SUBHARMONIC
Yu. I.
FUNCTIONS
Lyubich
and
ON
A
DIRECTED
GRAPH
M . I. T a b a c h n i k o v
UDC
517.516
Ii. I N T R O D U C T I O N
The principal topic of the present paper is a class of functions on a directed graph, where this class
is delineated so as to have an analog to the classical principle of the maximum. The translation of this
principle to graph theory requires that the concept of the boundary of a graph be defined beforehand. This
definition is given in w2, along:wR h some illustrative examples. It is imoortant to mention that, apart from
terminological and interpretative differences, the concept has already been encountered in the literature
for other reasons (see [1-5]). The fundamental properties of the boundary of a graph are considered in w3.
Then in | 4 theiprinciple of the }'naximum and some related theorems are set forth. The remaining sections
(|w 5-7) ar e given over to the applications.
| 2 . D E F I N I T I O N O F T H E BOUNDARY. E X A M P L E S
Consider an arbitrary directed graph G= (X, r). The vertex y is said to be accessible from the vertex
x if there is a path from x to y or if y-- x. The set of vertices accessible from x is denoted I(x). It is invariant, i.e., FI(x)z I(x), and it is the minimum invariant set containing x.
Definition I. A vertexxis caUeda boundary vertex if the subgraph (I(x);r) is strongly connected, i.e.,
if x is accessible from all vertices that are accessible from it.
Definition 2. The set of boundary vertices is called the boundary of the graph G and is denoted ~CeX,
or just ~X.
Clearly the boundary is an invariant set.
Definition 3. The subgraph (aX, F) is called the boundary subgraph of the graph G.
Definition 4. Vertices that are not boundary vertices are called interior vertices. The set Int X of
interior vertices is called the interior of the graph G.
W e now present s o m e examples.
1. Markov Chains [1, 2]. Let C be a homogeneous discrete Markov chain with the set of states X.
A state x is said to be nonessential if there is a state y such that the transition probability from x to y during a certain time is positive, but the reverse transition is unallowed.
--e
Let us compare the chain C with the graph G = (X, r), defining Fx as the set of states in which a transition from x per unit time is possible. The boundary of that graph is the set of essential states. The limiting probabilities of the ~Larkov chain are concentrated on the boundary.
2. Ashby Systems [3]. Every Ashby system comprises a certain number of "blocks" B v B 2 ..... Bin.
Between the blocks there are effect-transmission arrows. If an effect-transmission arrow runs from Bi
to Bk, block B i is said to act directly on block Bk. The blocks and transmission arrows form a diagram
(graph) of immediate effects. Block B i is said to dominate B k if in the diagram of immediate effects there
is a path from B i to B k but not in the opposite direction.
Consider the graph representing the inverse of the diagram of immediate effects. The boundary of
this graph is the set of blocks that have no dominant blocks. In other words, the boundary blocks are those
which cannot be controlled without feedback, i.e., the boundary in this example is the "top level" of a control system.
3. Sets of Equations. W e define the graph of the set of equations
,/,(lh .....
~,,) =
o (l =
i .....
,,)
Translated from Sibirskii Matematicheskii Zhurnal, Vol. I0, No. 3, May-June,
Original article submitted October 11, 1967.
432
(I)
pp. 600-613, 1969.
as the graph G-- (X, r) with a set o f v e r t i c e s X= {1. . . . , n} in which J{! r i if, andonlyif, the unknow~ ~j occurs in the i~-thequntion (this means that the function fi is not a constnnt relative to tj), The unknowns o r
equations whose indices belong to the boundary of this graph are called interior unknowns and equations.
If the system (I) is a standard difference approximation of the first boun(hry-value problem for a partial
dif|erentlal equation, then the boundary (interior) unknowns and equations c o r r e s p o n d to the boundary {int e r i o r ) nodes of a net (see [6]).
If a set of equations has the form
(2)
~, -- ~',(|, ..... |.) (t---- f ..... ,0,
the definition of its graph is analogous to the preceding one, except fo r gi on the right-hand sides. The
boundary of this graph coincides with the boundary of the system (1) with fi = ~ i - gi.
The in variance of the boundary makes it possible to t r e a t a set of boundary equations (boundary subsystem) independently of the other equations, because it contains only the boundary unknowns. The graph
of this subsystem coincides with the boundary subgraph of the system.
4. ~htrices_[4]. Let A be an n-th m a t r i x over an a r b i t r a r y s e m i r i n g R. We define the graph of the
m a t r i x A as the graph G=(X, F), where X= {1 . . . . . nL JE F i .r aij ~ 0. A m a t r i x is said to be noadecomposable Lf its graph i s strong b" connected. A decomposable m a t r i x can be reduced by transposition of rows
to a partitioned t r i a n g u l a r m a t r i x of the f o r m
/o,,
o ~
""
A=/
'D
.........
t
\
!
0
~
1 ..............
*
i D"''''
f,
~
]
where DI! are nondecomposable m a t r i c e s and e v e r y partitioned row below the horizontal dividing line contains a nonzero off-diagonal block. In the m a t r i x ~, the boundary subgraph corresponds to the s y s t e m of
blocks DII . . . . . Dkk. It f o r m s the boundary submatrix &~. F o r the set of linear equations
R
~a,~
.
b, = 0
(i = l . . . . , n)
(3)
(i = ] . . . . . n)
(4)
kzi
o r for the set
R
~ ----- ~_j p , k ~ "~- q~
the boundary submatrix c o r r e s p o n d s to the boundary subsystem.
i3.
FUNDAMENTAL
PROPERTIES
OF THE
BOUNDARY
OF A GRAPH
The graph G= (X, F) is said to be p-finite ff all simple paths in it are finite. Clearly all finite graphs
:ar e p-finite, and it is a simple m a t t e r to give an example of a p-fin!re graph that is not finite.
THEOREM 1. F o r e v e r y v e r t e x of a p-finite graph t h e r e is a boundary v e r t e x accessible from that
vertex.
Proof. Let all v e r t i c e s accessible f r o m a certain v e r t e x x0EX not be boundary v e r t i c e s . Then t h e r e
is a v e r t e x x!EI(x0) f r o m which x 0 is inaccessible. But x 1 is also not a boundary vertex, hence there is a
v e r t e x xzE I(xl) from which x 1 is inaccessible, etc. We obtain a simple infinite path x0xlx2. . . . but this contradicts the p-finiteness of the graph.
Remark. It is apparent f r o m the proof that it would be sufficient to a s s u m e finiteness on the part of
simple paths, all of whose v e r t i c e s w e r e i n t e r i o r [i.e., p-finiteness of the subgraph Int {X, r)]. This r e m a r k also applies to certain o t h e r situations.
COROLLARY 1. The boundary of a p-finite graph is nonempty.
The condition of p - f i n i t e n e s s is e s s e n t i a l here.
433
COROLLARY 2, Every nonempty Invarlant set of vertices of a p-finite graph contains at least one
boundarypoint.
By T h e o r e m I we call A c X an accumulating set if for every veI~e• xEX there is a vertex yE All I(x).
COROLLARY 3. The boundary of a p-flnite graph is the minimum accumulatlng Invarlant set of v e rt/ces.
The following proposition descrlbe~ the topology of the boundary subgraph (~X, r).
THEOREM 2. The connected components of a boundary subgraph a r e strongly connected.
ProoL Let x ~ x i . . . x m be a chain In (~X, 1~, I.e., f o r every k=0, 1. . . . . m suppose that either Xk+ i
EI'~ k o r XkE FXk+ l- In either case Xk+lEI(Xk), since Xk+ ! is a boundary vertex. Consequently XmEI(x~),
Le., for e v e r y chain there is a path having the same ends.
Thus a boundary subgraph is either strongly connected or is an unconnected amalgamation of strongly
connected graphs.
We s ay that the boundary of. a graph is trivial ff it coincides with the set of all vertices.
COROLLARY 4. F o r the boundary of a graph to be trivial it i s n e c e s s a r y and sufficient that it he
strongly conne~ed or an unconnected amalgamation of strongly connected graphs.
COROLLARY 5. The boundary of a sym m et ri c graph is triviaL
14. P R I ' N C I P L E
OF T H E M A X I M U M F O R
SUBHARMONIC
FUNCTIONS
ON A G R A P H
The role of the concept of the boundary of a graph is largely exemplified by the fact that it can be used
to formulate a di s c r et e analog of the classical principle of the maximum.
Let G= fX, I') be an a r b i t r a r y graph, M= X.
Definition 5. A real function q~(x) (xE ~ is said to be subharmonic on the set * M ff fo'r every x E M
such that I'x ~ ~ the following inequality holds:
~(z) < s u p {+(y)I~ ~ r~},
where the equal sign holds only in the ease
+(~) = ~(~) (~, ~r~).
x~map!e
E
' I_ T h e solution ~(i) = ~t (i ~ 1. . . . . n) of the system (4), given the assumptions
is subharmonic on M,
E___xam~!e2. Consider the homogeneous system
l
i, = F, p , ~
(i = t , . . . , n)
k-=t
under the assumption
I
Y, Ip,kl < i
0 ~ M).
k--i
The function ~ i ) = [~i[ (I=1 . . . . , n) is subharmonie on M.
THEOREM 3 (Principle of the Maximum). In the graph G = (X, I') let all simple paths whose vertices
belong to the set M c X be finite. We put
*If M = X , w e can s a y t o n the graph G. w
434
where ~M Is the set of boundary points of thc subgr~ph ~ t , I~ from which points of the set X \ M a r e I n a c c e s s i b l e in G. If the function ~ x ) is subharmonlc on M, then
,,,p {~{z) Ix ~ x} ~ ,,p {~ (z) Iz e .~}.
F o r the p r o o f we define t h e graph GM = CX, FM), setting
rN,~
r~(zEM), rMz ~ z (z r ~0.
T h e graph G M is p-finite, and its boundary coincides with I~L The function r
X r e l a t i v e to the graph GM.
is subharmonic on aH of
Thus T h e o r e m 3 reduces to the following special f o r m :
THEOREM 4. If the function r
is subharmonic on a p-finite graph G= (X, 1"5, then
sup {r
e x} -- ,,,e {~(~)lze ax}.
We p r e f a c e the proof of this t h e o r e m with some a n c i l l a r y considerations. We r e g a r d the graph G at
all times as p-finite.
Let us call the chain* xoxix2.... a relaxation path if
and a s t r i c t relax~Uon path if at least one of the inequ,'dities (5) is strict. We call the v e r t e x a constancy
point of the function ~ if ~{yl = ~ x ) (yE rx). The existence of a constancy point ensues from the follow,.'ng
lemma.
LEM~L~ I. A constancy point is a c c e s s i b l e f r o m e v e r y v e r t e x x by some relaxation path.
F o r the proof we'~onstruct the mapping 7:X---X such that
vz E rz, ~ (x) ~ <p(vz) (x ~ X),
the equals sign holding only if x is a constancy point (henceforth ~'e a s s u m e without loss of generality that
rx~0
Oce x)).
Let x, 7xo 7~x. . . . be a relaxation path, and by virtue of p - f i n i t e n e s s let it contain a loop. This loop
consists of constancy points.
We now verLfy the following basic lemma.
LEMMA 2. Let
-oo <.
< ~.p {~(~)Iz ~ x}.
Then there i~ a v e r t e x x0 such that ~(x0) > p and all v e r t i c e s a c c e s s i b l e from it by relaxation paths a r e constancT points.
Proof. We set L={x[c(x) >/~}. If t h e s t a t e m e n t of the l e m m a is false, then for e v e r y v e r t e x xEL t h e r e
is a v e r t e x Yx that is accessible from x by a relaxation path, but that is not a constancy point. Then
~(~y~) > ~(y.).
Consequently, from e v e r y v e r t e x x~ L the v e r t e x Zx= 7Yx is a c c e s s i b l e by a s t r i c t relaxation path. Obviously zx ~L. Iterating the raapping x ~ z x, we obtain a relaxation path with an infinite set of s t r i c t tnequaIRies.
This is inconsistent with the p-finiteness of the graph.
LEMMA 3. Let the vertexx0( satisfy the r e q u i r e m e n t s of L e m m a 2. Then the se~ J(x~) of v e r t i c e s
a c c e s s i b l e from x~ by relaxation paths is lnvariant.
This statement is obvious.
F o r the proof of T h e o r e m 4 we a s s u m e the opposite and c o n s i d e r
-- sup {q~(~)I~ ~x}.
* Finite o r infinite.
435
N o w L e m m a 2 is applicable, The corresponding set J(x 0) I~ clearly nonempty and by L e m m a 3 is Invarlant.
According to Corollary 2, it contains u boundary point x t. But, clearly, ~(x t) >/a, contradicting the definition
of the n u m b e r / J .
The p r i n c i p l e of the m a x i m u m has thus been completely proved. We cite a s p e c i a l case.
THEOREM 5. In the graph G-- iX, 1"3 let all i n t e r i o r (i.e., c o m p r i s i n g only interior points) s i m p l e
paths be finite. If t h e function ~ x ) is s u b h a r m o n i c on the interior of the graph G, then
~,,v {~(~) i~ e x} -- ..p {~(~) Iz e 0.u
F o r the p r o o f it suffices to set M = Int X in T h e o r e m 3. Then 5M = 0, and ~ = 8X.
In connection with the p r i n c i p l e of the m a x i m u m we look into the p r o b l e m of the abundance of subh a r m o n i c functions on a graph. C l e a r l y all constants a r e subharmonic.
THEOREM 6. If a p - f i n R e g r a p h G= iX, r ) is strongly connected, then a l l s u b h a r m o n i c functions on
X are constants.
Proof. Let us apply L e m m a 2, setting p-- -=. The corresponding set J(x 0) is nonempty ancl invariant.
Inasmuch as the graph is strongly connected, J(x~) = X. Consequently all vertices are constancy points.
Making use once again of the strong connectivity property, we arrive at the required assertion.
COROLLARY
6. If the graph G = iX, F) is p-finite, then a function subharmonic on X is constant on
every connected component of the boundary 9X.
W e note that on every nonstrc~ngly connected graph there is a subharmonie function that is not a constant. An example of such a function is the characteristic function of any nontrivial invariant subset, in
particular the characteristic function of the boundary if the boundary is nonempty and nontrivial.
In concluding this section we stress that the formulated theory is significantly simplified if we confine ourselves to finite graphs, because in this case the accessibility of the upper boundary can be utilized
in systematic fashion. Finite graphs are used exclusively in the ensuing applications (w1675 and 6).
|5.
TAUSSKYrS
THEOREM
AND THE
THEORY
OF
ITERATIONS
The t h e o r e m of O. T a u s s kT [7] with which we a r e concerned s t a t e s that if in a nondecomposable m a t r i x
~,,_,
A = (a,~)
the diagonal is dominant, i.e., ff aii * 0 (I.... , n) and
F, l=.,l < la,,I
(~ = I. . . . .
,,),
(6)
w h e r e at least one of the inequalities (6) is s t r i c t , then
det A r 0.
This t h e o r e m has a long history, originating with H a d a m a r d (see the bibliography in [7]). We wish to show
that the p r i n c i p l e of the m a x i m u m leads directly to the following m o r e g e n e r a l t h e o r e m .
THEOREM 7. Let the set of equations
~, = ~,p,,~,
( i = t . . . . . n)
(7)
be such that
~, IP,~l< t
(8)
kwl
in all i n t e r i o r equations, ff the boundary s u b s y s t e m of the s y s t e m (7) h a s only a t r i v i a l solution, then the
e n t i r e s y s t e m also has this p r o p e r t y .
Proof. The modulus of the solution is a s u b h a r m o n t c function on the i n t e r i o r of the g r a p h of the s y s t e m . By T h e o r e m 5 the m a x i m u m of the modulus is attained on the boundary, but on the boundary the solution is equal to zero by stipulation.
436
F o r the proof of T a u s s l ~ " s t h e o r e m we t r a n s f o r m the h o m o g e n e o u s s y s t e m
A
to the form (7), expanding the l-thequation in ~1 (t= I , . . , ,
equations, and in at least one of them
n). Then the condition (8) is satisfied in all the
gt
Y, Ip,,! < I.
(lo~
However, the bouadary of t h e graph of the resulting s y s t e m is t r i v i a l ? due to the nondecomposabllity of the
matrix. In o r d e r to obviate this difficulty, we add one m o r e unknown ~a+ 1 and the additional equation
|,,+,
---
O.
(H)
F u r t h e r m o r e , In the equation in which (10) holds we introduce the additional t e r m P i , n + l ~ + 1 with o b s e r vance of the condition
It
Ip~,,,+,l < i - ~ IP,Jl&=-I
The new s y s t e m is equivalent to the original one, but its boundary- is no longer trivial, r a t h e r it c o m p r i s e s
the one point n+ 1. The boundary subsystem (11) has only a trivial solution. By T h e o r e m 7 the original
s y s t e m is endowed with the same p r o p e r t y .
It is reasonable to suggest the d i r e c t derivation ~f TausskT's t h e o r e m from the theory" of subharmonic
functions. The essential fact is that the modulus of the solution of the homogeneous s y s t e m (8) under the
conditions of Taussk.-y's t h e o r e m is a subharmonic function on the set of v e r t i c e s of a strongly connected
graph and by T h e o r e m 6 is constant. But then it foltows from 'the equation with the s t r i c t inequality (6) that
the modulus is equal to zero.
We now consider a set of linear equations of the iterative type (,t) in vector notation
|=~+q,
and the conventional p r o c e s s of s u c c e s s i v e approximations for it:
~.<'+~) ------~ ' ~
-F q
(m ---- O, t, 2 . . . . ).
(12)
Global convergence of the p r o c e s s (i.e., convergence for all q and ~(0)) is equivalent to the inequality p(P)
< 1, w h e r e p(-) is the s p e c t r a l radius.
THEOREM 8. Let
~t
in all i n t e r i o r equations. Then global convergence of the p r o c e s s is equivalent to its global convergence for
the boundary subsystem.
This t h e o r e m is equivalent to the following:
THEOREM 9. Under the condition of T h e o r e m 8 the inequality p(aP) < 1 implies* p(P) < 1.1"
Proof. If p(aP) < 1, the boundary subsystem of the system
has only a trivial solution. By T h e o r e m 7 the e n t i r e s y s t e m has the same p r o p e r t y , l.e,, the value k = 1
does not enter into the s p e c t r u m of the matrix P. Applying this result to t h e m a t r i x pP for any # ~ 0, ~ I -< 1,
w e a r r i v e at the required inequality.
~-~"other words, there a r e no interior equations in the new s y s t e m .
? T h e c o n v e r s e is trivial, b e c a u s e always p(aP)-<p(P).
437
We note th,lt if the m a t r i x P l e nondecompoeable and inequality (13) holds for 1= t . . . . . n and is strict
for at l e a s t one I, then p(P) < I. Thls t h e o r e m of Collatz {81 follows at once f r o m T a u s s k y ' s t h e o r e m , We
e l a b o r a t e it In the next p a r a g r a p h .
16. SPECTRAL
PROPERTIES
OF
CONTRACTIONS
(lfi
.... ,n)
IN c n
The condition
I
~lP,~l<l
(14}
liml
admits a almple operator Interpretation,. Specifically tt means that the m a t r i x P as a linear operator is
a contraction tn the n-dimensional arithmetic space cn, In which the norm is
mai I~1-
lU:
Thus. i n e~
2.
|eU=~ m~,x Z le".lC l e a r l y p(P) < 1 for contractions. We investigate the eigenvalues k for which IkI = 1 and the c o r r e sponding p r o p e r substances.* We f i r s t consider the c a s e when the m a t r i x P is nondecomposable and X = 1.
F o r Pik ~ 0 we let
,,,~k -- p., / [ p.,, lOn the set of a r c s of the g r a p h of the m a t r i x P this d e t e r m i n e s a function ~' with values on the unit circle.
THEO[~EM: tA~. Let the m a t r i x P be noadecomposable and lip [ ~ 1. In o r d e r for )~ to be an eigeavalue
it is n e c e s s a r y :lad s:zfficient to have
F , Ip,~! = t
(i = s . . . . .
=)
(Is)
and to h~ve for any loop K in the graph of the m a t r i x P
1-[ 0,,~ = s.
(!e)
(L ~)EJr
Under conditions (15) and (16) the dimension of the corresponding p r o p e r s u b s p a c e E(1) is equal to 1.
The n e c e s s i t 3" of condition (15) is contained in the aforementioned t h e o r e m of Collatz. M o r e o v e r , if
f e E ( l ) , i.e., if
then [~[= coast by T h e o r e m 6, Consequently, if ~k = 0 fo.r at least one k, then ~ = 0. I t is c l e a r f r o m this that
any two eigen~'ectors a r e proportional. We pick an eigenvector ~ for which [~k[= 1 (k= 1. . . . . n). Since the
t r i a n g l e inequality
! ~ , ! < ~,lp,,,llhl
(i=l
.....
,,)
Is actually an equality, it follows that
,,,,,,~ = ~t
for all a r c s (i, k). The function w turns out to be "integrable," and ~ is its " i n t e g r a l . * t
once to (16).
(17)
This brings us at
We now p r o v e the sufficiencY of conditions (15) and (16). It follows f r o m (16) that for any path Wj l e a d big f r o m a fixed v e r t e x i 0 to an arbLtrary v e r t e x j the "integral"
They coincide with the r a d i c a l s u b s p a c e s , b e c a u s e f o r contractions the n o r m s of p o w e r s | Pmll a r e bounded:
IlpmU < 1 (m: 1 , 2 . 3 , . . . ) .
t W e could also s a y that the one-dimensional cochain w is the cobound~ry of the cochain ~, o r that It is c o h o m o l o g o u s to z e r o .
438
11
u~
depends only on the beginning and end of the path. In~89
is a path W0 from j to i 0. and by (16)
{IS)
due to the strong connectivity of fl~e graph, t h e r e
Expression (18) determines the function ~jfor all J (again, due to strong connectivity}, and c|early (17) holds.
It is also clear that [~j[---I (J 1, .,.., n). MI that remains is to note that
m
Y,,p,,~,=
t--I
~
Ip,,lo,,,~=~, ~ l p , , l = ~ ,
&ipil~-@
(~=~ .... ,,,).
k'.,!
We say that a nondec~ral~sable matrix satisfying conditions (t5} and (16) is integrable and we go o v e r
to arbitrary contractions.
THEOREM 11, If | P ~
1. then the dimensionallty r the space of solutions of the system
is equal to the number of connected components of the boundary subgraph of the matrix P for which the corresponding submatriees are lntegrable.
Pr._~ f . By virtue of subharmonicity a solution of the system {19) that vanishes at the boundary is equal
to zero. Hence there exists a linear operator R that translates the boundary subvector ~D into an interior
subvector ~I:
Inasmuch as this renders the submatrix of coefficients of the interior unknowns in the system (19) nondegenei'ate, .'my solution of the boun~kary subsystem is continued to a solution of the system. Therefore the
dimensionality we seek is equal to the dimensionality of the space of solutions of the boundary subsystem.
By Theorem 2 the boundary subsystem decomposes into independent subsystems, and Theorem 10 is applicable to each of the latter.
Turning now to the investigation of arbitrary eigenvalues on the unit circle, We set
[[ ~,~
I(K)=
for any loop K in the graph of the matrix P. We denote the length o~ the loop K by l(K).
L E_.Mb.IA 4_. Let P be a nonde.composable matrix satisfying condition (I5). The eigenvatues ~f P on the
unit.elrele a r e the roots of the set of equations
x~
--- I ( K ) ,
(2s)
where K spans the set of all loops in the graph of P. The corresponding proper subspaces are one-dfraensionaL
F o r the proof it suffices to go from the matrix P r o the matrix k ' l P (1;~[= l) and apply Theorem 10.
We investigate the system (21). First, obviously, we need consider only simple loops K. This reduces the system to a finite system of the form
h t = !,
(= = l . . . . .
r).
(~)
We denote the largest common divisor of the numbers ls (s = 1. . . . . r) by h. The number h is called the primltivity index o f t h e m a t H x P (see [9, 4]). We define the integers ms(s= 1. . . . . r) so that
F,t~.ffih.
l~)
P--i
Then f r o m (22) w e obtain
$*
ffi [ I I.-..
(~)
439
However, the converse is only true under certain conditions, which are equivalent to solvability of the system, as well as independence of the right-hand side of F.q. (24) from the choice of the number's m s satisfying (23).
LEMMA 5, In o r d e r for the system (22) to be solvable it Is necessary and sufficient to have
P
P
]1 !.-.
t~. = o-~
a~,l
~ t.
C~s)
a-q
If (25) is fulfilled, the system (22)is equivalent to Eq. (24).
Proof. The necessity is Obvious. F o r the proof of the remaining assertions we need only verify that
under (25) Eq. (22) follows from (24). We set
I,' = -~- (t ---- t, .... r).
Equation (24) leads to
r
But by virtue of (23)
I,'m, = t -- ~ lJm~.
Consequently, in order to obtain (22) it is sufficient to verify that
J, o.',
fl J~',~ = t.
e'~S
but this follows from (25), since
,,(-
§
,c,.,oo,-- o
Thus we 1-u,~veestablished the following:
THEOREM 12. Let P be a nondecompesable matrix,
~ Ip,~l~< t
(i----t . . . . . n),
kml
let {Ks}l r be all the simple loops in its graph, J s = J t N s ) . a n d / s = l ( K s ) . In order for the matrix P to have
elgenvalues on the unit circle it is nece.ssary and sufficient to ha~,e
R
Y, l P , h [ = t
(i = t . . . . . n)
and
Y,l,n, -- O ~
a'--l
1~. = 1.
m~l
iI this condition holds, the eigenvalues on the unit circle are the roots of the equation
f
a--!
where h Is the primltivlty index,
P
l.m. --- h;
and all proper subspaces are one-dimensional.
44O
For stochastic matrices ~:lk* I for .ll 1, k(Pik ,~ 0); therefore Js = 1 for all s, and we arrive ~t the
classical result of Frobenius (see [41), whereby the spectrum of a nondec~mposable stochastic matrix on
the unit circle consists of the roots of the equation
and the proper subspaces are one-dimensionaL
Thegeneral case of a decomposable matrix Is [mmedlately reducible to the nondecomposable case
{see T h e o r e m
II).
17. T H E D E P T H AND R E L A T E D
ESTIMATORS
Let G= IX, F) be a p-finite graph. We define the depth of a vertex x as the length l(x) of the shortest
path from x to the boundary* 0X. For x E ~ we put l(x)= 0. We refer to the set {xl/(x)=l} as the layer of
depth l. We define the depth of a graph by the formula
L -= sup/(z).
The possibility of L= ~ is not excluded.
Clearly,
,.t. It(v)Iv ~ r.} -- t(~) - i
Let ~x) be a subharmonic function on Int X. We set
(r~ § ~).
= 8,p{~(x) It(z) =0.
THEOREM 13. po>-~i_>,~_>...
For the proof we denote by Xk the amalgamation of layers of depth t _>k (k= 0, 1, 2 . . . . ), and by Gk
the graph (Xk, Fk), w h e r e
r.z= {r,.
(l(z)>k),
(t(,)= k).
The limitation of function ~(x) on X k |s subharmonic with reference to the graph of Gk. As the boundary of this graph Is a layer of depth k on the original graph, according to the maximum principle
p, = sup{~p(x)It(x) ~ k},
w h e n c e / ~ Pk+ lThe concept of ~]epth permits a certain refinement of Theorem 9. For the matrix P involved in that
theorem we introduce the coupling coefficient ~I of t h e / - t h layer in the graphofthe matrixwith the ( / - 1)-th
layer:
,~ = mln{~, I I(O = 0 (t~ I),
where
O i ffi=
~'
Ip, I.
JllO)-.~O- t
THEOREM 14. Under the condition of Theorem 8
p (P) ~ max {p (oP),
(,- b-[L
UL
}.
where L Is the depth of the graph of the matrix P.
Proof. Let }, be an eigenvalue of P, a n d let {(1}1n is the corresponding eigenvector. We set
It ffi maxl1,1,
From the equation
I~ = max { 1 ~ [ 1 1 ( 0 = I}.
I
m
* S e e [6], pp. 4 3 2 - 4 3 3 , w h e r e t h e depth concept (under the t e r m WpositlonW) Is introduced in a special situation. The estimator from [61 based on this concept is generalized below (Theorem 14).
441
with l(l)- ! e I we deduce
IM I11,l
< ~,,..-, + (I - ,,',1~, < ..~,,_, + (! - ,,,1~.
Hence
(27)
If ~k~> p(aP), the boundary subsystem of the system (26) has only a trivial solution. Therefore
=0. Iterating (27) with regard for the a priori estimator Ixl-<l, we obtain
(28}
The estimator IxI < 1 follows from the particular Eq. (261 in which ~il=/~
If
z.
IM" > t - l 'p..,|
[ *~,
then
!
IM'> t-l'[
~
t l = t . . . . . L).
and by (28)
m<a
(t----- o, t . . . . . L),
which is inadmissible.
LITERATURE
1.
2.
3,
4.
5.
6.
7.
8.
9.
442
CITED
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F . R . Gantmakher, Theory of Matrices [in Russian], Gostekhizdat, Moscow (1953).
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J., 1~0' No. 2, 181-196 (1960).
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(1959).
O. Taussky, ~A recurring theorem of determinants, ~ Am. ~[ath. Monthly, 56, 672-676 (1949).
L. Collatz, ~Uber die konvergenzkriterien bei iterations verfahren ffir lineare gleichungssysteme,~
Math~ Zeitschr., 53, No. 2, 149-161 (1950).
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4, 496-501 (1958).