Matrix Generalizations of Some Theorems on Trees, Cycles and Cocycles in Graphs
Author(s): Stephen B. Maurer
Source: SIAM Journal on Applied Mathematics, Vol. 30, No. 1 (Jan., 1976), pp. 143-148
Published by: Society for Industrial and Applied Mathematics
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SIAM J. APPL. MATH.
Vol. 30, No. 1, January1976
MATRIX GENERALIZATIONS OF SOME THEOREMS
TREES, CYCLES AND COCYCLES IN GRAPHS*
ON
STEPHEN B. MAURERt
Abstract.We extendto arbitrary
matricesfourtheoremsof graphtheory,one about projections
onto the cycle and cocyclespaces, one about the intersectionof these spaces, and two matrix-tree
theorems.The squares of certaindeterminants,
not apparentforgraphs,appear in the extensions.
1. Introduction.We present here four theorems about matrices and
associated vectorspaces. When the matricesare chosen in a special way, the
theoremsreduceto knownresultsingraphtheory.This paper arose froma desire
to generalizethose graphtheoremsto the,generalindependencesystemscalled
matroids.Since everyfiniteset of vectors(e.g., the columnsof a matrix)is a
matroid,and since manymatroidscan be representedin thisway,we have been
partlysuccessful.Our theoremscan also be looked at fromotherpointsof view.
For instance,Theorems1 and 2 maybe interpreted
intermsofdual linearcodes.
We willfirststateour theorems,thenstatethegraphtheoremstheyinclude,
thengive proofs.
2. The theorems.To stateour resultsconciselywe need considerablenotation.For easy referencewe introducethisnotationin a list:
9 is a fieldof characteristic
p.
M is an m x n matrixover S.
R and 16are the row and columnspaces of M.
X is the nullspace of M, i.e., the orthogonalcomplementof R.
, en} is the standardbasis of 3?/.
{e1,e2,
J={1, 2, . , n}. For KcJ, K=J-K.
$Y9
is thecollectionof subsetsofJ indexingcolumnsetsof M whichare bases
of T6.Bo is a fixedset in X, B rangesover all of 3. Bo and B willoftenreferto the
bases indexedas well as the indexsets.
RBej, forj E B, is theunique vectorE a1e, in R such thataj = 1, and ai = 0 for
B
i,e U {j}. (Uniquenessis well knownand willbe verifiedlater.)For j E B, RBe, is
definedto be 0, thezero vectorof
RB is the linearmap from 9In to R such thatRB(ej) = RBej forall j E J.
R is the matrixwithrowsRBoej. j E Bo.
= 0 for
NBe,, forJE B, is theunique vectorE fBiei
in .X suchthat1,j= 1, and 13i
i/ B U{j}. Forj E B, NBej =0.
gn __>
is the linearmap withNB(ej) = NBej, j E J.
NB:
N is the matrixwithrowsNB0eJ,
j E Bo.
E
is
element
in
or
an
variable
over S.
Xj,j J,
XK =IK Xjforj E K c J.
X: gn _>in is the linearmap withwithX(ej) = Xjej.
X is the matrixof the mapX.
Sn.
-f
* Received by the editorsDecember 14, 1973, and in revisedformJuly1, 1974.
t Departmentof Combinatoricsand Optimization,Universityof Waterloo,Waterloo,Ontario,
Canada. Now at MathematicsDepartment,PrincetonUniversity,
Princeton,New Jersey08540.
143
144
STEPHEN
B. MAURER
ofmatrix
pt is thetranspose
P.
ofsquarematrix
P. Wedo notorderrowsandcolumns,
IPIisthedeterminant
so someofourresults
arecorrect
determinants
involving
onlyuptosign.Thenwe
write =
i
ratherthan =.
M(A, K), whereA is anybasisof16andK c J,is thematrix
whosecolumns
givethecoordinates
relativetoA ofthecolumn-vectors
ofM indexedbyK.
[A : K]= IM(A,K)J2I assumingM(A, K) is square. Note there is no
ambiguity
aboutthesignof[A K].
rj F=>ja[Bo: B].
1. RB isa projection
onto24andX ZB[Bo: B]XBRBisself-adjoint.
- & [BO: B]RB is theorthogonal
In particular,
if 1$ 0, qr
projection
ontoR with
X. Similarly,
kernel
NBprojects
ontofN,X & [Bo: B]XBNBis self-adjoint,
and if
onto
with
kernel
X
R.
7 $0, N-1'Z [Bo: B]NB is theorthogonal
projection
THEOREM 2. R n.A={O} ifand onlyif 1 $ 0.
THEOREM 3. JRX Rtl=
[Bo: B]XB and IN X Ntl= [Bo: B]]XB
THEOREM
4.
= ?
+
R=
[Bo: B]XB and NX
t>2[BO:
B]X .
We saidthatthesetheorems
includethegraphtheorems
tofollow.Actually,
twoofthegraphtheorems
a somewhat
haveas theirbest-known
forms
anomalous
specialcase notincludedin theabove.The above can be statedin evenmore
to includethesespecialcases,buttherestatements
generality
requireevenmore
notation.In theprocessof provingtheabove we willin factprovethefurther
andpointouthowtheycovertheanomalousgraphcases.
generalizations
THEOREM
RX
3. Graphs.Let graphG have m nodesand n edges.For convenience
we
assumeG connected.
Iftheedgesarenotalreadydirected,
directthemarbitrarily.
DefineM = [ai,]byai, = 1 ifedgej entersnodei,-1 ifedgej leavesnodei,and0
otherwise.If G was directedto beginwith,the usual fieldis the rationals;
otherwise
theusualchoiceis {0, 1}. However,whatfollowsis wellknownto be
trueforanyi. X is justthespacegenerated
of G, i.e.,thecycle
bythecircuits
space.2 is thespacegenerated
bythecut-sets;
graph-theorists
nowadays
usually
callthisthecocyclespace,buttraditionally
itwasthespaceofcoboundaries,
with
cocyclesmeaning
else.A setofcolumns
ofM isa basisof'6 ifandonly
something
ifitisindexedbya spanning
tree,andfortwosuchbases,whichinthissectionwe
call TO,T insteadof Bo, B, IM(T0,7T)= ?1. Finally,NTej,j e T, is justwhatis
knownas thefundamental
ofcyclesforT,and RTe,,jE T,thefundamental
system
ofcocycles.TheyarebasesofX and2, respectively.
system
Forgraphsthereis
anotherwayto geta basisof 2. Foreachvertextheedgesincident
to itforma
cut-set,
and thecorresponding
cocyclesforall butanyone vertexforma basis.
Sucha cocycle-basis
isalsocalleda fundamental
a vertex-isolating
system,
system.
THEOREM 1'. LetTbe inthecollection
trees
27ofspanning
ofgraphG. ThenRT
is a projection
ontothecocyclespace,and X E, XTRT iSself-adjoint.
Let7 be the
number
treesmodp, thecharacteristic
ofspanning
oftheunderlying
field.If q $ 0,
f'- EgRT is theorthogonal
ontothecocyclespacewiththecyclespaceas
projection
itskernel.
ontothecyclespace,X E, XrNTisself-adjoint,
Similarly,
NTprojects
and
ontothecyclespacewiththecocycle
if 1 $ 0, f-'ZgNTis theorthogonal
projection
spaceas kernel.
MATRIX GENERALIZATIONS
OF GRAPHS
145
Theorem1' is due to Nerode and Shank[3]. Theyused theself-adjointness
of
X E, XtNT to give a short proof of the famous formula,firstestablishedby
Kirchhoff
[2], forthecurrentsin a resistiveelectricalnetwork.In thatcontextXiis
the resistancein wirej.
THEOREM 2'. 7he cycleand cocyclespaces of G overa fieldof characteristic
p have nontrivial
intersection
ifand onlyifthenumberof spanningtreesequals 0
mod p.
Theorem2' is due to Shank (unpublished)butthespecial case p = 2 has been
publishedindependently
by Shank [4] and Chen [1]. Actually,Chen's argument
worksin general,as we show below.
THEOREM 3'. Let R (resp. N) have some fundamentalsystemof cocycles
(cycles) of G foritsrows.Then
IRXRtl=ZE.XT, INXNNI= ,XTr.
Theorem 3' is a generalmatrix-tree
theoremappearingin Trent[5], When
X = I and p = 0, eitherdeterminant
givesthenumberofspanningtrees.Whenthe
rowsof R are a vertex-isolating
fundamentalsystem,RR' is a principalminorof
the degree-minus-adjacency
matrix.This is thebest-knowncase of thetheorem,
but also the case not coveredby Theorem 3 as stated.
NowadaysTheorem3' is oftenattributedto Kirchhoff,
forit is claimedto be
setdowna systemoflinearequationsand showed,
implicitin hispaper. Kirchhoff
withoutusing determinants,
that the solutionscould be writtenwithcommon
denominator ,jXt.This certainlysuggeststhat the matrixof coefficients
has
determinantZ3XT,but thiscoefficient
matrixis not any of those in Theorem 3'.
The following,whichwe have not seen statedin the literature,is morerightfully
called Kirchhoff's
matrix-tree
theorem.
THEOREM 4'. WithR and N as in Theorem3,
N
RX
NXXT,
R
NX
matrixwas of the formon the right,withR arising
Specifically,Kirchhoff's
froma vertex-isolating
fundamentalsystemand N fromsome tree.
4. Proofs.Theorems 1-3 can all be proved by a few modificationsin the
proofscited forTheorems 1'-3', but forclarityand completenesswe give full
argumentsforall of Theorems 1-4. We presentthe proofsin severalsteps.The
purposeof the firstthreestepsis to cut the remainingworkin half.
One morebatchofnotation:M* is anymatrixwithrowspace X and thusnull
space R. (From the matroidpoint of view, M and M* representduals). The
columnsof M*, like thoseof M, are indexedbyJ.Let c6*be thecolumnspace of
M*. If A* is a basis of V, and Kc J, M*(A*, K) and [A* : K]* are defined
analogouslyto M(A, K) and [A : K]. Finally,( ,.) is the usual innerproductin
if u = E aej and v = Z3jej,then(u, v) = E a,31j.
9n
146
STEPHEN
B. MAURER
(a) K c Jindexesa basisoft* ifand onlyifK E $@.(Thiswas first
observedby
matroids[6].) Any two matriceswiththe same
Whitneyin his paper introducing
rowspace can be obtainedone fromtheotherbyelementaryrowoperations,and
theseoperationsdo not affectwhichsets of columnsare columnspace bases. So
foranyB E X3,we mayreplaceM with[I, P], whereI is theidentityindexedby B.
Indeed, [I, P] = M(B, J). It is theneasy to show that[-Pt, I'] is a choice forM*,
where1*is theidentity
indexedbyB. Thus B indexesa basis of c(*. Indeed,forany
choice of M*, [-PI' I'] = M*(B, J). Conversely,to show thateverybasis of 10 is
some B, repeat the above argumentwiththe roles of M and M* reversed.
(b) ForK c J,IM*(B,
K)J= ?IM(B, K)j, so [B: K]* = [B: K]. We have just
shown that M(B, J) = [I, P], M*(B, J) = [-P', F]. Thus if Q is the minorof P
indexedby rowsB n K and columnsB n K, thenIM(B, K)I = ? IQI.Likewise,if
Q*is the minorof -Pt indexedby (B n K, B n K), thenIM*(B, K)| = ?IQ*I.But
(c) Let A be any basis of '6 (notnecessarilyone in 2%7).
Let A * be any basis of
10. Foreveryresultbelowinvolving[A: B], XB,RB,X, .X,R and/orN, thereis an
analogous resultinvolving[A* B]*, XB,NB,X, X, N and/orR. Simplyapplythe
statedresultto M* insteadof M.
(d) RBej is well-defined.Let u, v be two vectors satisfyingthe defining
conditionsforRBej forsome j E B. Then u - v = E ate,E R, where ai = 0 forall
i E B. Since B indexesa columnbasis forM, all theothera, dependon thosewith
i E B. Thus u = v.
(e) RB is a projection
on R. Clearlytheimageof RB is in R, so we mustshow
RB is the identityon R. WritingRBu - u = E ajej, we have forany u E i9n that
ai = 0 forall i c B. If u e , thenRBU - u E R and the argumentin (d) applies.
(f) The vectors
RBeJ,
j E B, are therowsofM(B, J). In particularR = M(Bo, J).
Since M(B, J) = [I, P], with I indexed by B, row j meets the two conditions
definingRBej.
In the nextfewparagraphswe abbreviateB U {j} - {i} as B - i + j.
(g) (RBei, e1) ?0 if and only if B - i +j e 2. (RBei,e,) is the (i, j) entryof
M(B, J) = [I, P]. Clearly,ifthisentryis 0, columnj dependson columnsB - i,and
conversely.
(h) IfB' = B-i +j,then (RBej, ei) = (RBei, ej)-,. First,RB'ej = (RBei, ej)-'RBei,
fortherightside meetsthetwoconditionsdefining
theleft.Second,(RBei, ei) = 1.
(i) [A: B][B :K] = [A :K]. This follows because M(A, B) x M(B, K)
-M(A, K).
(j) X > [A: B]XBRBis self-adjoint.Call thisoperatorH forshort.We must
show (Hei, ei) = (ei,Hei) for all i, j. By (g) the only summandsin H that give
= {BIB - i + j E 2}, and on therightonly
nonzerotermson theleftare thosein /3ij
,. ClearlyB->B'= B-i +j is a bijectionbetween 4'ijand s?I,,so it sufficesto
prove
(X[A: B]XBRBei,e) = (ei,X[A: B']XBRBej).
The left simplifiesto XjXB[A: B](RBei,e,), the rightto XiXB'[A: B'](ei, RBe).
Since XjXB= XiXB,by (h) and (i) it sufficesto prove (RBei,ej)2=[B: B']. But
M(B, B') is an identitymatrix except for one column, so in light of (f),
IM(B, B')I = ?(RBei, e1).
MATRIX
OF GRAPHS
GENERALIZATIONS
147
(k) Define, =
[A : B] = [A : Bo]r1,and H = E[A : B]RB. If , #0 , then
ontoR withkernelfV.
the
-'1His
orthogonal
projection
By (e), IJJ-His a projection,
A
and by (j), it is self-adjoint.A projectionis self-adjointifand onlyifitskernelis
the orthogonalcomplementof its image, and by definitionX is the orthogonal
complementof R.
We have now provedTheorem 1. ChoosingBo forA, (e), (j) and (k) givethe
firsttwosentencesof thattheorem.Applying(b) and (c), we getthelastsentence.
A to Bo originally.Because of (b) it was
There were two reasons forrestricting
thenunnecessaryto introduceM* and relatednotationat thebeginning.Of more
mathematicalinterest,onlywhenA E X13is A a basis of the associatedmatroid.
Next we prove Theorem 3. It followsfrom(1) below in the same way that
Theorem 1 followedfrom(e), (j), (k).
(1) Let R be any matrixwhoserowsare a basis forR. Then R = M[A, J] for
somebasisA ofX, and IRX RI| = Z [A: B]XB. We have alreadynotedthatiftwo
matriceshave the same row space, e.g., R and R, theydifferonlyby elementary
rowoperations.If in additiontheybothhave k rows,theircolumnsetsrepresent
bases of 3;k, whichis the column
the same vectorswithrespectto two different
space whenevertherowsofeitherare independent.ThisprovesR = M(A, J). The
secondclaimis an immediateconsequenceof theChauchy-Binettheorem:ifP is
m x n and Q is n x m, then IPQI= |Pcl IQK|, wherePK rangesover all m x m
m x m minorsofQ. (See Trent
minorsofP, and QK rangesoverthecorresponding
[5] fora completedescription.)Now take P = RX, Q = Rt, and recall thatX is
diagonal and IM(A, K)I #0 ifand onlyifK E A.
(m) R n X = {0} ifand onlyif7 #0 (Theorem2). We followChen [1], using
Theorem 3 wherehe used Theorem 3' restrictedto the field{0, 1}. Considerthe
square matrixP =
[NjIt sufficesto show IPI= ?77,forclearlyR n X = {O}ifand
onlyifIPI #0 . Now
t= [RRt RNt]RRt
NNt
0
forcingIPI=
q.
NRt
Thus by Theorem 3, IPPtl
(n)
= 72,
01
NN'J
= Z [BoB]XB (Theorem4). Let P=
RNX
RRt
0 1
VN =[NXRt
Then
[N]Q= [NX]
NXNNtJ
so IQIIPI= 7q [Bo: B]XB by Theorem 3. Since IPI= ?, Theorem 4 follows
wheneverq #0 . To cover the case q = 0, recall thatR = M(BO, J) is of the form
[I, R'], and temporarilylet the entriesin R' be independentvariables over 9
ratherthanelementsin i. Then surelyq #0 and thetheoremholds,providinga
wheneverR' is made up of elementsof i..
formulawhichholds by substitution
(o) Let R = M(A, J). Then
NX
=
from(n) and the factRt= M(A, Bo) X Rl.
?
IM(A, Bo)IlZ [Bo: B]XB. This follows
148
STEPHEN
B. MAURER
The specialcase ofTheorem3' followsfrom(1).Recallthatinthiscase R of
fundamental
cocyclesystem.
thetheorem
(RIin(1))comesfroma vertex-isolating
Forgraphs,
[A: B] = ?1 whenbasisA comesfroma treeandinthisspecialcase.
Likewise(o) coversthespecialcasesofTheorem4', ofwhichtherearetwo:when
andwhenitcomesfroma
ofR isvertex-isolating,
thefundamental
cocyclesystem
treethanthefundamental
ofN.
different
cyclesystem
We havean alternate
here,which
proofofTheorem2, omitted
5. Remarks.
overthenumber
of
ofinduction
technique
usesthestandard
graph(andmatroid)
and deletions.The statement
of 2' does not
edgeswiththeaid of contractions
proof.
andneither,
inthisgraphcase,doesouralternate
involvematrices,
withsomecoefficients
to arbitrary
matrices
for
ThatTheorem3' generalizes
toanyonewhohasstudiedtheproof.Whatmaybe
beenevident
thex's hassurely
andthefact
ofthesecoefficients
description
newinourTheorem3 is thespecific
and INX N'l.
thattheyarethesameforJRX RWj
to
thenumberof bases in 0Yis a problemof interest
Finally,determining
when
matroidtheorists.
Theorems3 and 4 are mostinteresting
Consequently
M forwhichthe
p = 0, X = 1,and[Bo: B] = 1 forall B E $. The classofmatrices
matrices
unimodular
holdsin justthewell-known
lastcondition
(equivalently,
ofbasesforanyM by
thenumber
In theory
wemaydetermine
regular
matroids).
thenumber
eachXia different
computing
making
variable,
JRX R'J,andcounting
of distinctproductsXB whichoccur,ignoringtheircoefficients.
However,
variablesis,toputitmildly,
determinants
time-consuming.
containing
evaluating
to
Ifp = 2, Shankhasshown[4,Lemma5] thattofindallthedistinct
XBitsuffices
butitisverydoubtful
theproduct
ofthediagonalentries,
anysuchresult
compute
holdsforotherp.
us in these
We thankProfessor
Shankforinteresting
Acknowledgment.
the
for
several
valuable
We
the
referees
concerning
matters. thank
suggestions
ofthispaper.
organization
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