Doubly Stochastic Graph Matrices

Home , Adjacency matrix, Generalized inverse, Stochastic matrix

Univ. Beograd. Publ. Elektrotehn. Fak.

Ser. Mat. 8 (1997), 64{71.

DOUBLY STOCHASTIC GRAPH MATRICES

Russel l Merris

Let L(G) b e the n n Laplacian matrix of graph G: This note intro duces

the p ositive de nite doubly sto chastic matrix (G)= I +L(G) :

Let G =(V; E) b e a (simple) graph with vertex set V = fv ;v ;:::;v g and

1 2 n

edge set E of cardinality o(E )=m: Let D (G) b e the diagonal matrix whose (i; i)-

entry is d(v ); the degree of vertex v : Denote by A(G) the n n adjacency matrix

i i

whose (i; j )-entry is 1 if fv ;v g2E; and 0 otherwise. The Laplacian matrix is

i j

L(G)=D(G)A(G):By Gersgorin's Theorem, L(G) is p ositive semide nite.

Because its rows sum to zero, it is a singular M -matrix. Denote the eigenvalues of

L(G)by =0: Then, either from Kirchhoff's Matrix-

1 2 n1 n

Tree Theorem [2; 16] or the general theory of M -matrices [1, p.156; 10], > 0

if and only if G is connected, if and only if L(G) is irreducible. This observation

led Fiedler [7] to de ne a(G)= ; calling it the algebraic connectivity of G:

Numerous variations on the theme if \inverting" L(G)have app eared in the

recent literature. (See, for example, [9], [13], [14], [17], and [21].) The purp ose of

this note is to intro duce another.

1. De nition. If G isagraph on n vertices, de ne

I +L(G) (G)= :

2. Prop osition. If G isagraph on n vertices, then (G) is a positive de nite

doubly stochastic matrix that is entrywise positive if and only if G is connected.

Pro of. Because M (G)=I +L(G) is a symmetric p ositive de nite matrix, it is

invertible and its inverse is p ositive de nite. Indeed, the eigenvalues of (G) are

1 1

> 0: (1) 1

1+ a(G) 1+

Because M (G)isanM-matrix, its inverse is entrywise nonnegative. Because M (G)

is irreducible if and only if G is connected [8, Chap. 3], it follows from the general

1991 Mathematics Sub ject Classi cation: Primary, 05C50. Secondary, 15A48, 15A51, 15A57. 64

Doubly sto chastic graph matrices 65

theory of M -matrices [1, p.141] that (G)isentrywise p ositive if and only if G is

connected.

Let e b e the n-by-1 matrix each of whose entries is 1. Because e is an eigen-

vector of L(G) corresp onding to =0;it is an eigenvector of (G) corresp onding

to the Perron eigenvalue 1. (Indeed, the eigenvectors of (G) and L(G) are iden-

tical.) Hence, (G)e = e; proving that (G)isrow sto chastic. Beacuse (G)is

symmetric, the pro of is complete 2

Recall that a symmetric matrix is doubly p ositive (nonnegative) if it is en-

trywise p ositive (nonnegative) and p ositive (semi)de nite. Thus, (G) is a doubly

nonnegative, doubly sto chastic matrix which is doubly p ositive if and only if G is

connected. It turns out that doubly nonnegative, doubly sto chastic matrices have

b een the sub ject of several recent articles (see, e.g., [11] and [19]); M -matrices

whose inverses are sto chastic were studied in [20]. Moreover, (G) is not the rst

doubly sto chastic matrix to b e asso ciated with G: If is the largest vertex degree

of (a nontrivial graph) G; then I L(G) is doubly sto chastic.

Denote by G _ u the graph on n +1 vertices and m + n edges obtained from

G by joining each of its vertices to a new vertex u: (A vertex adjacenttoevery

other vertex of a graph is said to b e dominating vertex.) Observe that I + L(G)

is the n n principal submatrix of L(G _ u) obtained by deleting the row and

column corresp onding to u: Let us take a moment to consider Prop osition 2 from

this p ersp ective.

Let H b e a xed but arbitrary graph on n +1 vertices, fv ;v ;:::;v g:

1 2 n+1

Denote by L (H ) the principal submatrix of L(H ) obtained by deleting row and

column i: (It seems that

n+1

L (H )

i=1

is known in the chemical literature as an \Eichinger matrix" [15].)

A real matrix P is said to b e doubly superstochastic if there exists a doubly

sto chastic matrix S such that every elementofP is greater than or equal to the cor-

resp onding elementofS: Evidently, for a nonnegative matrix to b e sup ersto chastic

each of its row and column sums must b e at least 1. As the matrix

0 1

0 0 1

@ A

0 0 1

A =

1 1 0

shows, however, this necessary condition is not sucient. Indeed [4], a nonnegative

n n matrix is doubly sup ersto chastic if and only if the sum of the elements of

every p q submatrix is at least p + q n:

3. Prop osition. Let H bea connectedgraph on n +1 vertices, fv ;v ;:::;v g:

1 2 n+1

Then L (H ) isapositive de nite doubly superstochastic matrix. It is doubly

stochastic if and only if v is a dominating vertex. It is entrywise positive if and

only if the graph H v (obtained by deleting vertex v and al l the edges incident

i i

with it) is connected.

66 Russell Merris

Pro of. Because H is connected, L (H ) is p ositive de nite. Hence, it is invertible

and its inverse is p ositive de nite. Let H b e the graph obtained from H by adding

edges, if necessary, so that v b ecomes a dominating vertex. then L (H )=D+

i i

L(H); where D is an n n matrix whose only nonzero entries (if any) are 1's

on the diagonal. In particular, eachentry of the M -matrix B = L (H ) is bigger

than or equal to the corresp onding entry of the M -matrix A = L (H ): It follows

1 1

from the general theory of M -matrices [10; 12, Thm 2.5.4] that A B (in

the entrywise sense), with equality if and only if D =0:Hence, the result is a

consequence of Prop osition 2 and the fact that A is irreducible if and only if H v

is connected. 2

If H is a tree and d(v )= 1;then eachentry of A = L (H ) is a p ositive

i i

integer. In particular, A S for every n n doubly sto chastic matrix S:

A general graph{theoretic interpretation for the entries of L (H ) can b e

obtained from the \all minors Matrix-Tree Theorem" [2]. Sp eci cally, the (r;s)-

entry of the classical adjoint adj L (H ) is the numb er of 2-tree spanning forests of

H in which v and v lie in one tree and v in the other. (If r = s and " = fv ;v g

r s i r i

is an edge of H; this is the numb er of spanning trees of H that contain "; an

observation rst made in [16, Thm. 2].) Of course, det L (H ) = t(H ); the

numb er of spanning trees in H:

4. Corollary. Suppose H is a connectedgraph on n +1 vertices. Fix i and let

0 be the eigenvalues of L (H ): Then 1 with equality if

1 2 n i n

and only if v is a dominating vertex.

Pro of. 1= is the largest eigenvalue of the doubly sup ersto chastic matrix

L (H ) : 2

Wenow return to (G):

5. De nition. Let G =(V; E) beagraph with vertex set V = fv ;v ;:::;v g

1 2 n

and edge set E: Denote by F the family consisting of al l disconnectedspanning

forests of G: For each F 2F ; let (F ) be the product of the numbers of vertices

in the connectedcomponents of F and (F ) the product of the numbers of vertices

in the connectedcomponents of F that do not contain vertex v : Final ly, de ne

F (i; j )=fF 2F : v and v belong to the same component of F g:

G G i j

6. Prop osition. Suppose G is a graph with n 2 vertices and t(G) spanning

trees. Then the (i; j )-entry of (G) is a fraction whose numerator is

(2) t(G)+ (F );

F2F (i;j )

and whose denominator is

(3) nt(G)+ (F ):

F2F

Pro of. Let H = G _ u; vhere u is viewed as the (n +1)st vertex of H: Then, as

wehave observed, L (H )=I +L(G): Thus, (G) = adj L (H ) =t(H ):

n+1 n n+1

Doubly sto chastic graph matrices 67

To each spanning tree of G there corresp ond n di erent spanning trees of

H (obtained by joining u to each of the n vertices of G). Thus, of the spanning

trees T of H; exactly nt(G)have the prop erty that T u is connected. If T is a

spanning tree of H such that F = T u is a disconnected spanning forest of G;

then u can/must b e adjacentinT to exactly one vertex in each comp onentofF:

This gives (F ) p ossibilities for T: Hence, t(H ) is given byFormula (3).

To each spanning tree T of G; there corresp onds one 2-comp onent spanning

forest of H in which u is an isolated vertex, namely, T + u: To each F 2F (i; j );

there corresp ond (F ) 2-comp onent spanning forests of H in which v and v lie in

i i j

one comp onent and u in the other. (They arise by joining u to one vertex in each

of the comp onents of F that do not contain v and v :) Hence, by the all minors

i j

Matrix-Tree Theorem, the numerator of (G) is given byFormula (2). 2

7. Corollary. Suppose G is a graph on vertex set fv ;v ;:::;v g; where n 2:

1 2 n

Let (G)= (! ): Then

(i) ! =0 if and only if v and v lie in di erent connectedcom-

ij i j

ponents of G;

(ii) ! >! ; for al l j 6= i; 1 i n;

ii ij

(iii) ! 1= 1+ d(v ) ; and

ii i

P P P

(iv) (F )= (F ):

j =1

F2F

F 2F (i;j )

Pro of. A consequence of Formula (2), part (i) is a strengthening of the \entrywise

p ositive" part of Prop osition 2. Because F (i; j ) is a prop er subset of F (i; i);

G G

part (ii) follows from Formula (2). Part (iii) follows from a result of Fiedler

[6] relating the corresp onding diagonal entries of twomutually inverse p ositive

de nite matrices. While part (iv) can b e proved directly by means of an elementary

counting argument, it is an immediate consequence of Prop osition 6 and the fact

that (G) is doubly sto chastic. 2

Consider the characteristic p olynomial of L(G);

p (x) = det xI L(G) =(x )(x ) (x ):

G n 1 2 n

The sum of the absolute values of the co ecients of p (x)is

(1) p (1)=(1+ )(1 + ) (1 + )

G 1 2 n

(4) = det I + L )

n G

(5) = det L (G _ u) :

n+1

8. Corollary. Let G bea graph on n 2 vertices. Then

68 Russell Merris

(i) det (G) =(1) =p (1); and

(ii) (1) p (1) = nt(G)+ (F ):

F2F

Pro of. Part (i) follows from Equation (4). Part (ii) is a consequence of Equation

(5) and the fact that Formula (3) is the numb er of spanning trees in G _ u: (Part

(ii) also follows from a result of Kel'mans [5, Thm. 1.4].) 2

The second largest eigenvalue of (G)is1= 1+ a(G) ; where a(G)isthe

algebraic connectivityofG: In particular,

= max x (G)x ; (6)

1+ a(G)

where the maximum is over all real unit vectors x the sum of whose co ordina-

tes is 0.

9. Corollary. Let G beagraph on n 3 vertices with doubly stochastic graph

matrix (G)=(! ): If 1 i

1 ! + ! 2!

ii jj ij

(7) :

1+ a(G) 2

Pro of. Let x =(e e )= 2;where e is the

1 2 3 4

i j i

ith standard basis vector in R : Then (7) is

Figure l

equivalentto1= 1+ a(G) x (G)x :

10. Example. If G is the graph illustrated in Figure 1, then

0 1

32 12 4 2 2

B C

12 24 8 4 4

B C

(G)= 4 8 20 10 10

B C

@ A

2 4 10 31 5

2 4 10 5 31

and the value of the right-hand side of (7) is exhibited in p osition (i; j ) of the matrix

0 1

32 44 59 59

B C

32 28 47 47

B C

B = 44 28 31 31 :

B C

104

@ A

59 47 31 52

By Corollary 9, each o diagonal entry of B is a lower b ound for 1= 1+a(G) =

0:658: (The biggest lower b ound emerging from this calculation is 59=104 =0:567:

In general, it seems that the b est choice for i and j corresp onds to the p osition

of a smallest entry in (G):) The reason for exhibiting b oth the lower and upp er

triangular parts of matrix B is that it is the analog of the \resistance distance"

matrix [14], in which (G) replaces a generalized inverse of L(G):

Doubly sto chastic graph matrices 69

11. De nition. Let G bea graph. De ne F [i; j ]= F nF (i; j )=fF 2F : v

G G G G i

and v lie in di erent components of F g: For each F 2F [i; j ]; denote by o(F ) the

j G i

number of vertices in the component of F containing v and by (F ) the product

i i;j

of the numbers of vertices in the connectedcomponents of F that contain neither

v nor v (with the understanding that the empty product is 1).

i j

12. Corollary. Let G beagraph on n 3 vertices. If 1 i

2(1) p (1)

: (8) o(F )+ o(F ) (F )

i j i;j

1+ a(G)

F 2F [i;j ]

Pro of. Inequality (8) is just a rearrangement of terms when the values from

Prop osition 6 and Corollary 8 (ii) are substituted into Inequality (7). 2

If A =(a ) is a real n n matrix, its permanent is de ned by

X n

per (A)= a :

i (i)

i=1

Dating from the publications of [22], there has b een a great deal of interest

in p ermanents of doubly sto chastic matrices.

13. Prop osition. Let G beagraph on n vertices. Then

[n!e]

per (G) ;

(n +1)

with equality if and only if G = K ; the complete graph, where [] is the greatest

integer function and e is the base of the natural logarithms.

Pro of. As the result is easily veri ed for n 2; we assume n>2: Supp ose u and

v are nonadjacentvertices of G: Let G b e the graph obtained from G by adding

a new edge, fu; v g: Then L(G )= L(G)+ Q; where Q is p ermutation similar to

1 1

the direct sum of and the (n 2)-square zero matrix. In particular,

1 1

0 0

L(G ) L(G) in the p ositive semide nite sense. Therefore, (G) (G ); and [3]

per (G) > per (G ) : Thus, it remains to show that p er (K ) =[n!e]=(n+

1) : Because, I + L(K )= (n+1)I J ; (K )= (I +J )=(n+1); where J

n n n n n n n n

is the n n matrix each of whose entries is 1.

Let Q b e the set of all strictly increasing sequences of length k chosen

k;n

from f1; 2;:::;ng: If A and B are n n matrices, then [18, p.17]

X X

per (A + B )= per A[ j ] per B ( j ) ;

k=0 ; 2Q

k;n

where A[ j ] is the submatrix lying in rows and columns ; and B ( j )is

submatrix of B obtained by deleting rows and columns : In particular,

(n +1) per (K ) = p er (I + J )

n n n

70 Russell Merris

n n

X X X

= per J ( j ) = n! 1=k ! : 2

k =0 2Q k=0

k;n

The same pro of will show that d (G) >d (K ) for any of a whole

family of \generalized matrix functions".

Added in pro of. The author is grateful to D. J. Klein for p ointing out that

p ortions of Prop osition 3 and Corollary 8 overlap results previously obtained in V.

E. Golender, V. V. Drboglav, A. B. Rosenblit: Graph potentials method

and its application for chemical information processing. J. Chem. Inf. Comput.

Sci. 21 (1981), 126{204.

REFERENCES

1. A. Berman, R. J. Plemmons: Nonnegative Matrices in the Mathematical Sciences.

Academic Press, New York, 1979.

2. S. Chaiken: Acombinatorial proof of the al l minors matrix tree theorem. SIAM J.

Alg. Disc. Meth. 3 (1982), 319{329.

3. G. H. Chan, M. H. Lim: Aconjecture involving generalized matrix functions. Linear

Algebra Appl. 36 (1981), 157{163.

4. S. C. K. Chu, C. K. Li: Characterization of doubly superstochastic matrices by network

ows. Intl. J. Math Ed. in Sci & Tech. 17 (1986), 127{129.

5. D. M. Cvetkovic, M. Doob, H. Sachs: SpectraofGraphs. Academic Press, New

York, 1979.

6. M. Fiedler: Relations between the diagonal elements of two mutual ly inverse positive

de nite matrices. Czech. Math. J. 89 (1964), 39{51.

7. M. Fiedler: Algebraic connectivityofgraphs. Czech. Math. J. 23 (1973), 298{305.

8. M. Fiedler: Special Matrices and Their Applications in Numerical Mathematics. Mar-

tinus Nijho , Boston, 1986.

9. M. Fiedler: Moore{Penrose involutions in the classes of Laplacians and simplices.

Linear & Multilinea r Algebra 39 (1995), 171{178.

10. M. Fiedler, V. Ptak: On matrices with non-positive o -diagonal elements and pos-

itive principal minors. Czech. Math. J. 12 (1962), 382{400.

11. R. Grone, R. Loewy, S. Pierce: Nonchordal positive semi-de nite stochastic matri-

ces. Linear & Multilinear Algebra 32 (1992), 107{113.

12. R. A. Horn, C. R. Johnson: Topics in Matrix Analysis. Cambridge University Press,

1991.

13. S. Kirkland, M. Neumann, B. L. Shader: Characteristic vertices of weightedtrees

via Perron values. Linear & Multilin ear Algebra 40 (1996), 311{325.

14. D. J. Klein, M. Randic: Resistance distance. J. Math. Chem. (1993), 81{95.

Doubly sto chastic graph matrices 71

15. M. Kunz: On topological and geometrical distance matrices. J. Math. Chem. 13

(1993), 145{151.

16. M. Lewin: A generalization of the matrix-tree theorem. Math. Z. 181 (1982), 55{70.

17. R. Merris: Anedge version of the matrix-tree theorem and the Wiener index. Linear

& Multilinea r Algebra 25 (1989), 291{296.

18. H. Minc: Permanents. Addison{Wesley, Reading, 1978.

19. N. Shaked{Monderer: Extreme chordal doubly nonnegative matrices with given row

sums, manuscript.

20. R. L. Smith: M-matrices whose inverses are stochastic. SIAM J. Alg. Disc. Meth. 2

(1981), 259{264.

21. G. P. H. Styan, G. E. Subak{Sharpe: Inequalities and equalities associated with

the Campbel l{Youla generalized inverse of the inde nite-admittance matrix of resistive

networks. Linear Algebra Appl., to app ear.

22. B. L. van der Waerden: Aufgabe 45. Jb er. Deutsch. Math. Verein. 35 (1926), 117.

California State University, (Received Octob er 7, 1996)

Hayward, CA 94542,

USA

[email protected]