Determinant of Laplacian removing $2$ (or more) rows and columns
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Based on Kirchoff's theorem, I'm wondering what can be said about the determinant of the Laplacian, after removing the columns and rows corresponding to two vertices. The motivation would be to see if iterating this process would give insight on the factorization or growth of the number of spanning trees as a function of the vertices.
I would appreciate any references to the literature! Thank you.
linear-algebra combinatorics graph-theory graph-laplacian
edited Nov 18 at 19:52
Rócherz
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asked Jul 3 at 2:31
emilia_alv
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up vote
1
down vote
favorite
Based on Kirchoff's theorem, I'm wondering what can be said about the determinant of the Laplacian, after removing the columns and rows corresponding to two vertices. The motivation would be to see if iterating this process would give insight on the factorization or growth of the number of spanning trees as a function of the vertices.
I would appreciate any references to the literature! Thank you.
linear-algebra combinatorics graph-theory graph-laplacian
edited Nov 18 at 19:52
Rócherz
2,7062721
asked Jul 3 at 2:31
emilia_alv
61
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Based on Kirchoff's theorem, I'm wondering what can be said about the determinant of the Laplacian, after removing the columns and rows corresponding to two vertices. The motivation would be to see if iterating this process would give insight on the factorization or growth of the number of spanning trees as a function of the vertices.
I would appreciate any references to the literature! Thank you.
linear-algebra combinatorics graph-theory graph-laplacian
edited Nov 18 at 19:52
Rócherz
2,7062721
asked Jul 3 at 2:31
emilia_alv
61
Based on Kirchoff's theorem, I'm wondering what can be said about the determinant of the Laplacian, after removing the columns and rows corresponding to two vertices. The motivation would be to see if iterating this process would give insight on the factorization or growth of the number of spanning trees as a function of the vertices.
I would appreciate any references to the literature! Thank you.
linear-algebra combinatorics graph-theory graph-laplacian
linear-algebra combinatorics graph-theory graph-laplacian
edited Nov 18 at 19:52
Rócherz
2,7062721
asked Jul 3 at 2:31
emilia_alv
61
edited Nov 18 at 19:52
Rócherz
2,7062721
asked Jul 3 at 2:31
emilia_alv
61
edited Nov 18 at 19:52
Rócherz
2,7062721
edited Nov 18 at 19:52
Rócherz
2,7062721
edited Nov 18 at 19:52
Rócherz
2,7062721
2,7062721
asked Jul 3 at 2:31
emilia_alv
61
asked Jul 3 at 2:31
emilia_alv
61
asked Jul 3 at 2:31
emilia_alv
61
61
add a comment |
add a comment |
1 Answer
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Induction over the number of vertices whose corresponding rows and columns you are removing from the Laplacian matrix (hence removing those vertices from the graph): whatever is left from the original Laplacian matrix is guaranteed to be positive definite, if and only if at least one of the remaining vertices was connected to any of the vertices you removed.
answered Aug 5 at 17:10
Rócherz
2,7062721
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
Induction over the number of vertices whose corresponding rows and columns you are removing from the Laplacian matrix (hence removing those vertices from the graph): whatever is left from the original Laplacian matrix is guaranteed to be positive definite, if and only if at least one of the remaining vertices was connected to any of the vertices you removed.
answered Aug 5 at 17:10
Rócherz
2,7062721
add a comment |
up vote
0
down vote
Induction over the number of vertices whose corresponding rows and columns you are removing from the Laplacian matrix (hence removing those vertices from the graph): whatever is left from the original Laplacian matrix is guaranteed to be positive definite, if and only if at least one of the remaining vertices was connected to any of the vertices you removed.
answered Aug 5 at 17:10
Rócherz
2,7062721
add a comment |
up vote
0
down vote
up vote
0
down vote
Induction over the number of vertices whose corresponding rows and columns you are removing from the Laplacian matrix (hence removing those vertices from the graph): whatever is left from the original Laplacian matrix is guaranteed to be positive definite, if and only if at least one of the remaining vertices was connected to any of the vertices you removed.
answered Aug 5 at 17:10
Rócherz
2,7062721
Induction over the number of vertices whose corresponding rows and columns you are removing from the Laplacian matrix (hence removing those vertices from the graph): whatever is left from the original Laplacian matrix is guaranteed to be positive definite, if and only if at least one of the remaining vertices was connected to any of the vertices you removed.
answered Aug 5 at 17:10
Rócherz
2,7062721
edited Nov 18 at 19:50
answered Aug 5 at 17:10
Rócherz
2,7062721
answered Aug 5 at 17:10
Rócherz
2,7062721
answered Aug 5 at 17:10
Rócherz
2,7062721
2,7062721
add a comment |
add a comment |
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