : Q orthogonal } . The pattern manifold is S cl ( H ) = { M : M i , j = 0 if { i , j } ∈ E ( H ) } . Also define S cl y ( H , G ) = { M ∈ S cl ( G ) : M i , j = y if { i , j } ∈ E ( G ) \ E ( H ) } such that S cl ( H ) = S cl 0 ( H , G ) � S cl y ( H , G ) ⊂ S cl ( G ) . IEPG: Multiplicities and minors 7/14 Math & Stats, University of Victoria
Transversality and Strong Arnold Property Two manifolds intersect transversally at a point A if their normal spaces only have trivial intersection. � M 1 ∩ M 2 ⇐ ⇒ Nor M 1 . A ∩ Nor M 2 . A = { 0 } Let A ∈ S ( H ). Then A has the Strong Spectral Property if � ∩ S cl ( H ) . E A transversal not transversal IEPG: Multiplicities and minors 8/14 Math & Stats, University of Victoria
Theorem (BFHHLS 2017) Let H be a spanning subgraph of G. If A ∈ S ( H ) has the Strong Spectral Property, then there is B ∈ S ( G ) with spec( B ) = spec( A ) . E A S cl ( H ) = S cl 0 ( H , G ) � ∩ S cl ( H ) A ∈ E A IEPG: Multiplicities and minors 9/14 Math & Stats, University of Victoria
Theorem (BFHHLS 2017) Let H be a spanning subgraph of G. If A ∈ S ( H ) has the Strong Spectral Property, then there is B ∈ S ( G ) with spec( B ) = spec( A ) . � ∩ S cl ( G ) B ∈ E A E A S cl ( G ) ⊃ S cl y ( H , G ) S cl ( H ) = S cl 0 ( H , G ) � ∩ S cl ( H ) A ∈ E A IEPG: Multiplicities and minors 9/14 Math & Stats, University of Victoria
Tangent spaces Let A ∈ S ( H ). Then the tangent spaces are Tan E A . A = { K ⊤ A + AK : K skew-symmetric } Tan S cl ( H ) . A = S cl ( H ) . Let Q ( t ) be an orthogonal matrix with Q (0) = I . Then d � dt [ Q ( t ) ⊤ AQ ( t )] t =0 = ˙ Q (0) ⊤ AA ˙ Q (0) . � IEPG: Multiplicities and minors 10/14 Math & Stats, University of Victoria
Tangent spaces Let A ∈ S ( H ). Then the tangent spaces are Tan E A . A = { K ⊤ A + AK : K skew-symmetric } Tan S cl ( H ) . A = S cl ( H ) . Let Q ( t ) be an orthogonal matrix with Q (0) = I . Then d � dt [ Q ( t ) ⊤ AQ ( t )] t =0 = ˙ Q (0) ⊤ AQ (0) + Q (0) ⊤ A ˙ Q (0) . � IEPG: Multiplicities and minors 10/14 Math & Stats, University of Victoria
Tangent spaces Let A ∈ S ( H ). Then the tangent spaces are Tan E A . A = { K ⊤ A + AK : K skew-symmetric } Tan S cl ( H ) . A = S cl ( H ) . Let Q ( t ) be an orthogonal matrix with Q (0) = I . Then d � dt [ Q ( t ) ⊤ AQ ( t )] t =0 = ˙ Q (0) ⊤ AQ (0) + Q (0) ⊤ A ˙ Q (0) . � IEPG: Multiplicities and minors 10/14 Math & Stats, University of Victoria
Tangent spaces Let A ∈ S ( H ). Then the tangent spaces are Tan E A . A = { K ⊤ A + AK : K skew-symmetric } Tan S cl ( H ) . A = S cl ( H ) . Let Q ( t ) be an orthogonal matrix with Q (0) = I . Then d � dt [ Q ( t ) ⊤ AQ ( t )] t =0 = ˙ Q (0) ⊤ A + A ˙ Q (0) . � IEPG: Multiplicities and minors 10/14 Math & Stats, University of Victoria
Tangent spaces Let A ∈ S ( H ). Then the tangent spaces are Tan E A . A = { K ⊤ A + AK : K skew-symmetric } Tan S cl ( H ) . A = S cl ( H ) . Let Q ( t ) be an orthogonal matrix with Q (0) = I . Then d � dt [ Q ( t ) ⊤ AQ ( t )] t =0 = ˙ Q (0) ⊤ A + A ˙ Q (0) . � Q (0) ⊤ Q (0) + Q (0) ⊤ ˙ ˙ Q ( t ) ⊤ Q ( t ) = I = ⇒ Q (0) = 0 , so ˙ Q (0) is skew-symmetric. IEPG: Multiplicities and minors 10/14 Math & Stats, University of Victoria
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