Total Unimodular Matrices
and Graphs
Sharon Hutton and Eric Staron
Advisors: Charles R. Johnson
Aneta Sawikowska
October 16, 2005
1
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Total Unimodular Matrices
Definition
A matrix is totally unimodular if every minor is in the set { 1, 0, 1}. Definition
Let A be mbyn matrix. The negation of any number of entries of A is called a signing of A. 2
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Lemma
If A is totally unimodular, then the transpose of A is also totally unimodular. Lemma
If A is a totally unimodular matrix,
and A'
is obtained by negating row i of A, then A'
is totally unimodular. 3
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Corollary
If A is totally unimodular, and A'
is obtained by negating column i of A, then A'
is totally unimodular .
Corollary
If A is totally unimodular, and A'
is obtained by the negation of an arbitrary number of rows and columns of A, then A'
is totally unimodular. 4
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Lemma
If A is a totally unimodular matrix, and A'
is formed by permuting rows and columns of A, then A'
is totally unimodular.
Theorem
Let A be a matrix where row i has at most one nonzero entry. Let A'
be the matrix A with row i removed. Then A is totally unimodular iff A'
is totally unimodular.
Corollary
Rows and columns composed of at most one nonzero entry can be deleted from a matrix when determining total unimodularity.
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Theorem
Let A be a matrix of the form
C 0
0 C
or
0 D
D 0
where C and D are totally unimodular matrices, not necessarily square. Then A is totally unimodular.
Corollary
If B is a totally unimodular matrix, then
0
T
B
B
0
is totally unimodular.
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Definition
Let A = A
A12
,
A22
in which A11 is nonsingular. A
The Schur Complement of A with respect to A11, denoted by [A/A11], is equal to
11
21
−1
A22 − A21 A11 A12
Theorem
If A is a totally unimodular matrix, and [A/A(i,i)] is the Schur Complement of A with respect to A(i,i), where i∈{1,2,... ,n} then [A/A(i,i)] is totally unimodular.
Proof is based on equality
det( A) = det( A11 ) det([ A / A11 ]).
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Total Unimodular Graphs
Definition
Let G be an undirected graph on n vertices. The adjacency matrix A of graph G is a n × n matrix such that the nondiagonal entry aij is the number of edges joining vertex i and vertex j. Each diagonal entry aii is twice the number of loops on vertex i. 8
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Definition
A graph is totally unimodular if there exists
totally unimodular signing of its adjacency
matrix. Theorem
The total unimodularity of a graph is independent of its vertex labeling. 9
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Definition
An induced subgraph is a subset of the vertices of a graph together with any edges whose endpoints are both in this subset. Theorem
Let G be a graph, and G'
be an induced subgraph of G. If G is totally unimodular, G'
is totally unimodular. 10
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Table of graphs which are not TU
4 vertices: 5 vertices:
6 vertices:
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Theorem
Let G be a graph on n vertices with a vertex of degree one, and let G'
be the graph of G with the degree one vertex and corresponding edge removed. Then G is totally unimodular iff G'
is totally unimodular. Corollary
All trees are totally unimodular. Furthermore,
any signing of the adjacency matrix of tree is
totally unimodular. 12
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Definition
Let G be a graph and T be a tree. If a new graph G is formed where a vertex of G is identified with a degree one vertex of T, then T it called a branch of G. Corollary
Let G'
be a graph which contains branches. Let G be the graph of G'
with the branches removed. Then G is totally unimodular iff G'
is totally unimodular. 13
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Definition
A graph G is called bipartite if the vertices of G can be decomposed into two disjoint sets such that no two vertices in a given set are connected by an edge. Remark
The adjacency matrices of bipartite graphs have the general form 0 B
T
0 B
Therefore, bipartite graph is totally unimodular
iff matrix B is totally unimodular.
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Theorem
If G is a 4ncycle, n = 1, 2, … , then G is totally unimodular. Furthermore, G is totally unimodular
without changing the signs in adjacency matrix. Theorem
Let G be a 2ncycle where 2n is not divisible by 4. Then G is totally unimodular, but requires an adjacency matrix signing with two minus signs. Theorem
Let G be an odd cycle. Then G is totally unimodular, and its adjacency matrix requires one negative 1 in order to be totally unimodular. 15
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The Schur Complement and TU Graphs
Definition
Let G be a connected graph, and let edge eij∈ G. Edge eij is said to be a cut edge if G becomes disconnected when eij is removed. Definition
An edge eij is a neighbors not connected (NNC) edge if the neighbors of vi and vj are not connected by an edge. This is equivalent to saying eij is not part of any 4cycle. An edge eij is said to be a no common vertex (NCV) edge if vi and vj are not connected to a common vertex. This is equivalent to eij not being part of any 3cycle.
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An edge which is both NNC and NCV 02/08/06
Remark
All cut edges are squeezable edges. Definition
Let G be a graph, vi, vj, eij ∈ G. If eij is a squeezable edge, and A is the adjacency matrix of G, then the squeeze of G on eij, denoted Sij(G), is the graph of the adjacency matrix obtained by taking the Schur Complement with respect to edge e ij . 17
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Theorem
If G is a totally unimodular graph on n vertices, then Sij(G) is totally unimodular. Theorem
Let G be a graph on n vertices and eij ∈ G be a cut edge. If Sij(G) is totally unimodular, the G is totally unimodular.
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Example
5
1
G:
2
1
3
4
i
j
6
0
0
1
A=
0
0
0
0
0
1
0
0
0
1
1
0
1
0
0
0
0
1
0
1
1
0
0
0
1
0
1
0
0
0
1
1
0
5 (3)
−1
[ A / A(i, j )] = A22 − A21 A11 A12
Sij(G):
2
6 (4)
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Example
0
1
1
A=
1
0
0
5
3
G:
4
3 (1)
1
2
i
j
6
5 (3)
Sij(G):
4 (2)
6 (4)
1
0
0
0
1
1
1
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
1
0
1
0
0
1
0
−1
[ A / A(i, j )] = A22 − A21 A11 A12 =
0
0
=
0
0
0
0
=
1
1
0 1
0 1
−
0 0 −1 0
0 1 0 0
0 0
0 0
1 1
0 1 1
1 0 1
1 1 0
0
−1
0 0 1 1 1 0 0
=
1 1 0 0 0 1 1
1
0
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References
R. Horm, C. R. Johnson, Matrix Analysis, Cambridge University Press, New York, 1985
R. C. Read, R. J. Wilson, An Atlas of Graphs, Clarendon Press, Oxford, 1998
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