Connections between the matching and chromatic polynomials

Sci. VOL. 15 NO. 4 (1992) 757-766

757

CONNECTIONS BETWEEN THE

MATCHING

AND CHROMATIC POLYNOMIALS

E.J.FARRELL

Department

of Mathematics The University of the

West

Indies

St.

Augustine,Trinidad

and

EARLGLENWHITEHEAD,JR.

Department

of Mathematics

&

Statistics University ofPittsburgh Pittsburgh,

PA

15260

U.S.A.

(Received

June

8, 1990 andin revisedform

June

6,

1991)

ABSTRACT.

The main results established are

(i)

a connection between the matching and chromaticpolynomialsand

(ii)

aformula for thematching polynomialofa

general

complementofa

subgraph ofagraph. Somedeductionsonmatching andchromaticequivalence and uniquenessare maie.

KEY WORDS

AND

PHRASES.

Matching polynomial, chromatic polynomial, matching equivalence,chromaticequivalence, triangle-free graphs.

1991 AMS SUBJECT

CLASSIFICATION

CODE. 05A99, 05C99.

1.

INTRODUCTION.

The graphsconsideredherearefinite, undirected,andcontainnoloopsandnomultiple edges. Let Gbesuchagraph.

A

matchinginGisaspanningsubgraphof

G,

whosecomponentsarenodes andedges only. Let _ak be the number of matchings inGwithkedges and letnbethe number of nodes in

G.

Thenthematching polynomialofGis

k=0

wherew and_{w2 are}indeterminatesor weightsassociatedwith each node andedge respectively,in G. Ifweputw _w2 w, thenweobtain

[./21

M(G; to)

akw

n-k,

k=0

which is called the simple matching polynomial of G. The basic properties of the matching polynomialcanbe foundinthe introductory paperbyFarrell

[1].

colors. The basicproperties of

P(G;A)

canbefound in Read

[10].

In

thispaper, we willassume

that

P(G;A)

isexpressedinthe completegraphbasi____s, i.e.,

P(a;)=

%()n-,

k=0

where _ckis the number of color partitions of the nodesofGinton-k non-emptyindistinguishable

classes and

(A)r

A(A-

1)(A- 2)...(A-

r4-

1).

Let

rn(a)

denote thevectorof thenonzerocoefficientsof

M(G;

_w),

written inascending powers ofw_2. Thisvectoriscalled thematchingvectorof

G.

Graphs Gand

H

arematchingequivalent if

_re(G)

_re(H).

Analogously, let

c(G)

denotethevectorof thenonzerocoefficientsof

P(G;$).

This vector is called the chromatic vector of G. Graphs G and

H

are chromatically equ.ivalent if

c(G)

c(H).

Let

V(G)

denote the node set of G and

IV(G)

denote the cardinality of

V(G).

Graphs G and

H

are called co-matching if

IV(G)[

IV(H)[

and G and

H

are matching equivalent. Graphs G and

H

are called co-chromatic if

IV(G)

IV(H)

,d ,d e

chromatically equivalent.

A

graph Gis matching uniqueif

M(G;_w)= M(H;_w)

implies that

H

is

isomorphictoG. The termchromaticallyunique isanalogouslydefined.

A

chain isa tree withnodes of valencies and2only. The chain withnnodes will be denoted by

’Pn.

The notations

Kp,

Krn,

n and

Cv

will be used for the complete graph with p nodes, the

complete m by n bipartite graph and the cycle

(circuit)

with p nodes, respectively. Let 5’ bea subset of

V(G).

ThenG- 5’willdenote thegraphobtainedfromGby removing the nodesinS. If S isasubset of

E(G),

then

G- S

will denote thegraphobtainedfrom

G

by removing theedgesin

S.

Weextendaresult ofFruchtand Giudici

[3],

by showing thattheir necessary conditionfor the matching vector to be equal to the chromatic vectoris also a sufficient condition.

A

formulafor the matching polynomial ofacomplement ofasubgraphofalabeledgraph is then derived. Also,

we obtain connections between matching and chromaticequivalence and uniqueness. Finally, we

deducesomeresults for0-graphs.

In

thematerial whichfollows,the upper and lower limits of summations willbeomittedwhen theyare obvious from the contents of the summand. For example, the lower summationlimitis zero and the upper summation limit is

[n/2]

in all matchingpolynomials of graphs with n nodes. The notation

G

UG₂U..-U

Gr

will denote the disjoint union ofgraphs

G1, G2,...

and

Gr.

The

notation

(G,H)

will be used forapair of two equivalent,co-matching orco-chromaticgraphs; this pair will be referredtoasanequivalent, co-matchingorco-chromaticpair,asappropriate.

2.

CONNECTIONS BETWEEN THE

MATCHING AND THE CHROMATIC POLYNOMIALS.

Thefollowinglemmascanbe easilyproved.

LEMMA

1.

Every

propercoloring ofagraph Gwithvcolorsinducesapartition of

V(G)

into rpartsinwhichnodeszandtbelongtoapartonlyif zt

_

E(G).

LEMMA

2. Let Gbeagraph

(without

loopsormultiple

edges).

Then

P(O;

)

y

ak(,)l

k, k

where

at:

is the number of partitions of

V(G)

into exactly p-k parts such that nodes z _{and t}

belong to the same part only if zt

:

E(G);

and the summation is taken over all (nonnegative integral)values ofk less thanp.

CONNECTIONS BETWEEN THE MATCHING AND CHROMATIC POLYNOMIALS 759

LEMMA

3. Let

{R1,R2,...,Rn}

be the set of all color partitions of

V(G),

where

R

_k

{SI,k,

S2,/,...,Srk,

_k

}.

Then

Si,

_j

<

2 for all integers and j such that 1

< <

_{rj and}

1

<

j

<

n, if andonlyif is /k-free (triangle

free).

Hence,

if

P(G;$)=

_ak()Op_

k. Then a/ris k

the number of matchingsinGwith p-k componentsifandonlyif

G

is A-free.

PROOF.

Suppose

that

G

is /k-free. Let us assume that

]Si,

j]

>

2 for some and j.

Let

x,l,z

_

Si,

_j. Then x, / and z are nonadjacent in G. Thus, ztz is a triangle in G. This is a

contradiction. Therefore,ourassumptionisfalse.

[Si,

_j

<

2 for all and j.

Conversely, suppose that

]Si,

_j

<

2 for all and j. Then G does not contain a proper

coloring in which three nodes can be coloredthe same. Thus, G does not contain a set of three mutually nonadjacent nodes. Therefore, G is /k-free. This proves the first part of the theorem. When

Si,

_j

-<

2for all and j, thesubgraphofGinducedby

Si,

_jis eitheranode

(if

Si,

]

1)

oran edge

(if

Si,

_j

2).

Itfollows that each color partition of

V(G)

induces auniquematching

inG.

Since the number ofmatchings in

G

with rcomponentsis thecoefficientof wr in thesimple matchingpolynomial of

G,

weimmediately obtain thefollowing theorem.

THEOREM

1. Let

G

be a graph. Then

P(G;A)=

M(;w’)

(where

w’

means that w in

M(G;w)

is replaced by

(A)/)and

dually,

M(G;_w)=

P(G;A’) (where

A’

means that

(A)

isreplaced bythe monomial

w

2 p

w

andp isthenumber of nodesin

G)

ifandonlyif is A-free.

COROLLARY

1.1.

_re(G)

cG)

if andonlyifGis A-free.

This corollaryis animprovement of the mainresult in

[5],

since it gives bothanecessary and sufficient condition for the matching vector ofagraph to beequal to the chromatic vector of the complementgraph.

Suppose

that

G

isnot A-free. Thenthe sets

S

_i,may havemorethan two elements.

In

this case, for some k, there will be other k-color partitions of

V(G)

(viz

those which contain

Si, j’s

for

which

Si,

_j

>

2).

Thisobservationand Theorem1yield thefollowingresult.

THEOREM

2. Let Gbeagraph;

itssimplematchingpolynomial and

M(G;

w)

bkwt’

k k

P);

)

a()-k

the chromatic polynomial of its complement. Then a/r

> bk,

for all values of k. Furthermore, equalityholds

(for

all values of

k)

if andonlyif

G

is /k-free.

We

notethat Theorem 2, notonlycharacterizesgraphswhosematchingvectoris equaltothe chromatic vector of its complement, but also gives

(i)

a lower bound for the coefficients of the

chromatic polynomial and

(ii)

a criterion for determining whether ornot agiven graph contains

triangles.

3.

MATCHING POLYNOMIALS OF COMPLEMENTS

OF

GRAPHS.

DEFINITION. Let H

be agraph and

G

asubgraph of

H.

A

complement of

G

in is a

graphobtained from

H

byremovingtheedgesofanisomorphof

G.

Thus,

if

R

is acomplementofGin

H,

then

V(R)= V(H)

and

E(R)

E(H)- E(G’),

whereG’

inH.

In

this case, we denote the uniquecomplement ofGin

H

byG_it. Ifwetake

H

tobe the complete graph

K

p, then again the complementis unique and is denoted by

Gp,

or simply by

G,

when it is unnecessary to specify the number of nodes. Notice that ifin addition, G also has p

nodes, then the complement is the usual graph complement. If

H

is the labeled bipartitegraph

Kin,

n, thenwe denote thecomplement by

Gm,

n, or G_B,where

B

is acompletebipartitegraph,if

itis unnecessary tospecify the cardinalities of the disjoint node sets.

Thefollowing theoremgivesthe matching polynomial of thecomplement ofagraph t7relative to anylabeledgraph

H

which containsGas asubgraph.

THEOREM 3. Let Gbe agraph with eedges. Let

H

bea labeledgraph containing

G

as a

subgraph. Then

M(GH;W-)=

(-1)kw2

k

_,M(H-V(Sk);w_

),

k=l

where S_kis aset of kedges ofG belongingtoamatchingin

H

and the secondsummation istaken

overallsuch matchingsin

H

containing kedgesofG.

PROOF. Our proof is based on the Principle of Inclusion and Exclusion. Let

Pi

be the propertythat theedge ofGisusedinthe amatchingofH. Thenthereare eproperties onefor

eactl

edge in G.

Let

N(pl,P2

",Pie)

denote the contribution to

M(H;_w)

of the matchings containing the edges 1,2,...,k of G. We must find

N(p’l,P’2,...,P’e),

i.e., the contribution to

M(H;_w)

of the matchings which do not contain any of the e edges of G. Therefore, this is the

contributionof all thematchingsin

GH,

i.e.,

M(GH;W_

).

Let

D

_k be amatching in

H

which contains the set S_k

{i1,i2,...,ik}

of kedgesofG. The edgesin S_k areindependent and therefore cover2k nodes ofH.

Suppose

that

H

contains nnodes. Then the remainingn-2k nodes of

H

canbecoveredbyamatchingc_kof

H-

V(Sk).

Clearly,no

edgeofer_kcanbeincident to anyof the kedgesinG.

Therefore the matchings which cover the n-2k nodes can be enumerated independently. They will be all the matchings in

H-

V(Sk).

Therefore, thecontribution of D_k to the matching polynomial of

H

is

N(Pl,

P2,"

",Pk)

wk2M(

H-

V(Sk);w-)"

Thecontributionof all suchmatchingsof

H

withkedges belongingtoGis therefore

,

wk2M(H

V(Sk);w_

),

where the summation is taken over all such matchings

D

_k of H. The theorem follows by the Principleof InclusionandExclusion.

Suppose,

in the above theorem,

H

is the complete graph

K

n. Then

H-V(Sk)

will be the completegraph

Kn_

2k" IfGcontainsakmatchingswith kedges, then the second summation will

become

akM(Kn_2k;w_).

Thus wehave thefollowingresult

(also

essentially givenin Zaslavsky

[4]

andWahid

[51.

COROLLARY

3.1. Let Gbeagraphwith pnodesandn

>

p,apositive integer. Alsolet

THE MATCHING AND CHROMATIC POLYNOMIALS 761

Then

M(n;_W)

"(

1)k

akwk2M(K,_

2k;w_

).

In

the case where

H

is the labeled complete bipartite graph

Kin,

n, we obtain the following

result,also given byGutman

[6].

COROLLARY

3.2. Let

M(G; w_)=

,

_ak

w-2k

w,

whereGisabipartite graph. Then

M(m,n;

’(--1)kak

wk2

M(gm_k,n_k;w_).

We note from Corollary 3.2 that the matching polynomial of

Gm,

n depends only on the

coefficients of the terms in

M(G;w_).

It

follows that all the complements of

G

in

Km,

_n are

co-matching. We therefore speak of the matching polynomial of a complement of a graph in the completebipartitegraph. Thegraphisnot unique but thematchingpolynomialis.

Corollary 3.1 can be improved as shown in the following result which is essentially given in

Codsi

[7].

THEOREM

4. Let Gbeagraphwith pnodes. Then

ifandonlyif

M(G;w_)

y

_ak

w[-2k

wk2

k

M(;_w)

(-

1)kak

wk2M(Kp_2k;w_).

k

PROOF.

Corollary3.1 establishes the sufficiency of the condition. Assuming the formula for

M(G;_w),

it canbededucted

(after

carefulmanipulations),that

M(G;_w)

isthe desiredformula.

Corollaries 3.1 and3.2 canbe used to obtain results concerning equivalence, co-matchingand matchinguniqueness. Thefollowingaredeductionsfrom these two corollaries.

THEOREM

5. If

(G,H)

is a matchingequivalent pair, thensois

(Gn,

Hn).

Ifin addition,

G

and

H

arebipartite,then

(GB,

HB)

isalsoamatching equivalentpair.

THEOREM

6. Gismatchingunique if andonlyif

G

is.

Theorems 5 and 6 can be used to substantially increase the presently known families of equivalent

graphs,

co-matching graphsandmatchinguniquegraphs.

The following lemmaiseasy to prove.

LEMMA

4. Let

G

and

H

be matching equivalent graphs with p nodes and m nodes respectively. Then

M(G;w_)

w

M(H;w_),

wherer p m.

THEOREM

7.

Let G

beagraphwhichismatching equivalenttoa A-free graph

H.

Then

p(r;

A)

M(G;

w*),

where w* is the transformation in which wk in

M(G;w)

is replaced by

(A)r+k

where

r--IV(H)I- V(G)I.

Dually,

M(G;_w)=

p(r;,v),

where ,V _represents the transformation in

Pw

₂ p

which

(A)r

+

kisreplaced by Wl2r

+

2k- p-r-k_{and isthe}

number of nodesin

G.

PROOF.

SinceGand

H

arematchingequivalent,wehavefrom Lemma 4,

From

Theorem 1,wehave

M(H;w’)=

P(H;A).

FromEquation

(1),

it isclear that the polynomial

M(H;w’)

is essentially the polynomial

M(G;w)

with wk replaced by

(A)r+k"

Hence the result follows. The dual part of theproofcanbe similarlyestablished.

Weillustrate the theorem with the followingexample. Let G and

H

be thegraphs shownin

Figure1.

In

thisexample,r 0 becausebothG and

H

have5nodes. FIGURE1:

The following polynomials areeasilycomputed.

and

P(G;)

()5

+

5()4

+

4()3

M(a;w*).

lIotice

thatG

9.

Also,itcanbeverifiedthat

and

P(G;

A)

P(H;

A)

(A)5

+

5(A)4

-4-

5(A)3

+

(A)2

M(H; _w)=

M(G;

_w)

w51

+

5w31w2

+

4WlW

Thus

G

and

H

arematchingequivalentgraphs.

Hence

G

ismatching equivalenttoa A-free

graph

H.

However,

M(G;w*):

P(G;$).

Thus,ifGismatchingequivalent toa A-freegraph,it isthe complementof the /k-freegraph

(and

notthecomplementof

G)

that hasequalchromaticvector.

It is well known

(for

example, see

[1])

that the characteristic polynomial ofa tree coincides

with itsmatchingpolynomial,exceptfor thealternationof the coefficient signs. Thisobservation,

combined with Corollary 1.1, yields the following theorem which gives a relation between the matching,characteristicandchromaticpolynomials.

THEOREM 8. For anytree

T,

_m(T)= _(T)=

_c(),

where

_q,(T)

i the vector oftheabsolute values ofthecoefficientsofthecharacteristicpolynomial ofT.

Suppose

thatwetake

T

tobe thechain

Pn

with nnodes. From

[1]

(Theorem 9),

wehave

Itiswell known

(see

Farrell

[8]

Corollary

7.1)

that

FromTheorem7,wehave

763

Theorem 7 can be used to obtain combinatorial identities when explicit formulas for

P(H;A)

and

M(G;

w*)

areknown. Wegiveanexampleofthis.

EXAMPLE.

()m().

_!

()m+

n-*" k=0

PROOF. Since KmO

Kn,

Kin,

n,itfollows

(Theorem

8)

that

M(Km,

n;w*)

P(Km,

n;+X)

P(Km

O

P(Km;A)P(Kn;A)=

(A)m(A)n.

The results followsbyusing the formula for

M(Km,

n;w_)

givenin

[1] (Theorem 20).

Wenote that thisidentitywasalso derivedbyGoldmanetal.in

[10]

(Corollary

12).

4.

DEDUCTIONS FOR

CHROMATIC

EQUIVALENCE AND

UNIQUENESS.

The following theorem shows that for /k-freegraphs, chromaticequivalence of complementsis

completelydeterminedbymatchingequivalence.

THEOREM

9. Let G and

H

be /k-freegraphs. Then

(G,H)

is a chromatically equivalent pair if andonlyif

(G, H)

isamatching equivalentpair.

PROOF. Suppose

that

(G,H)

is matching equivalent. Then

_re(G)= _re(H).

SinceG and

H

are /X-free,

c()=

_re(G)

and

c(r)=

_re(H).

This implies that

c(O)=

c(r),

which implies that

(G,

H)

isachromatically equivalent pair.

Conversely, suppose that

(O,r)is

achromatically equivalent pair. Then

c(O)= c(r).

Since Gand

H

are /\-free, byTheorem 5wehave

_re(G)=

_re(H),

i.e.,

(G,H)

ismatchingequivalent.

It isclear that if

(G,H)

is aco-matching pair, then (trivially) it isalsoamatchingequivalent pair. If, in addition, G and

H

are A-

free,

then by Theorem 9,

(O,r)

is a chromatically equivalent pair. But Gand

H

must have thesamenumber ofnodes because theyareco-matching graphs. It follows that Gand

H

have thesamenumberof nodes.

Hence

(G, H)

isaco-chromatic pair. Theconverse situation isalsotrue,i.e., if

(G,H)

is aco-chromaticpair, then

(G,H)

must be

aco-matching pair.

Hence,

wehave proven the following theorem.

THEOREM

10. Let

G

and

H

be /\-freegraphs. Then

(O,r)

is.aco-chromatic pair ifand onlyif

(G,

H)

isaco-matching pair.

In

Farrell and Wahid

[4],

many families of co-matching graphs have been identified, and

constructions are given for generalfamilies.

Also,

bysuitably choosing the lengths of the chains and cycles, these graphs can be made to be /k-free. Therefore, by applying Theorem 10, many familiesof co-chromatic graphs can be identified.

For

example, it canbe easily verified that the graphs G and

H

shown in Figure 2are co-matching for alla,b

>

2; this was proven in

[9].

Note that

G

and

H

are /k-free. Therefore, byTheorem 10,

(G,H)

isaco-chromaticpair.

An

interesting question at this point is the following: Is it possible for two x-free graphsto have different chromatic polynomials, but have chromatically equivalent complements?

Suppose

that there exist /x-free graphs G and

H

such that

(G,H)

is matching equivalent but

no..t

chromatically equivalent. Since G and

H

are /x-free,

(G,H)

will be chromatically equivalent by Theorem 9. Our question would be answered if we can find a pair of matching equivalent but chromatically nonequivalent /-free graphs. The graphs G

P2n

+

and H

Pn

LIC_n

+

form

such a pair, for all n>2. Thus, the answer toour questionis yes. We note thatG and Hwere

shown to be co-matchingin

[4].

Their complementswere showntobe chromatically equivalent in

Suppose that G is /k-free and matching unique. Does it follow that

G

will be chromatically unique? Theanswerisnoingeneral. Let Gbe thechain

P4"

Ithasbeen shown

(Farrell

andGuo

[3])

that

Pn,

forn even,ismatching unique. Therefore

P4

ismatchingunique.

However,

P4

P4"

But

P4

is notchromatically unique becauseit ischromatically equivalentto

K1,

_3.

Notice that we cannot say that ifG is matching unique thenG is not chromatically unique. For suppose that G is a graph consisting of isolated nodes and independent edges. Then G is

knownto be matchingunique.

However,

in

[9],

itwasshown thatGischromaticallyunique.

At this point, let us consider the converse of the above question.

Suppose

that

G

is

chromatically unique, does it follow that Gis matching unique? The following theorem answers thisquestion.

THEOREM 11. IfGis /X-freeandGischromatically unique, thenGis matchinguniqueup to /X-freeness. That is,theredoes notexistanother /k-freegraphwhichismatching equivalentto

G.

PROOF. Let G satisfy theconditionsgivenin the theorem.

Suppose

that thereexists a

/X-free G such that

M(G1;_w

M(G;_w).

Since

G

is /x-free,

M(G;w’)=

P(G;$).

Since

G

is A-free,

M(G1;w’

P(G1;,

).

Therefore,

P(G1;)=

P(G;$). Since G is chromatically unique, we musthaveG Gwhichimplies thatG G.

5.

SOME DEDUCTIONS

FOR

THE

THETA

GRAPH.

We can use Theorems and 7 in order to derive formulas for the chromatic polynomials of complements of some graphs. For example, if G consists ofrn isolatednodes and n independent edges,then

P(Fg,

A)

(nr)(A)m

+

n

+

r

Thisfollows from Theorem and theformula for the matching polynomial ofG.

Wecanalsoobtainthechromaticpolynomial of thecomplementof thegraphconsisting of two components

Pn

andCn

+

(n

>

2).

In

this case,

P(PnUCn+

;A)=r

In

[1],

formulas are given for the maching polynomiMs of he basic graphs with cyclomtic number 2. Thesegraphs e &-free forsuitable choicesof he lengths ofthe chinsd cycles of

CONNECTIONS BETWEEN THE MATCHING AND CHROMATIC POLYNOMIALS 765

M(G(r,s,t);u2)

M(P

r

+

s

+

t-1;

-w)

+

w2

[M(Pr-

1;

)M(Ps

+

t-2

;t-v)

+

M(Ps-

l;W-)M(Pr

+

t-2;

t-v)

+

M(Pt- l;u-OM(Pr

+

-

2;kv)

M(P

r_

I’Lo)M(P

s_

1;w_)M(Pt_

1;

_w)],

wherer,s,t

>

2.

Clearly,

G(r,s,t)

is A-free, forr,s,t

>

2.

By

Theorem 1, thuswehave

THEOREM

12.

The following theorem shows that the matching polynomial of the

0-graph

G(2,s,t)

depends onlyonthesums

+

t.

THEOREM 13.

M(G(2,s,t);w_)

WlM(P

s

+ t;w_)

+

2w2M(P

s

+

t-

;-w)

+

WlW2M(Ps

+

t-

2;-w)

PROOF.

Applying thefundamental edge theorem

(Theorem

1 in

[1])

successively tothetwo edgesofthepathoflength2,weobtain

M(G(2,s,t);w_)

WlM(C

s

+ t;w_)

+

2w2M(P

s

+

t-

l;-W)

Applyingthe fundamentaledgetheorem tooneoftheedgesofthe cycle

C

s

+

t, wehave

M(Cs

+

t;w-)

M(Ps

+

t;-

w)

+

w2M(Ps

+

2;-w)

Theproofiscompletedby combining these two equations.

COROLLARY

13.1. Thegraphs

G(2,Sl,tl)

and

G(2,sl,t2)

areco-matchingifandonlyif

s

+t

=s2+t

_2.

PROOF.

If s

i+t

=s

2+t

2, then

G(2,s2,t2)

are co-matching by Theorem 14. If

s

+t

#

s

2+t2,then

V(G(2,s,tl))[

# [v(G(2,s2,t2))[

which implies that

G(2,s,tl)

and

G(2,s2,

t2)

arenotco-matching.

Thefollowingcorollary appearsasTheorem 3.2.4 in

[9].

COROLLARY

13.2. The graphs

(G(2,sl, tl)

and

G(2,s2,t2)

are co-chromatic if and only if

s

+

_s2

+

_2.

PROOF.

Since 2_<s

i<t

for

iE{1,2},

G(2,si, ti)

is A-free. If

s+t

_l=s

2+t

2, then

then

v(G(2,s,t))l

#

V(G(2,s2,t2))iwhich

implies that

G(2,sl,tl)

and

G(2,s2,t2)

are not co-chromatic.

ACKNOWLEDGEMENT. The first author, E.J. Farrell, would like to thank the CIES and the University ofPittsburghfor theirsupport during the periodofresearch.

REFERENCES

1.

FARRELL,

E.

J., An

Introduction to Matching Polynomials, J. Combin. Theory

B,

27

(1979)

75-86.

2.

FARRELL,

E.

J.,

On a Class of Polynomials Obtained from the Circuits in a Graph andits

Applicationto CharacteristicPolynomials ofGraphs,DiscreteMath., 28

(1979)

121-133.

3.

FARRELL,

E. J. and

GUO,

J.

M.,

On the Characterizing Properties of the Matching Polynomial, submitted.

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FARRELL,

E. J. and

WAHID,

S.

A.,

Some General ClassesofComatching Graphs, Internat.

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Math. andMath.,Vol. 11 No.

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W. and

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WHITE,

D.

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GODSIL,

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GUTMAN, I.,

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LEORINC,

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R. C., An

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52-71.

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