Graphs and Matrices

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R.B. Bapat

Graphs and Matrices

ABC

Sheldon Axler, San Francisco State University Carles Casacuberta, Universitat de Barcelona Angus MacIntyre, Queen Mary, University of London Kenneth Ribet, University of California, Berkeley Endre Süli, University of Oxford

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Preface

This book is concerned with results in graph theory in which linear algebra and matrix theory play an important role. Although it is generally accepted that linear algebra can be an important component in the study of graphs, traditionally, graph theorists have remained by and large less than enthusiastic about using linear alge- bra. The results discussed here are usually treated under algebraic graph theory, as outlined in the classic books by Biggs [20] and by Godsil and Royle [39]. Our em- phasis on matrix techniques is even greater than what is found in these and perhaps the subject matter discussed here might be termed linear algebraic graph theory to highlight this aspect.

After recalling some matrix preliminaries in the first chapter, the next few chap- ters outline the basic properties of some matrices associated with a graph. This is followed by topics in graph theory such as regular graphs and algebraic connectiv- ity. Distance matrix of a tree and its generalized version for arbitrary graphs, the resistance matrix, are treated in the next two chapters. The final chapters treat other topics such as the Laplacian eigenvalues of threshold graphs, the positive definite completion problem and matrix games based on a graph.

We have kept the treatment at a fairly elementary level and resisted the temptation of presenting up to date research work. Thus several chapters in this book may be viewed as an invitation to a vast area of vigorous current research. Only a beginning is made here with the hope that it will entice the reader to explore further. In the same vein, we often do not present the results in their full generality, but present only a simpler version that captures the elegance of the result. Weighted graphs are avoided, although most results presented here have weighted, and hence more general, analogs.

The references for each chapter are listed at the end of the chapter. In addition, a master bibliography is included. In a short note at the end of each chapter we indicate the primary references that we used. Often, we have given a different treat- ment, as well as different proofs, of the results cited. We do not go into an elaborate description of such differences.

It is a pleasure to thank Rajendra Bhatia for his diligent handling of the manuscript. Aloke Dey, Arbind Lal, Sukanta Pati, Sharad Sane, S. Sivaramakrishnan

v

and Murali Srinivasan read either all or parts of the manuscript, suggested changes and pointed out corrections. I sincerely thank them all. Thanks are also due to the anonymous referees for helpful comments. Needless to say I remain responsible for the shortcomings and errors that persist. The facilities provided by the Indian Statis- tical Institute, New Delhi, and the support of the JC Bose Fellowship, Department of Science and Technology, Government of India, are gratefully acknowledged.

Ravindra Bapat

New Delhi, India

Contents

Preface . . . . v

1 Preliminaries . . . . 1

1.1 Matrices . . . . 1

1.2 Eigenvalues of symmetric matrices . . . . 5

1.3 Generalized inverses . . . . 7

1.4 Graphs . . . . 9

2 Incidence Matrix . . . 11

2.1 Rank . . . 12

2.2 Minors . . . 13

2.3 Path matrix . . . 15

2.4 Integer generalized inverses . . . 16

2.5 Moore–Penrose inverse . . . 17

2.6 0 − 1 Incidence matrix . . . 19

2.7 Matchings in bipartite graphs . . . 21

3 Adjacency Matrix . . . 25

3.1 Eigenvalues of some graphs . . . 26

3.2 Determinant . . . 28

3.3 Bounds . . . 31

3.4 Energy of a graph . . . 36

3.5 Antiadjacency matrix of a directed graph . . . 37

3.6 Nonsingular trees . . . 39

4 Laplacian Matrix . . . 45

4.1 Basic properties . . . 46

4.2 Computing Laplacian eigenvalues . . . 47

4.3 Matrix-tree theorem . . . 48

4.4 Bounds for Laplacian spectral radius . . . 50

4.5 Edge–Laplacian of a tree . . . 51

vii

5 Cycles and Cuts . . . 57

5.1 Fundamental cycles and fundamental cuts . . . 57

5.2 Fundamental matrices . . . 59

5.3 Minors . . . 60

6 Regular Graphs . . . 65

6.1 Perron–Frobenius theory . . . 65

6.2 Adjacency algebra of a regular graph . . . 70

6.3 Complement and line graph of a regular graph . . . 70

6.4 Strongly regular graphs and friendship theorem . . . 73

6.5 Graphs with maximum energy . . . 76

7 Algebraic Connectivity . . . 81

7.1 Preliminary results . . . 81

7.2 Classification of trees . . . 83

7.3 Monotonicity properties of Fiedler vector . . . 88

7.4 Bounds for algebraic connectivity . . . 89

8 Distance Matrix of a Tree . . . 95

8.1 Distance matrix of a graph . . . 96

8.2 Distance matrix and Laplacian of a tree . . . 99

8.3 Eigenvalues of the distance matrix of a tree . . . 104

9 Resistance Distance . . . 111

9.1 The triangle inequality . . . 112

9.2 Network flows . . . 113

9.3 A random walk on graphs . . . 116

9.4 Effective resistance in electrical networks . . . 118

9.5 Resistance matrix . . . 119

10 Laplacian Eigenvalues of Threshold Graphs . . . 125

10.1 Majorization . . . 125

10.2 Threshold graphs . . . 129

10.3 Spectral integral variation . . . 131

11 Positive Definite Completion Problem . . . 137

11.1 Nonsingular completion . . . 137

11.2 Chordal graphs . . . 138

11.3 Positive definite completion . . . 140

12 Matrix Games Based on Graphs . . . 145

12.1 Matrix games . . . 145

12.2 Vertex selection games . . . 147

12.3 Tournament games . . . 148

12.4 Incidence matrix games . . . 151

Contents ix

Hints and Solutions to Selected Exercises . . . 159

Bibliography . . . 165

Index . . . 169

Chapter 1

Preliminaries

In this chapter we review certain basic concepts from linear algebra. We consider only real matrices. Although our treatment is self-contained, the reader is assumed to be familiar with the basic operations on matrices. Relevant concepts and results are given, although we omit proofs.

1.1 Matrices

Basic definitions

An m × n matrix consists of mn real numbers arranged in m rows and n columns.

The entry in row i and column j of the matrix A is denoted by a

. An m × 1 matrix is called a column vector of order m; similarly, a 1 × n matrix is a row vector of order n. An m × n matrix is called a square matrix if m = n.

Operations of matrix addition, scalar multiplication and matrix multiplication are basic and will not be recalled here. The transpose of the m × n matrix A is denoted by A

.

A diagonal matrix is a square matrix A such that a

= 0, i 6= j. We denote the diagonal matrix











λ

0 · · · 0 0 λ

· · · 0 .. . .. . . . . .. . 0 0 · · · λ











by diag(λ

, . . . , λ

). When λ

= 1 for all i, this matrix reduces to the identity matrix of order n, which we denote by I

or often simply by I if the order is clear from the context. The matrix A is upper triangular if a

= 0, i > j. The transpose of an upper triangular matrix is lower triangular.

1

Trace and determinant

Let A be a square matrix of order n. The entries a

, . . . , a

are said to constitute the (main) diagonal of A. The trace of A is defined as

trace A = a

+ · · · + a

.

It follows from this definition that if A, B are matrices such that both AB and BA are defined, then

trace AB = trace BA.

The determinant of an n × n matrix A, denoted by det A, is defined as det A = ∑

σ

sgn(σ )a

· · · a

,

where the summation is over all permutations σ (1), . . . , σ (n) of 1, . . . , n, and sgn(σ ) is 1 or −1 according as σ is even or odd. We assume familiarity with the basic properties of determinant.

Vector spaces associated with a matrix

Let IR denote the set of real numbers. Consider the set of all column vectors of order n (n × 1 matrices) and the set of all row vectors of order n (1 × n matrices).

Both of these sets will be denoted by IR

. We will write the elements of IR

either as column vectors or as row vectors, depending upon whichever is convenient in a given situation. Recall that IR

is a vector space with the operations matrix addition and scalar multiplication.

Let A be an m × n matrix. The subspace of IR

spanned by the column vectors of A is called the column space or the column span of A. Similarly the subspace of IR

spanned by the row vectors of A is called the row space of A.

According to the fundamental theorem of linear algebra, the dimension of the column space of a matrix equals the dimension of the row space, and the common value is called the rank of the matrix. We denote the rank of the matrix A by rank A.

For any matrix A, rank A = rank A

. If A and B are matrices of the same order, then rank(A + B) ≤ rank A + rank B. If A and B are matrices such that AB is defined, then rank AB ≤ min{rank A, rank B}.

Let A be an m × n matrix. The set of all vectors x ∈ IR

such that Ax = 0 is easily seen to be a subspace of IR

. This subspace is called the null space of A, and we denote it by N (A). The dimension of N (A) is called the nullity of A. Let A be an m × n matrix. Then the nullity of A equals n − rank A.

Minors

Let A be an m × n matrix. If S ⊂ {1, . . . , m}, T ⊂ {1, . . . , n}, then A[S|T ] will denote

the submatrix of A determined by the rows corresponding to S and the columns cor-

responding to T. The submatrix obtained by deleting the rows in S and the columns

1.1 Matrices 3

in T will be denoted by A(S|T ). Thus, A(S|T ) = A[S

|T

], where the superscript c denotes complement. Often, we tacitly assume that S and T are such that these ma- trices are not vacuous. When S = {i}, T = { j} are singletons, then A(S|T ) is denoted A(i| j).

Nonsingular matrices

A matrix A of order n × n is said to be nonsingular if rank A = n; otherwise the matrix is singular. If A is nonsingular, then there is a unique n × n matrix A

, called the inverse of A, such that AA

= A

A = I. A matrix is nonsingular if and only if det A is nonzero.

The cofactor of a

is defined as (−1)

det A(i| j). The adjoint of A is the n × n matrix whose (i, j)th entry is the cofactor of a

. We recall that if A is nonsingular, then A

is given by 1

det A times the adjoint of A.

A matrix is said to have full column rank if its rank equals the number of columns, or equivalently, the columns are linearly independent. Similarly, a matrix has full row rank if its rows are linearly independent. If B has full column rank, then it admits a left inverse, that is, a matrix X such that X B = I. Similarly, if C has full row rank, then it has a right inverse, that is, a matrix Y such that CY = I.

If A is an m × n matrix of rank r then we can write A = BC, where B is m × r of full column rank and C is r × n of full row rank. This is called a rank factorization of A. There exist nonsingular matrices P and Q of order m × m and n × n, respectively, such that

A = P  I

0 0 0

 Q.

This is the rank canonical form of A.

Orthogonality

Vectors x, y in IR

are said to be orthogonal, or perpendicular, if x

y = 0. A set of vectors {x

, . . . , x

} in IR

is said to form an orthonormal basis for the vector space S if the set is a basis for S, and furthermore x

x

is 0 if i 6= j, and 1 if i = j. The n × n matrix P is said to be orthogonal if PP

= P

P = I. One can verify that if P is orthogonal then P

is orthogonal.

If x

, . . . , x

are linearly independent vectors then by the Gram–Schmidt orthog- onalization process we may construct orthonormal vectors y

, . . . , y

such that y

is a linear combination of x

, . . . , x

; i = 1, . . . , k.

Schur complement

Let A be an n × n matrix partitioned as

A =  A

A

A

A



, (1.1)

where A

and A

are square matrices. If A

is nonsingular then the Schur com- plement of A

in A is defined to be the matrix A

− A

A

A

. Similarly, if A

is nonsingular then the Schur complement of A

in A is A

− A

A

A

.

The following identity is easily verified:

 I 0

−A

A

I

  A

A

A

A

  I −A

A

0 I



=  A

0

0 A

− A

A

A



. (1.2)

The following useful fact can be easily proved using (1.2):

det A = (det A

) det(A

− A

A

A

). (1.3) We will refer to (1.3) as the Schur complement formula, or the Schur formula, for the determinant.

Inverse of a partitioned matrix

Let A be an n × n nonsingular matrix partitioned as in (1.1). Suppose A

is square and nonsingular and let A/A

= A

− A

A

A

be the Schur complement of A

. Then

A

=  A

+ A

A

(A/A

)

A

A

−A

A

(A/A

)

−(A/A

)

A

A

(A/A

)

 .

Note that if A and A

are nonsingular, then A/A

must be nonsingular. Equiva- lent formulae may be given in terms of the Schur complement of A

.

Cauchy–Binet formula

Let A and B be matrices of order m × n and n × m respectively, where m ≤ n. Then det(AB) = ∑ det A[{1, . . . , m}|S] det B[S|{1, . . . , m}],

where the summation is over all m-element subsets of {1, . . . , n}.

To illustrate by an example, let

A =  2 3 −1 4 0 2

 , B =



 1 −2 0 3 5 1



 .

Then det(AB) equals det  2 3

4 0

  1 −2 0 3



+ det  2 −1 4 2

  1 −2 5 1



+ det  3 −1 0 2

  0 3 5 1



.

1.2 Eigenvalues of symmetric matrices 5

1.2 Eigenvalues of symmetric matrices

Characteristic polynomial

Let A be an n × n matrix. The determinant det(A − λ I) is a polynomial in the (com- plex) variable λ of degree n and is called the characteristic polynomial of A. The equation

det(A − λ I) = 0

is called the characteristic equation of A. By the fundamental theorem of algebra the equation has n complex roots and these roots are called the eigenvalues of A.

We remark that it is customary to define the characteristic polynomial of A as det(λ I − A) as well. This does not affect the eigenvalues.

The eigenvalues might not all be distinct. The number of times an eigenvalue occurs as a root of the characteristic equation is called the algebraic multiplicity of the eigenvalue.

We may factor the characteristic polynomial as

det(A − λ I) = (λ

− λ ) · · · (λ

− λ ).

The geometric multiplicity of the eigenvalue λ of A is defined to be the dimension of the null space of A − λ I. The geometric multiplicity of an eigenvalue does not exceed its algebraic multiplicity.

If A and B are matrices of order m × n and n × m, respectively, where m ≥ n, then the eigenvalues of AB are the same as the eigenvalues of BA, along with 0 with a (possibly further) multiplicity of m − n.

If λ

, . . . , λ

are the eigenvalues of A, then det A = λ

· · · λ

, while trace A = λ

+ · · · + λ

.

A principal submatrix of a square matrix is a submatrix formed by a set of rows and the corresponding set of columns. A principal minor of A is the determinant of a principal submatrix. A leading principal minor is a principal minor involving rows and columns 1, . . . , k for some k.

The sum of the products of the eigenvalues, of A, taken k at a time, equals the sum of the k × k principal minors of A. When k = 1 this reduces to the familiar fact that the sum of the eigenvalues equals the trace.

If λ

, . . . , λ

are the eigenvalues of the n × n matrix A, and if q(A) is a polynomial in A, then the eigenvalues of q(A) are q(λ

), . . . , q(λ

).

If A is an n × n matrix with the characteristic polynomial p(A), then the Cayley–

Hamilton theorem asserts that p(A) = 0. The monic polynomial q(A) of minimum degree that satisfies q(A) = 0 is called the minimal polynomial of A.

Spectral theorem

A square matrix A is called symmetric if A = A

. The eigenvalues of a symmetric

matrix are real. Furthermore, if A is a symmetric n × n matrix, then according to the

spectral theorem there exists an orthogonal matrix P such that

PAP

=











λ

0 · · · 0 0 λ

· · · 0 .. . .. . . . . .. . 0 0 · · · λ









 .

In the case of a symmetric matrix the algebraic and the geometric multiplicities of any eigenvalue coincide. Also, the rank of the matrix equals the number of nonzero eigenvalues, counting multiplicities.

Let A and B be symmetric n × n matrices such that they commute, i.e., AB = BA.

Then A and B can be simultaneously diagonalized, that is, there exists an orthogonal matrix P such that PAP

and PBP

are both diagonal, with the eigenvalues of A (respectively, B) along the diagonal PAP

(respectively, PBP

).

Positive definite matrices

An n × n matrix A is said to be positive definite if it is symmetric and if for any nonzero vector x, x

Ax > 0. The identity matrix is clearly positive definite and so is a diagonal matrix with only positive entries along the diagonal. Let A be a symmetric n × n matrix. Then any of the following conditions is equivalent to A being positive definite:

(i) the eigenvalues of A are positive;

(ii) all principal minors of A are positive;

(iii) all leading principal minors of A are positive;

(iv) A = BB

for some matrix B of full column rank;

(v) A = T T

for some lower triangular matrix T with positive diagonal entries.

A symmetric matrix A is called positive semidefinite if x

Ax ≥ 0 for any x. Equiv- alent conditions for a matrix to be positive semidefinite can be given similarly. How- ever, note that the leading principal minors of A may be nonnegative and yet A may not be positive semidefinite. This is illustrated by the example  0 0

0 −1



. Also, in (v), the diagonal entries of T need only be nonnegative.

If A is positive semidefinite then there exists a unique positive semidefinite matrix B such that B

= A. The matrix B is called the square root of A and is denoted by A

.

Let A be an n × n matrix partitioned as A =  A

A

A

A



, (1.4)

where A

and A

are square matrices.

1.3 Generalized inverses 7

The following facts can be easily proved using (1.2):

(i) If A is positive definite then A

− A

A

A

is positive definite;

(ii) Let A be symmetric. If A

and its Schur complement A

− A

A

A

are both positive definite then A is positive definite.

Interlacing for eigenvalues

The following result, known as the Cauchy interlacing theorem, finds considerable use in graph theory.

Let A be a symmetric n × n matrix and let B be a principal submatrix of A of order n − 1. If λ

≥ · · · ≥ λ

and µ

≥ · · · ≥ µ

are the eigenvalues of A and B, respectively, then

λ

≥ µ

≥ λ

≥ · · · ≥ λ

≥ µ

≥ λ

. (1.5) A related interlacing result is as follows. Let A and B be symmetric n × n matrices and let A = B + xx

for some vector x. If λ

≥ · · · ≥ λ

and µ

≥ · · · ≥ µ

are the eigenvalues of A and B respectively, then

λ

≥ µ

≥ λ

≥ · · · ≥ λ

≥ µ

. (1.6) Let A be a symmetric n × n matrix with eigenvalues λ

(A) ≥ · · · ≥ λ

(A), ar- ranged in nonincreasing order. Let ||x|| denote the usual Euclidean norm, (∑

x

)

. The following extremal representation will be useful:

λ

(A) = max

||x||=1

{x

Ax}, λ

(A) = min

||x||=1

{x

Ax}.

Setting x to be the ith column of I in the above representation we see that λ

(A) ≤ min

i

{a

} ≤ max

i

{a

} ≤ λ

(A).

1.3 Generalized inverses

Let A be an m × n matrix. A matrix G of order n × m is said to be a generalized inverse (or a g-inverse) of A if AGA = A. If A is square and nonsingular then A

is the unique g-inverse of A. Otherwise, A has infinitely many g-inverses, as we will see shortly.

Let A be an m × n matrix and let G be a g-inverse of A. If Ax = b is consistent then x = Gb is a solution of Ax = b.

Let A = BC be a rank factorization. Then B admits a left inverse B

and C admits a right inverse C

. Then G = C

B

is a g-inverse of A, since

AGA = BC(C

B

)BC = BC = A.

Alternatively, if A has rank r then there exist nonsingular matrices P, Q such that A = P  I

0

0 0

 Q.

It can be verified that for any U,V,W of appropriate dimensions,

 I

U V W



is a g-inverse of

 I

0 0 0

 . Then

G = Q

 I

U V W

 P

is a g-inverse of A. This also shows that any matrix that is not a square, nonsingular matrix admits infinitely many g-inverses.

Another method that is particularly suitable for computing a g-inverse is as fol- lows. Let A be of rank r. Choose any r × r nonsingular submatrix of A. For conve- nience let us assume

A =  A

A

A

A

 ,

where A

is r × r and nonsingular. Since A has rank r, there exists a matrix X such that A

= A

X , A

= A

X . Now it can be verified that the n × m matrix G defined as

G =  A

0 0 0



is a g-inverse of A. (Just multiply AGA out to see this.) We will often use the notation A

to denote a g-inverse of A.

A g-inverse of A is called a reflexive g-inverse if it also satisfies GAG = G. Ob- serve that if G is any g-inverse of A then GAG is a reflexive g-inverse of A.

Let A be an m × n matrix, G be a g-inverse of A and y be in the column space of A. Then the class of solutions of Ax = y is given by Gy + (I − GA)z, where z is arbitrary.

A g-inverse G of A is said to be a minimum norm g-inverse of A if, in addition to AGA = A, it satisfies (GA)

= GA. If G is a minimum norm g-inverse of A, then for any y in the column space of A, x = Gy is a solution of Ax = y with minimum norm.

A proof of this fact will be given in Chapter 9.

A g-inverse G of A is said to be a least squares g-inverse of A if, in addition to

AGA = A, it satisfies (AG)

= AG. If G is a least squares g-inverse of A then for any

x, y, ||AGy − y|| ≤ ||Ax − y||.

1.4 Graphs 9

Moore–Penrose inverse

If G is a reflexive g-inverse of A that is both minimum norm and least squares then it is called a Moore–Penrose inverse of A. In other words, G is a Moore–Penrose inverse of A if it satisfies

AGA = A, GAG = G, (AG)

= AG, (GA)

= GA. (1.7) We will show that such a G exists and is, in fact, unique. We first show unique- ness. Suppose G

, G

both satisfy (1.7). Then we must show G

= G

. The deriva- tion is as follows.

G

= G

AG

= G

G

A

= G

G

A

G

A

= G

G

A

AG

= G

AG

AG

= G

AG

= G

AG

AG

= G

AA

G

G

= A

G

A

G

G

= A

G

G

= G

AG

= G

.

We will denote the Moore–Penrose inverse of A by A

. We now show the exis- tence. Let A = BC be a rank factorization. Then it can be easily verified that

B

= (B

B)

B

, C

= C

(CC

)

and then

A

= C

B

.

Let A be a symmetric n × n matrix and let P be an orthogonal matrix such that A = P diag(λ

, . . . , λ

)P

.

If λ

, . . . , λ

are the nonzero eigenvalues then A

= P diag  1

λ

, . . . , 1 λ

, 0, . . . , 0

 P

.

In particular, if A is positive semidefinite, then so is A

.

1.4 Graphs

We assume familiarity with basic theory of graphs. A graph G consists of a finite set of vertices V (G) and a set of edges E(G) consisting of distinct, unordered pairs of vertices. We usually take V (G) to be {1, . . . , n} and E(G) to be {e

, . . . , e

}.

We may refer to edges j

, j

, ... when we actually mean edges e

, e

, .... We

consider simple graphs, that is, graphs without loops and parallel edges. Our

emphasis is on undirected graphs. However, we do consider directed graphs as

well.

If e

is an edge with end-vertices i and j, then we say that e

and i or e

and j are incident. We also write e

= {i, j}. The notation i ∼ j is used to indicate that i and j are joined by an edge, or that they are adjacent.

Notions such as connected graph, subgraph, degree, path, cycle and so on are standard and will not be recalled here. The complement of the graph G will be de- noted by G

. The complete graph on n vertices will be denoted by K

. The complete bipartite graph with partite sets of cardinality m, n, will be denoted by K

. Note that K

is called a star. Further notions will be recalled as and when the need arises.

Exercises

1. Let A be an m × n matrix. Show that A and A

A have the same null space. Hence conclude that rank A = rank A

A.

2. Let A be a matrix in partitioned form:

A =











A

0 · · · 0 A

A

· · · 0 .. . . . . .. . A

A

· · · A









 .

Show that rank A ≥ rank A

+ · · · + rank A

, and that equality holds if A

= 0, i > j.

3. Let P be an orthogonal n × n matrix. Show that a

and det A(1|1) have the same absolute value.

4. Let A and G be matrices of order m ×n and n ×m, respectively. Show that G = A

if and only if A

AG = A

and G

GA = G

.

5. If A is a matrix of rank 1, then show that A

= αA

for some α. Determine α.

It would be difficult to list the many excellent books that provide the necessary background outlined in this chapter. A few selected references are indicated below.

References and Further Reading

1. R.B. Bapat, Linear Algebra and Linear Models, Second ed., Hindustan Book Agency, New Delhi, and Springer, Heidelberg, 2000.

2. Adi Ben–Israel and Thomas N.E. Greville, Generalized Inverses. Theory and Applications, Second ed., Springer, New York, 2003.

3. J.A. Bondy and U.S.R. Murty, Graph Theory, Graduate Texts in Mathematics, 244, Springer, New York, 2008.

4. S.L. Campbell and C.D. Meyer, Generalized Inverses of Linear Transformation, Pitman, 1979.

5. R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.

6. D. West, Introduction to Graph Theory, Second ed., Prentice–Hall, India, 2002.

Chapter 2

Incidence Matrix

Let G be a graph with V (G) = {1, . . . , n} and E(G) = {e

, . . . , e

}. Suppose each edge of G is assigned an orientation, which is arbitrary but fixed. The (vertex-edge) incidence matrix of G, denoted by Q(G), is the n × m matrix defined as follows.

The rows and the columns of Q(G) are indexed by V (G) and E(G), respectively.

The (i, j)-entry of Q(G) is 0 if vertex i and edge e

are not incident, and otherwise it is 1 or −1 according as e

originates or terminates at i, respectively. We often denote Q(G) simply by Q. Whenever we mention Q(G) it is assumed that the edges of G are oriented.

Example 2.1. Consider the graph shown. Its incidence matrix is given by Q.

•1

e₂

 •2

e₁

>>~

~ ~

~ ~

~ ~

~ ~

~ ~

~ ~

~ ~

~ •3

e₅



•4

e₃

``@@@

@@@ @@@ @@@ @@@ @

•5

e4

``@@@

@@@ @@@ @@@

@@@ @

e₆

>>~

~ ~

~ ~

~ ~

~ ~

~ ~

~ ~

~ ~

~

Q =













−1 1 −1 0 0 0

1 0 0 −1 0 0

0 −1 0 0 1 0

0 0 1 0 0 −1

0 0 0 1 −1 1













11

2.1 Rank

For any graph G, the column sums of Q(G) are zero and hence the rows of Q(G) are linearly dependent. We now proceed to determine the rank of Q(G).

Lemma 2.2. If G is a connected graph on n vertices, then rank Q(G) = n − 1.

Proof. Suppose x is a vector in the left null space of Q := Q(G), that is, x

Q = 0.

Then x

− x

= 0 whenever i ∼ j. It follows that x

= x

whenever there is an i j-path.

Since G is connected, x must have all components equal. Thus, the left null space of Q is at most one-dimensional and therefore the rank of Q is at least n − 1. Also, as observed earlier, the rows of Q are linearly dependent and therefore rank Q ≤ n − 1.

Hence, rank Q = n − 1. u t

Theorem 2.3. If G is a graph on n vertices and has k connected components then rank Q(G) = n − k.

Proof. Let G

, . . . , G

be the connected components of G. Then, after a relabeling of vertices (rows) and edges (columns) if necessary, we have

Q(G) =











Q(G

) 0 · · · 0

0 Q(G

) 0

.. . . . . .. . 0 0 · · · Q(G

)









 .

Since G

is connected, rank Q(G

) is n

− 1, where n

is the number of vertices in G

, i = 1, . . . , k. It follows that

rank Q(G) = rank Q(G

) + · · · + rank Q(G

)

= (n

− 1) + · · · + (n

− 1)

= n

+ · · · + n

− k = n − k.

This completes the proof. u t

Lemma 2.4. Let G be a connected graph on n vertices. Then the column space of Q(G) consists of all vectors x ∈ IR

such that ∑

x

= 0.

Proof. Let U be the column space of Q(G) and let

W = (

x ∈ IR

:

n i=1

∑

x

= 0 )

.

Then dimW = n − 1. Each column of Q(G) is clearly in W and hence U ⊂ W. It follows by Lemma 2.2 that

n − 1 = dimU ≤ dimW = n − 1.

Therefore, dimU = dimW. Thus, U = W and the proof is complete. u t

2.2 Minors 13

Lemma 2.5. Let G be a graph on n vertices. Columns j

, . . . , j

of Q(G) are linearly independent if and only if the corresponding edges of G induce an acyclic graph.

Proof. Consider the edges j

, . . . , j

and suppose there is a cycle in the correspond- ing induced subgraph. Without loss of generality, suppose the columns j

, . . . , j

form a cycle. After relabeling the vertices if necessary, we see that the submatrix of Q(G) formed by the columns j

, . . . , j

is of the form  B

0



, where B is the p × p incidence matrix of the cycle formed by the edges j

, . . . , j

. Note that B is a square matrix with column sums zero. Thus, B is singular and the columns j

, . . . , j

are linearly dependent. This proves the “only if” part of the lemma.

Conversely, suppose the edges j

, . . . , j

induce an acyclic graph, that is, a forest.

If the forest has q components then clearly k = n − q, which by Theorem 2.3, is the rank of the submatrix formed by the columns j

, . . . , j

. Therefore, the columns

j

, . . . , j

are linearly independent. u t

2.2 Minors

A matrix is said to be totally unimodular if the determinant of any square submatrix of the matrix is either 0 or ±1. It is easily proved by induction on the order of the submatrix that Q(G) is totally unimodular as seen in the next result.

Lemma 2.6. Let G be a graph with incidence matrix Q(G). Then Q(G) is totally unimodular.

Proof. Consider the statement that any k × k submatrix of Q(G) has determinant 0 or ±1. We prove the statement by induction on k. Clearly the statement holds for k = 1, since each entry of Q(G) is either 0 or ±1. Assume the statement to be true for k − 1 and consider a k × k submatrix B of Q(G). If each column of B has a 1 and a

−1, then det B = 0. Also, if B has a zero column, then det B = 0. Now suppose B has a column with only one nonzero entry, which must be ±1. Expand the determinant of B along that column and use induction assumption to conclude that det B must be

0 or ±1. u t

Lemma 2.7. Let G be a tree on n vertices. Then any submatrix of Q(G) of order n − 1 is nonsingular.

Proof. Consider the submatrix X of Q(G) formed by the rows 1, . . . , n − 1. If we add all the rows of X to the last row, then the last row of X becomes the negative of the last row of Q(G). Thus, if Y denotes the submatrix of Q(G) formed by the rows 1, . . . , n − 2, n, then det X = − det Y. Thus, if det X = 0, then det Y = 0. Continuing this way we can show that if det X = 0 then each (n − 1) × (n − 1) submatrix of Q(G) must be singular. In fact, we can show that if any one of the (n − 1) × (n − 1) submatrices of Q(G) is singular, then all of them must be so. However, by Lemma 2.2, rank Q(G) = n − 1 and hence at least one of the (n − 1) × (n − 1) submatrices

of Q(G) must be nonsingular. u t

We indicate another argument to prove Lemma 2.7. Consider any n − 1 rows of Q(G). Without loss of generality, we may consider the rows 1, 2, . . . , n − 1, and let B be the submatrix of Q(G) formed by these rows. Let x be a row vector of n − 1 components in the row null space of B. Exactly as in the proof of Lemma 2.2, we may conclude that x

= 0 whenever i ∼ n, and then the connectedness of G shows that x must be the zero vector.

Lemma 2.8. Let A be an n × n matrix and suppose A has a zero submatrix of order p × q where p + q ≥ n + 1. Then det A = 0.

Proof. Without loss of generality, suppose the submatrix formed by the first p rows and the first q columns of A is the zero matrix. If p ≥ q, then evaluating det A by Laplace expansion in terms of the first p rows we see that det A = 0. Similarly, if p < q, then by evaluating by Laplace expansion in terms of the first q columns, we

see that det A = 0. u t

We return to a general graph G, which is not necessarily a tree. Any submatrix of Q(G) is indexed by a set of vertices and a set of edges. Consider a square submatrix B of Q(G) with the rows corresponding to the vertices i

, . . . , i

and the columns corresponding to the edges e

, . . . , e

. We call the object formed by these vertices and edges a substructure of G. Note that a substructure is not necessarily a sub- graph, since one or both end-vertices of some of the edges may not be present in the substructure.

If we take a tree and delete one of its vertices, but not the incident edges, then the resulting substructure will be called a rootless tree. In view of Lemma 2.7, the in- cidence matrix of a rootless tree is nonsingular. Clearly, if we take a vertex-disjoint union of several rootless trees, then the incidence matrix of the resulting substruc- ture is again nonsingular, since it is a direct sum of the incidence matrices of the individual rootless trees.

Example 2.9. The following substructure is a vertex-disjoint union of rootless trees.

The deleted vertices are indicated as hollow circles.

•1 // ◦2 // •3

•4



•5 //

A A A A A A A A



◦6

◦7 •8 •9

The incidence matrix of the substructure is given by

















1 0 0 0 0 0 0 −1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 0 0 0 0 −1 0 0 0 0 0 0 −1

















2.3 Path matrix 15

and is easily seen to be nonsingular. Note that the rows of the incidence matrix are indexed by the vertices 1, 3, 4, 5, 8, and 9, respectively.

Let G be a graph with the vertex set V (G) = {1, 2, . . . , n} and the edge set {e

, . . . , e

}. Consider a submatrix X of Q(G) indexed by the rows i

, . . . , i

and the columns j

, . . . , j

. It can be seen that if X is nonsingular then it corresponds to a substructure which is a vertex-disjoint union of rootless trees. A sketch of the ar- gument is as follows. Since X is nonsingular, it does not have a zero row or column.

Then, after a relabeling of rows and columns if necessary, we may write

X =











X

0 · · · 0

0 X

0

.. . . . .

0 0 X









 .

If any X

is not square, then X must have a zero submatrix of order p × q with p + q ≥ k + 1. It follows by Lemma 2.8, that det X = 0 and X is singular. Hence, each X

is a square matrix. Consider the substructure S

corresponding to X

. If S

has a cycle then by Lemma 2.5 X

is singular. If S

is acyclic then since, it has an equal number of vertices and edges, it must be a rootless tree.

2.3 Path matrix

Let G be a graph with the vertex set V (G) = {1, 2, . . . , n} and the edge set E(G) = {e

, . . . , e

}. Given a path P in G, the incidence vector of P is an m × 1 vector defined as follows. The entries of the vector are indexed by E(G). If e

∈ E(G) then the jth element of the vector is 0 if the path does not contain e

. If the path contains e

then the entry is 1 or −1, according as the direction of the path agrees or disagrees, respectively, with e

.

Let G be a tree with the vertex set {1, 2, . . . , n}. We identify a vertex, say n, as the root. The path matrix P

of G (with reference to the root n) is defined as follows.

The jth column of P

is the incidence vector of the (unique) path from vertex j to n, j = 1, . . . , n − 1.

Theorem 2.10. Let G be a tree with the vertex set {1, 2, . . . , n}. Let Q be the inci- dence matrix of G and let Q

be the reduced incidence matrix obtained by deleting row n of Q. Then Q

= P

.

Proof. Let m = n − 1. For i 6= j, consider the (i, j)-element of P

Q

, which is

∑

p

q

. Suppose e

is directed from x to y, and e

is directed from w to z. Then q

= 0 unless k = w or k = z. Thus,

m

∑

k=1

p

q

= p

q

+ p

q

.

As i 6= j, we see that the path from w to n contains e

if and only if the path from z to n contains e

. Furthermore, when p

and p

are nonzero, they both have the same sign. Since q

= 1 = −q

, it follows that ∑

p

q

= 0.

If i = j, then we leave it as an exercise to check that ∑

p

q

= 1. This com-

pletes the proof. u t

2.4 Integer generalized inverses

An integer matrix need not admit an integer g-inverse. A trivial example is a matrix with each entry equal to 2. Certain sufficient conditions for an integer matrix to have at least one integer generalized inverse are easily given. We describe some such conditions and show that the incidence matrix of a graph belongs to the class.

A square integer matrix is called unimodular if its determinant is ±1.

Lemma 2.11. Let A be an n × n integer matrix. Then A is nonsingular and admits an integer inverse if and only if A is unimodular.

Proof. If det A = ±1, then 1

det A adj A is the integer inverse of A. Conversely, if A

exists and is an integer matrix, then from AA

= I we see that (det A)(det A

) = 1

and hence det A = ±1. u t

The next result gives the well-known Smith normal form of an integer matrix.

Theorem 2.12. Let A be an m × n integer matrix. Then there exist unimodular ma- trices S and T of order m × m and n × n, respectively, such that

SAT =  diag(z

, . . . , z

) 0

0 0

 ,

where z

, . . . , z

are positive integers (called the invariant factors of A) such that z

divides z

, i = 1, 2, . . . , r − 1.

In Theorem 2.12 suppose each z

= 1. Then it is easily verified that T  I

0 0 0

 S is an integer g-inverse of A.

Note that if A is an integer matrix which has integer rank factorization A = FH, where F admits an integer left inverse F

and H admits an integer right inverse H

, then H

F

is an integer g-inverse of A.

We denote the column vector consisting of all 1s by 1. The order of the vector will be clear from the context. Similarly the matrix of all 1s will be denoted by J.

We may indicate the n × n matrix of all 1s by J

as well.

In the next result we state the Smith normal form and an integer rank factorization of the incidence matrix explicitly.

Theorem 2.13. Let G be a graph with vertex set V (G) = {1, 2, . . . , n} and edge set

{e

, . . . , e

}. Suppose the edges e

, . . . , e

form a spanning tree of G. Let Q

be

2.5 Moore–Penrose inverse 17

the submatrix of Q formed by the rows 1, . . . , n − 1 and the columns e

, . . . , e

. Let q = m − n + 1. Partition Q as follows:

Q =

 Q

Q

N

−1

Q

−1

Q

N

 . Set

B =  Q

0 0 0

 ,

S =  Q

0 1

1



, T =  I

−N 0 I

 ,

F =

 Q

−1

Q



, H =  I

N  . Then the following assertions hold:

(i) B is an integer reflexive g-inverse of Q.

(ii) S and T are unimodular matrices.

(iii) SQT =  I

0 0 0



is the Smith normal form of Q.

(iv) Q = FH is an integer rank factorization of Q.

The proof of Theorem 2.13 is by a simple verification and is omitted. Also note that F admits an integer left inverse and H admits an integer right inverse.

2.5 Moore–Penrose inverse

We now turn our attention to the Moore–Penrose inverse Q

of Q. We first prove some preliminary results. The next result is the well-known fact that the null space of A

is the same as that of A

for any matrix A. We include a proof.

Lemma 2.14. If A is an m × n matrix, then for an n × 1 vector x, Ax = 0 if and only if x

A

= 0.

Proof. If Ax = 0 then A

Ax = 0 and hence x

(A

A)

= 0. Since A

A is symmetric, it follows that x

A

A = 0. Hence, x

A

AA

= 0, and it follows that x

A

= 0. The

converse follows since (A

)

= A. u t

Lemma 2.15. If G is connected, then I − QQ

=

J.

Proof. Note that (I − QQ

)Q = 0. Thus, any row of I − QQ

is in the left null space of Q. Since G is connected, the left null space of Q is spanned by the vector 1

. Thus, any row of I − QQ

is a multiple of any other row. Since I − QQ

is symmetric, it follows that all the elements of I − QQ

are equal to a constant. The constant must be nonzero, since Q cannot have a right inverse. Now using the fact that I − QQ

is

idempotent, it follows that it must equal

J. u t

Let G be a graph with V (G) = {1, 2, . . . , n} and E(G) = {e

, . . . , e

}. Suppose the edges e

, . . . , e

form a spanning tree of G. Partition Q as follows:

Q =  U V  ,

where U is n × (n − 1) and V is n × (m − n + 1). Also, let Q

be partitioned as Q

=  X

Y

 , where X is (n − 1) × n and Y is (m − n + 1) × n.

There exists an (n − 1) × (m − n + 1) matrix D such that V = U D. By Lemma 2.14 it follows that Y = D

X. Let M = I −

J. By Lemma 2.15

M = QQ

= U X +VY = U X +U DD

X = U (I + DD

)X . Thus, for any i, j,

U

(I + DD

)X

= M(i, j),

where U

is U with row i deleted, and X

is X with column j deleted.

By Lemma 2.7, U

is nonsingular. Also, DD

is positive semidefinite and thus I + DD

is nonsingular. Therefore, U

(I + DD

) is nonsingular and

X

= (U

(I + DD

))

M(i, j).

Once X

is determined, the jth column of X is obtained using the fact that Q

1 = 0.

Then Y is determined, since Y = D

X.

We illustrate the above method of calculating Q

by an example. Consider the graph

•4

e₃

@ @

@ @

@ @

@ @

@ @

@ @

@ @

@ @

•1

e₁

@ @

@ @

@ @

@ @

@ @

@ @

@ @

@ @

e₄

>>~

~ ~

~ ~

~ ~

~ ~

~ ~

~ ~

~ ~

~ •3

•2

e₅

OO

e2

>>~

~ ~

~ ~

~ ~

~ ~

~ ~

~ ~

~ ~

~

with the incidence matrix









1 0 0 1 0

−1 1 0 0 1

0 −1 −1 0 0

0 0 1 −1 −1









.

2.6 0 − 1 Incidence matrix 19

Fix the spanning tree formed by {e

, e

, e

}. Then Q =  U V  where U is formed by the first three columns of Q. Observe that V = U D, where

D =





1 0

1 1

−1 −1



 .

Set i = j = 4. Then Q

=  X Y

 where

X

= (U

(I + DD

))

M(4, 4) = 1 8





3 −2 −1 1 2 −3 1 0 −3



.

The last column of X is found using the fact that the row sums of X are zero. Then Y = D

X . After these calculations we see that

Q

=  X Y



= 1 8













3 −2 −1 0 1 2 −3 0 1 0 −3 2 3 0 −1 −2

0 2 0 −2











 .

2.6 0 − 1 Incidence matrix

We now consider the incidence matrix of an undirected graph. Let G be a graph with V (G) = {1, . . . , n} and E(G) = {e

, . . . , e

}. The (vertex-edge) incidence matrix of G, which we denote by M(G), or simply by M, is the n × m matrix defined as fol- lows. The rows and the columns of M are indexed by V (G) and E(G), respectively.

The (i, j)-entry of M is 0 if vertex i and edge e

are not incident, and otherwise it is 1. We often refer to M as the 0 − 1 incidence matrix for clarity. The proof of the next result is easy and is omitted.

Lemma 2.16. Let C

be the cycle on the vertices {1, . . . , n}, n ≥ 3, and let M be its incidence matrix. Then det M equals 0 if n is even and 2 if n is odd.

Lemma 2.17. Let G be a connected graph with n vertices and let M be the incidence matrix of G. Then the rank of M is n − 1 if G is bipartite and n otherwise.

Proof. Suppose x ∈ IR

such that x

M = 0. Then x

+ x

= 0 whenever the vertices i and j are adjacent. Since G is connected it follows that |x

| = α, i = 1, . . . , n, for some constant α. Suppose G has an odd cycle formed by the vertices i

, . . . , i

. Then going around the cycle and using the preceding observations we find that α = −α and hence α = 0. Thus, if G has an odd cycle then the rank of M is n.

Now suppose G has no odd cycle, that is, G is bipartite. Let V (G) = X ∪ Y be a

bipartition. Orient each edge of G giving it the direction from X to Y and let Q be

the corresponding {0, 1, −1}-incidence matrix. Note that Q is obtained from M by multiplying the rows corresponding to the vertices in Y by −1. Consider the columns j

, . . . , j

corresponding to a spanning tree of G and let B be the submatrix formed by these columns. By Lemma 2.7 any n − 1 rows of B are linearly independent and (since rows of M and Q coincide up to a sign) the corresponding rows of M are also linearly independent. Thus, rank M ≥ n − 1.

Let z ∈ IR

be the vector with z

equal to 1 or −1 according as i belongs to X or to Y, respectively. Then it is easily verified that z

M = 0 and thus the rows of M are linearly dependent. Thus, rank M = n − 1 and the proof is complete. u t A connected graph is said to be unicyclic if it contains exactly one cycle. We omit the proof of the next result, since it is based on arguments as in the oriented case.

Lemma 2.18. Let G be a graph and let R be a substructure of G with an equal num- ber of vertices and edges. Let N be the incidence matrix of R. Then N is nonsingular if and only if R is a vertex-disjoint union of rootless trees and unicyclic graphs with the cycle being odd.

We summarize some basic properties of the minors of the incidence matrix of an undirected graph.

Let M be the 0 − 1 incidence matrix of the graph G with n vertices. Let N be a square submatrix of M indexed by the vertices and edges, which constitute a sub- structure denoted by R. If N has a zero row or a zero column then, clearly, det N = 0.

This case corresponds to R having an isolated vertex or an edge with both endpoints missing. We assume this not to be the case.

Let R be the vertex-disjoint union of the substructures R

, . . . , R

. After a relabel- ing of rows and columns if necessary, we have

N =











N

0 · · · 0

0 N

0

.. . . . .

0 0 N









 ,

where N

is the incidence matrix of R

, i = 1, . . . , k.

If N

is not square for some i, then using Lemma 2.8, we conclude that N is singu- lar. Thus, if R

has unequal number of vertices and edges for some i then det N = 0.

If R

is unicyclic for some i, with the cycle being even, then det N = 0. This follows easily from Lemma 2.16.

Now suppose each N

is square. Then each R

is either a rootless tree or is uni- cyclic with the cycle being odd. In the first case, det N

= ±1 while in the second case det N

= ±2. Note that det N = ∏

det N

, Thus, in this case det N = ±2

, where ω

(R) is the number of substructures R

, . . . , R

that are unicyclic.

The concept of a substructure will not be needed extensively henceforth. It seems essential to use the concept if one wants to investigate minors of incidence matrices.

We have not developed the idea rigorously and have tried to use it informally.

2.7 Matchings in bipartite graphs 21

2.7 Matchings in bipartite graphs

Lemma 2.19. Let G be a bipartite graph. Then the 0 − 1 incidence matrix M of G is totally unimodular.

Proof. The proof is similar to that of Lemma 2.6. Consider the statement that any k × k submatrix of M has determinant 0 or ±1. We prove the statement by induction on k. Clearly the statement holds for k = 1, since each entry of M is either 0 or 1.

Assume the statement to be true for k − 1 and consider a k × k submatrix B of M. If B has a zero column, then det B = 0. Suppose B has a column with only one nonzero entry, which must be 1. Expand the determinant of B along that column and use the induction assumption to conclude that det B must be 0 or ±1. Finally, suppose each column of B has two nonzero entries. Let V (G) = X ∪ Y be the bipartition of G.

The sum of the rows of B corresponding to the vertices in X must equal the sum of the rows of B corresponding to the vertices in Y. In fact both these sums will be 1

. Therefore, B is singular in this case and det B = 0. This completes the proof. u t Recall that a matching in a graph is a set of edges, no two of which have a vertex in common. The matching number ν(G) of the graph G is defined to be the maximum number of edges in a matching of G.

We need some background from the theory of linear inequalities and linear pro- gramming in the following discussion.

Let G be a graph with V (G) = {1, . . . , n}, E(G) = {e

, . . . , e

}. Let M be the in- cidence matrix of G. Note that a 0 − 1 vector x of order m × 1 is the incidence vector of a matching if and only if it satisfies Mx ≤ 1. Consider the linear programming problem:

max{1

x} subject to x ≥ 0, Mx ≤ 1. (2.1) In order to solve (2.1) we may restrict attention to the basic feasible solutions, which are constructed as follows. Let rank M = r. Find a nonsingular r × r submatrix B of M and let y = B

1. Set the subvector of x corresponding to the rows in B equal to y and set the remaining coordinates of x equal to 0. If the x thus obtained satisfies x ≥ 0, Mx ≤ 1, then it is called a basic feasible solution. With this terminology and notation we have the following.

Lemma 2.20. Let G be a bipartite graph with incidence matrix M. Then there exists a 0 − 1 vector z which is a solution of (2.1).

Proof. By Lemma 2.19, M is totally unimodular and hence for any nonsingular

submatrix B of M, B

is an integral matrix. By the preceding discussion, a basic

feasible solution of x ≥ 0, Mx ≤ 1 has only integral coordinates. Hence there is

a nonnegative, integral vector z which solves (2.1). Clearly if a coordinate of z is

greater than 1, then z cannot satisfy Mz ≤ 1. Hence z must be a 0 − 1 vector. u t

A vertex cover in a graph is a set of vertices such that each edge in the graph is

incident to one of the vertices in the set. The covering number τ(G) of the graph G

is defined to be the minimum number of vertices in a vertex cover of G.