acyclic matrices with a small number of distinct...
TRANSCRIPT
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with a small number of
distinct eigenvalues
Presenter: Xavier Mart́ınez-Rivera
Iowa State University
April 26, 2017
Reshmi Nair, Bryan Shader.Acyclic matrices with a small number of distinct eigenvalues.Linear Algebra and its Applications, 438 (2013), 4075–4089.
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Outline
Basic terminology
Smith normal form technique
Acyclic matrices with few distinct eigenvalues
References
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
The graph of a symmetric matrix
Definition
Let A = [aij ] ∈ Rn×n be symmetric.The graph of A, denoted by G (A),whose vertex and edge set is V and E , respectively,is defined as follows:
1. V = {1, 2, . . . , n};2. For i 6= j in V , ij ∈ E ⇐⇒ aij 6= 0.
Definition
Let G be a simple, n-vertex graph. Then
S(G ) := {A ∈ Rn×n : AT = A and G (A) = G}.
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
The graph of a symmetric matrix
Definition
Let A = [aij ] ∈ Rn×n be symmetric.The graph of A, denoted by G (A),whose vertex and edge set is V and E , respectively,is defined as follows:
1. V = {1, 2, . . . , n};2. For i 6= j in V , ij ∈ E ⇐⇒ aij 6= 0.
Definition
Let G be a simple, n-vertex graph. Then
S(G ) := {A ∈ Rn×n : AT = A and G (A) = G}.
![Page 5: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/5.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
The graph of a symmetric matrix
Definition
Let A = [aij ] ∈ Rn×n be symmetric.The graph of A, denoted by G (A),whose vertex and edge set is V and E , respectively,is defined as follows:
1. V = {1, 2, . . . , n};2. For i 6= j in V , ij ∈ E ⇐⇒ aij 6= 0.
Definition
Let G be a simple, n-vertex graph. Then
S(G ) := {A ∈ Rn×n : AT = A and G (A) = G}.
![Page 6: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/6.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices
Definition
A matrix A is acyclic if it is symmetric and G (A) is a tree.
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Smith normal form technique
In 2009, Kim and Shader introduced a technique for studyingthe multiplicities of the eigenvalues of an acyclic matrix Abased on the Smith normal form of the matrix xI − A.
![Page 8: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/8.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Definition
Let B ∈ (R[x ])n×n.Then ∆k(B) is defined to bethe monic gcd of all k × k minors of B .
Notation: q(A) denotes the number of distinct eigenvalues ofa matrix A.
Theorem (Kim & Shader; 2009)
Let A ∈ Rn×n be symmetric and let B = xI − A.Then q(A) = n − degree
(∆n−1(B)
).
![Page 9: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/9.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Definition
Let B ∈ (R[x ])n×n.Then ∆k(B) is defined to bethe monic gcd of all k × k minors of B .
Notation: q(A) denotes the number of distinct eigenvalues ofa matrix A.
Theorem (Kim & Shader; 2009)
Let A ∈ Rn×n be symmetric and let B = xI − A.Then q(A) = n − degree
(∆n−1(B)
).
![Page 10: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/10.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Definition
Let B ∈ (R[x ])n×n.Then ∆k(B) is defined to bethe monic gcd of all k × k minors of B .
Notation: q(A) denotes the number of distinct eigenvalues ofa matrix A.
Theorem (Kim & Shader; 2009)
Let A ∈ Rn×n be symmetric and let B = xI − A.Then q(A) = n − degree
(∆n−1(B)
).
![Page 11: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/11.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Notation: Given A ∈ Rn×n and X ⊂ {1, . . . , n},A(X ) denotes the submatrix of A resulting fromdeleting the rows and columns indexed by X .
Theorem (Kim & Shader; 2009)
Let T be a tree, let A ∈ S(T ) and let B = xI − A;let G1, . . . ,Gk be vertex-disjoint paths, whereV := V (G1) ∪ · · · ∪ V (Gn) contains t vertices.Then
1. ∆n−k(B) divides det(B(V )
);
2. degree(∆n−k(B)) ≤ n − t.
![Page 12: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/12.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Notation: Given A ∈ Rn×n and X ⊂ {1, . . . , n},A(X ) denotes the submatrix of A resulting fromdeleting the rows and columns indexed by X .
Theorem (Kim & Shader; 2009)
Let T be a tree, let A ∈ S(T ) and let B = xI − A;let G1, . . . ,Gk be vertex-disjoint paths, whereV := V (G1) ∪ · · · ∪ V (Gn) contains t vertices.Then
1. ∆n−k(B) divides det(B(V )
);
2. degree(∆n−k(B)) ≤ n − t.
![Page 13: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/13.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Notation: Given A ∈ Rn×n and X ⊂ {1, . . . , n},A(X ) denotes the submatrix of A resulting fromdeleting the rows and columns indexed by X .
Theorem (Kim & Shader; 2009)
Let T be a tree, let A ∈ S(T ) and let B = xI − A;let G1, . . . ,Gk be vertex-disjoint paths, whereV := V (G1) ∪ · · · ∪ V (Gn) contains t vertices.Then
1. ∆n−k(B) divides det(B(V )
);
2. degree(∆n−k(B)) ≤ n − t.
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
A well-known fact
Theorem
Let T be a tree and let A ∈ S(T ).Then q(A) ≥ diam(T ) + 1.
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Another well-known fact
Proposition
If A ∈ S(Pn), then q(A) = n.
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Notation: mA(λ) denotes the multiplicity of λ as aneigenvalue of A.
Lemma (Johnson, Duarte, Saiago; 2013)
Let T be a tree and let A ∈ S(T ).Then mA(λmax) = 1 = mA(λmin).
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Notation: mA(λ) denotes the multiplicity of λ as aneigenvalue of A.
Lemma (Johnson, Duarte, Saiago; 2013)
Let T be a tree and let A ∈ S(T ).Then mA(λmax) = 1 = mA(λmin).
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 2 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 2 ⇐⇒ G (A) = K2.
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Type-I matrices
Definition (Nair & Shader; 2013)
Let n ≥ 4. The matrix A ∈ Rn×n is a type-I matrix if it issimilar via a permutation matrix to a symmetric matrix of theform
? ∗ ∗ · · · ∗∗ β∗ β...
. . .
∗ β
.
Observation (Nair & Shader; 2013)
The graph of a type-I matrix is a star (which is a tree).
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Type-I matrices
Definition (Nair & Shader; 2013)
Let n ≥ 4. The matrix A ∈ Rn×n is a type-I matrix if it issimilar via a permutation matrix to a symmetric matrix of theform
? ∗ ∗ · · · ∗∗ β∗ β...
. . .
∗ β
.
Observation (Nair & Shader; 2013)
The graph of a type-I matrix is a star (which is a tree).
![Page 21: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/21.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Remark (Nair & Shader; 2013)
Let A be an n × n type-I matrix. Then
1. q(A) = 3;
2. A has an eigenvalue of multiplicity n − 2.
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Another well-known fact
Remark
Let A ∈ Rn×n be symmetric with λ as an eigenvalue.Then one of the following holds for all j ∈ {1, . . . , n}:
1. mA({j})(λ) = mA(λ)− 1;
2. mA({j})(λ) = mA(λ);
3. mA({j})(λ) = mA(λ) + 1.
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
The Parter-Wiener Theorem
Theorem (P-W Theorem)
Let T be a tree on n vertices.Let A ∈ S(T ), and let λ be an eigenvalue with mA(λ) ≥ 2.Then there is a vertex j such that mA({j})(λ) = mA(λ) + 1.
j is called a Parter vertex for λ.
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
The Parter-Wiener Theorem
Theorem (P-W Theorem)
Let T be a tree on n vertices.Let A ∈ S(T ), and let λ be an eigenvalue with mA(λ) ≥ 2.Then there is a vertex j such that mA({j})(λ) = mA(λ) + 1.
j is called a Parter vertex for λ.
![Page 25: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/25.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.
Proof:
• Let A be an n × n acyclic matrix.
• =⇒ A ∈ S(T ) for some tree T .
• Suppose q(A) = 3.
• =⇒ n ≥ 3.
• Let α < β < γ be the distinct eigenvalues of A.
• Recall that q(A) ≥ diam(T ) + 1.
• =⇒ diam(T ) ≤ 2.
• What trees on n ≥ 3 vertices have diameter 2?
• Answer: Stars and P3.
![Page 26: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/26.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.
Proof:
• Let A be an n × n acyclic matrix.
• =⇒ A ∈ S(T ) for some tree T .
• Suppose q(A) = 3.
• =⇒ n ≥ 3.
• Let α < β < γ be the distinct eigenvalues of A.
• Recall that q(A) ≥ diam(T ) + 1.
• =⇒ diam(T ) ≤ 2.
• What trees on n ≥ 3 vertices have diameter 2?
• Answer: Stars and P3.
![Page 27: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/27.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.
Proof:
• Let A be an n × n acyclic matrix.
• =⇒ A ∈ S(T ) for some tree T .
• Suppose q(A) = 3.
• =⇒ n ≥ 3.
• Let α < β < γ be the distinct eigenvalues of A.
• Recall that q(A) ≥ diam(T ) + 1.
• =⇒ diam(T ) ≤ 2.
• What trees on n ≥ 3 vertices have diameter 2?
• Answer: Stars and P3.
![Page 28: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/28.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.
Proof:
• Let A be an n × n acyclic matrix.
• =⇒ A ∈ S(T ) for some tree T .
• Suppose q(A) = 3.
• =⇒ n ≥ 3.
• Let α < β < γ be the distinct eigenvalues of A.
• Recall that q(A) ≥ diam(T ) + 1.
• =⇒ diam(T ) ≤ 2.
• What trees on n ≥ 3 vertices have diameter 2?
• Answer: Stars and P3.
![Page 29: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/29.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.
Proof:
• Let A be an n × n acyclic matrix.
• =⇒ A ∈ S(T ) for some tree T .
• Suppose q(A) = 3.
• =⇒ n ≥ 3.
• Let α < β < γ be the distinct eigenvalues of A.
• Recall that q(A) ≥ diam(T ) + 1.
• =⇒ diam(T ) ≤ 2.
• What trees on n ≥ 3 vertices have diameter 2?
• Answer: Stars and P3.
![Page 30: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/30.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.
Proof:
• Let A be an n × n acyclic matrix.
• =⇒ A ∈ S(T ) for some tree T .
• Suppose q(A) = 3.
• =⇒ n ≥ 3.
• Let α < β < γ be the distinct eigenvalues of A.
• Recall that q(A) ≥ diam(T ) + 1.
• =⇒ diam(T ) ≤ 2.
• What trees on n ≥ 3 vertices have diameter 2?
• Answer: Stars and P3.
![Page 31: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/31.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.
Proof:
• Let A be an n × n acyclic matrix.
• =⇒ A ∈ S(T ) for some tree T .
• Suppose q(A) = 3.
• =⇒ n ≥ 3.
• Let α < β < γ be the distinct eigenvalues of A.
• Recall that q(A) ≥ diam(T ) + 1.
• =⇒ diam(T ) ≤ 2.
• What trees on n ≥ 3 vertices have diameter 2?
• Answer: Stars and P3.
![Page 32: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/32.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.
Proof:
• Let A be an n × n acyclic matrix.
• =⇒ A ∈ S(T ) for some tree T .
• Suppose q(A) = 3.
• =⇒ n ≥ 3.
• Let α < β < γ be the distinct eigenvalues of A.
• Recall that q(A) ≥ diam(T ) + 1.
• =⇒ diam(T ) ≤ 2.
• What trees on n ≥ 3 vertices have diameter 2?
• Answer: Stars and P3.
![Page 33: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/33.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.
Proof:
• Let A be an n × n acyclic matrix.
• =⇒ A ∈ S(T ) for some tree T .
• Suppose q(A) = 3.
• =⇒ n ≥ 3.
• Let α < β < γ be the distinct eigenvalues of A.
• Recall that q(A) ≥ diam(T ) + 1.
• =⇒ diam(T ) ≤ 2.
• What trees on n ≥ 3 vertices have diameter 2?
• Answer: Stars and P3.
![Page 34: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/34.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Acyclic matrices with exactly 3 distinct eigenvalues
Theorem (Nair & Shader; 2013)
Let A be an acyclic matrix. Thenq(A) = 3 ⇐⇒ A is a type-I matrix or G (A) = P3.
Proof:
• Let A be an n × n acyclic matrix.
• =⇒ A ∈ S(T ) for some tree T .
• Suppose q(A) = 3.
• =⇒ n ≥ 3.
• Let α < β < γ be the distinct eigenvalues of A.
• Recall that q(A) ≥ diam(T ) + 1.
• =⇒ diam(T ) ≤ 2.
• What trees on n ≥ 3 vertices have diameter 2?
• Answer: Stars and P3.
![Page 35: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/35.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• If T = P3, then we are done.
• So, suppose T is a star on (n ≥ 4 vertices).
• Previous lemma =⇒ mA(α) = 1 = mA(γ).
• =⇒ mA(β) = n − 2.
• =⇒ mA(β) ≥ 2.
• The Parter-Wiener Theorem implies that T contains aParter vertex for β.
• Let j be a Parter vertex for β.
• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.
• Which vertex of the star could j be?
• Answer: it must be the central vertex. Why?
![Page 36: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/36.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• If T = P3, then we are done.
• So, suppose T is a star on (n ≥ 4 vertices).
• Previous lemma =⇒ mA(α) = 1 = mA(γ).
• =⇒ mA(β) = n − 2.
• =⇒ mA(β) ≥ 2.
• The Parter-Wiener Theorem implies that T contains aParter vertex for β.
• Let j be a Parter vertex for β.
• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.
• Which vertex of the star could j be?
• Answer: it must be the central vertex. Why?
![Page 37: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/37.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• If T = P3, then we are done.
• So, suppose T is a star on (n ≥ 4 vertices).
• Previous lemma =⇒ mA(α) = 1 = mA(γ).
• =⇒ mA(β) = n − 2.
• =⇒ mA(β) ≥ 2.
• The Parter-Wiener Theorem implies that T contains aParter vertex for β.
• Let j be a Parter vertex for β.
• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.
• Which vertex of the star could j be?
• Answer: it must be the central vertex. Why?
![Page 38: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/38.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• If T = P3, then we are done.
• So, suppose T is a star on (n ≥ 4 vertices).
• Previous lemma =⇒ mA(α) = 1 = mA(γ).
• =⇒ mA(β) = n − 2.
• =⇒ mA(β) ≥ 2.
• The Parter-Wiener Theorem implies that T contains aParter vertex for β.
• Let j be a Parter vertex for β.
• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.
• Which vertex of the star could j be?
• Answer: it must be the central vertex. Why?
![Page 39: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/39.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• If T = P3, then we are done.
• So, suppose T is a star on (n ≥ 4 vertices).
• Previous lemma =⇒ mA(α) = 1 = mA(γ).
• =⇒ mA(β) = n − 2.
• =⇒ mA(β) ≥ 2.
• The Parter-Wiener Theorem implies that T contains aParter vertex for β.
• Let j be a Parter vertex for β.
• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.
• Which vertex of the star could j be?
• Answer: it must be the central vertex. Why?
![Page 40: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/40.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• If T = P3, then we are done.
• So, suppose T is a star on (n ≥ 4 vertices).
• Previous lemma =⇒ mA(α) = 1 = mA(γ).
• =⇒ mA(β) = n − 2.
• =⇒ mA(β) ≥ 2.
• The Parter-Wiener Theorem implies that T contains aParter vertex for β.
• Let j be a Parter vertex for β.
• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.
• Which vertex of the star could j be?
• Answer: it must be the central vertex. Why?
![Page 41: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/41.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• If T = P3, then we are done.
• So, suppose T is a star on (n ≥ 4 vertices).
• Previous lemma =⇒ mA(α) = 1 = mA(γ).
• =⇒ mA(β) = n − 2.
• =⇒ mA(β) ≥ 2.
• The Parter-Wiener Theorem implies that T contains aParter vertex for β.
• Let j be a Parter vertex for β.
• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.
• Which vertex of the star could j be?
• Answer: it must be the central vertex. Why?
![Page 42: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/42.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• If T = P3, then we are done.
• So, suppose T is a star on (n ≥ 4 vertices).
• Previous lemma =⇒ mA(α) = 1 = mA(γ).
• =⇒ mA(β) = n − 2.
• =⇒ mA(β) ≥ 2.
• The Parter-Wiener Theorem implies that T contains aParter vertex for β.
• Let j be a Parter vertex for β.
• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.
• Which vertex of the star could j be?
• Answer: it must be the central vertex. Why?
![Page 43: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/43.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• If T = P3, then we are done.
• So, suppose T is a star on (n ≥ 4 vertices).
• Previous lemma =⇒ mA(α) = 1 = mA(γ).
• =⇒ mA(β) = n − 2.
• =⇒ mA(β) ≥ 2.
• The Parter-Wiener Theorem implies that T contains aParter vertex for β.
• Let j be a Parter vertex for β.
• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.
• Which vertex of the star could j be?
• Answer: it must be the central vertex. Why?
![Page 44: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/44.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• If T = P3, then we are done.
• So, suppose T is a star on (n ≥ 4 vertices).
• Previous lemma =⇒ mA(α) = 1 = mA(γ).
• =⇒ mA(β) = n − 2.
• =⇒ mA(β) ≥ 2.
• The Parter-Wiener Theorem implies that T contains aParter vertex for β.
• Let j be a Parter vertex for β.
• =⇒ mA({j})(β) = mA(β) + 1 = (n − 2) + 1 = n − 1.
• Which vertex of the star could j be?
• Answer: it must be the central vertex. Why?
![Page 45: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/45.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• ? ∗ ∗ · · · ∗∗ ?∗ ?...
. . .
∗ ?
.
• ? ∗ ∗ · · · ∗∗ β∗ β...
. . .
∗ β
.
![Page 46: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/46.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
• ? ∗ ∗ · · · ∗∗ ?∗ ?...
. . .
∗ ?
.•
? ∗ ∗ · · · ∗∗ β∗ β...
. . .
∗ β
.
![Page 47: Acyclic matrices with a small number of distinct …orion.math.iastate.edu/butler/2017/spring/x95/...Reshmi Nair, Bryan Shader. Acyclic matrices with a small number of distinct eigenvalues](https://reader034.vdocuments.site/reader034/viewer/2022051802/5aec45297f8b9ac3619040be/html5/thumbnails/47.jpg)
Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
Fin
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Basic terminology Smith normal form technique Acyclic matrices with few distinct eigenvalues References
I.-J. Kim, B. L. Shader.Smith Normal Form and acyclic matrices.J. Algebraic Combin. 29 (2009), 63–80.
R. Nair, B. L. Shader.Acyclic matrices with a small number of distincteigenvalues.Linear Algebra Appl. 438 (2013), 4075–4089.
C. R. Johnson, A. Leal Duarte, C. M. Saiago.The Parter Wiener theorem: refinement andgeneralization.SIAM J. Matrix Anal. Appl. 25 (2003), 352–361.