Derivatives and real roots of graph polynomials

0.12 MB·

by  Xueliang Li

Derivatives and Real Roots of Graph Polynomials
XueliangLiandYongtangShi
CenterforCombinatorics andLPMC
NankaiUniversity, Tianjin300071, China
Email: [email protected], [email protected]
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Abstract
n
a
J Graphpolynomialsarepolynomialsassignedtographs. Interestingly,theyalsoarise

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in many areas outside graph theory as well. Many properties of graph polynomials
1
have been widely studied. In this paper, we survey some results on the derivative and

]
O real roots of graph polynomials, which have applications in chemistry, control theory
C and computer science. Related to the derivatives of graph polynomials, polynomial
.
h reconstruction ofthematchingpolynomial isalsointroduced.
t
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m Keywords: graphpolynomial; derivatives;realroots;polynomial reconstruction
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AMSSubjectClassification (2010): 05C31,05C90,05C35,05C50

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v
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4 1 Introduction
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0 Manykindsofgraphpolynomialshavebeen introducedand extensivelystudied,such as
.
1
characteristic polynomial, chromatic polynomial, Tutte polynomial, matching polynomial,
0
6 independencepolynomial,cliquepolynomial,etc.
1
: Let G be a simplegraph with n vertices and m edges, whose vertex set and edge set are
v
i V(G)andE(G),respectively. ThecomplementGofGisthesimplegraphwhosevertexset
X
r is V(G) and whose edges are the pairs of nonadjacent vertices of G. For terminology and
a
notationnotdefined here, werefer to[3].
Denote by A(G) the adjacency matrix of G. The characteristic polynomial of G is de-
fined as
n
φ(G,x) = det(λI −A(G)) = an−i.
i
i=0
X
Therootsλ ,λ ,...,λ ofφ(G,x) = 0arecalledtheeigenvaluesofG. Formoreresultson
1 2 n
φ(G,x), werefer to[10, 11].
1
Denotebym(G,k)thek-thmatchingnumberofGfork ≥ 0. Weassumethatm(G,0) =
1. For k ≥ 1, m(G,k) is defined as the number of ways in which k pairwise independent
edges can beselected inG. Thematchingpolynomialis defined as
α(G,x) = (−1)km(G,k)xn−2k.
k≥0
X
There isalsoan auxiliarypolynomialα(G,x,y),which isdefined as
α(G,x,y) = (−1)km(G,k)xn−2kyk.
k≥0
X
Note that α(G,x,y) = yn/2α(G,xy−1/2). In view of this fact, we may define an auxiliary
polynomialofφ(G,x,y):
φ(G,x,y) = yn/2φ(G,xy−1/2) = a xn−kyk/2.
k
k≥0
X
Notethatφ(G,x,y)isapolynomialiny ifand onlyifG isbipartite.
Denote by n(G,k) the k-th independence number of G for k ≥ 0. We assume that
n(G,0) = 1. For k ≥ 1, n(G,k) is defined as the number of ways in which k pairwise
independentverticescan beselectedin G. Theindependencepolynomialis defined as
ω(G,x) = (−1)kn(G,k)xn−k,
k≥0
X
which is also called independent set polynomial in [35] and stable set polynomial in [58].
Formoreresultsontheindependencepolynomials,werefer thesurveys[42, 61].
Denote by c(G,k) the k-th clique number of G for k ≥ 0. We assume that c(G,0) = 1.
For k ≥ 1, c(G,k) is defined as the number of ways in which k pairwise adjacent vertices
can be selected in G. Note that c(G,1) = n and c(G,2) = m. The clique polynomial is
defined as
c(G,x) = (−1)kc(G,k)xn−k.
k≥0
X
Note that the cliquepolynomialof a graph G is exactly the independence polynomialof the
complementG ofG,i.e., c(G,x) = α(G,x). Obviously,we alsohave
c(G,x)+c(G,x) = α(G,x)+α(G,x).
Thefollowingresultsare easilyobtained.
Theorem 1. Let G andG betwo vertex-disjointgraphs. Then wehave
1 1
c(G ∪G ,x) = c(G ,x)+c(G ,x)−1, α(G ∪G ,x) = α(G ,x)·α(G ,x);
1 2 1 2 1 2 1 2
c(G +G ,x) = c(G ,x)·c(G ,x), α(G +G ,x) = α(G ,x)+α(G ,x)−1.
1 2 1 2 1 2 1 2
2
In [35], theauthorsobtainedthefollowingsimilarresult.
Theorem 2. Let G and G be two vertex-disjoint graphs with n and n vertices, respec-
1 2 1 2
tively. Then
c(G ×G ,x) = n ·c(G ,x)+n ·c(G ,x)−(n +n +n n x)+1.
1 2 2 1 1 2 1 2 1 2
Formorepropertieson c(G,x) and α(G,x),werefer to[35].
Manypropertiesofgraphpolynomialshavebeenwidelystudied. Inthispaper,wesurvey
some results on the derivativeand real roots of graph polynomials, which have applications
in chemistry,control theory and computerscience. Related to the derivativesof graph poly-
nomials,polynomialreconstructionofthematchingpolynomialis alsointroduced.
2 Derivatives of graph polynomials
The derivatives of the characteristic polynomial were examined by Clarke [9] and the
followingresultwas showed.
Theorem 3. Let G be a simple graphs and φ(G,x) be the characteristic polynomial of G.
Then
dφ(G,x)/dx = φ(G−v,x).
v∈V(G)
X
GutmanandHosoya[28] gotasimilarresultforthematchingpolynomial.
Theorem 4. Let G bea simplegraphsandα(G,x) bethematchingpolynomialofG. Then
dα(G,x)/dx = α(G−v,x).
v∈V(G)
X
One can get the first derivative of the independence polynomial and clique polynomial,
which have similar expressions as the matching polynomial and characteristic polynomial.
That is,
dω(G,x)/dx = ω(G−v,x), dc(G,x)/dx = c(G−v,x).
v∈V(G) v∈V(G)
X X
Although the four first derivativesobey fully analogous expressions, their proofs existingin
the literatures, are quite dissimilar. Li and Gutman [44] provided a unified approach to all
theseformulasby introducingageneral graph polynomial.
Let f be a complex-valued function defined on the set of graphs G such that G ∼= G
1 2
impliesf(G ) = f(G ). LetGbeagraphonnverticesandS(G)bethesetofallsubgraphs
1 2
ofG. Define
S (G) = {H : H ∈ S(G) and |V(H)| = k}, p(G,k) = f(H).
k
H∈XSk(G)
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Then, thegeneral graphpolynomialofG isdefined as
n
P(G,x) = p(G,k)xn−k.
k=0
X
Actually,let
(−1)|V(H)|/2 ifH is 1-regular;
f(H) =
( 0 otherwise.
Then theresultingpolynomialisthematchingpolynomial. Let
(−1)|V(H)| ifH isno edges;
f(H) =
( 0 otherwise.
Then theresultingpolynomialistheindependencepolynomial. Let
(−1)r(H) ·2c(H) ifallcomponentsofH are 1-or2-regular;
f(H) =
( 0 otherwise,
wherer(H)isthenumberofcomponentsinH andc(H)isthenumberofcyclesinH. Then
theresultingpolynomialis thecharacteristicpolynomial. Let
(−1)|V(H)| ifH is acompletegraph;
f(H) =
( 0 otherwise.
Then theresultingpolynomialisthecliquepolynomial.
Thefollowingtheorem was obtainedbyLi andGutmanin [44].
Theorem 5. ForthegraphpolynomialP(G,x) ofG, wehave
d
(P(G,x)) = P(G−v,x).
dx
v∈V(G)
X
Furthermore, Gutman[24, 27]gotthefirst derivativeformulaforα(G,x,y):
∂α(G,x,y)/∂y = − α(G−u−v,x,y).
uv∈E(G)
X
To find an expression for ∂φ(G,x,y)/∂y of a bipartite graph was posed by Gutman as a
problemin [23]. Asolutionofthisproblemwas offered byLi and Zhang[48].
Theorem 6. Fora bipartitegraphG,
∂φ(G,x,y)/∂y = − φ(G−u−v,x,y)− n(C)yn(C)/2−1φ(G−C,x,y), (1)
uv∈E(G) C⊆G
X X
where C isa cycle, possessingn(C) vertices.
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Theabovetheorem was proved by using Sachs Theorem for thecoefficients of thechar-
acteristicpolynomialand byverifyingtheequalityoftherespectivecoefficientsofthepoly-
nomialsontheleft-and right-handsidesofEq. (1). In [29], theauthorsput forward another
routetoEq. (1), fromwhich itbecomeevidentthatEq. (1)holdsforan arbitrary graph.
Moreover,ifwedefine
P(G,x,y) = p(G,k)xiyj,
i+j=n
X
then wecan obtain
∂P(G,x,y)
= ny−1P(G,x,y)−xy−1 P(G−v,x,y).
∂y
v∈V(G)
X
Derivativesofothergraph polynomialshavealso been studied,such as thecubepolyno-
mial[5], theTuttepolynomial[15], theWienerpolynomial[39], etc.
3 Polynomial reconstruction of the matching polynomial
Thederivativeofagraphpolynomialisrelatedtheproblemofpolynomialreconstruction.
This section aims to prove that graphs with pendant edges are polynomial reconstructible
and,ontheotherhand,todisplaysomeevidencethatarbitrarygraphsarenot,whichisgiven
in [47].
The famous (and still unsolved) reconstruction conjecture of Kelly [38] and Ulam [63]
states that every graph G with at least three vertices can be reconstructed from (the isomor-
phismclassesof)its vertex-deletedsubgraphs.
Withrespect to agraph polynomialP(G),thisquestionmaybeadapted as follows: Can
P(G) of a graph G = (V,E) be reconstructed from the graph polynomials of the vertex
deleted-subgraphs, that is from the collection P(G ) for v ∈ V ? Here, this problem
−v
is considered for the matching polynomial of a graph. For results about the polynomial
reconstruction of other graph polynomials, see the article by Bresˇar, Imrich, and Klavzˇar
[4, Section 1] and the references therein. For additional results, see [59, Section 7] [60,
Subsection 4.7.3].
The matching polynomial we considered here is the generating function of the number
of its matchings with respect to their cardinality, denoted by M(G,x,y), which is different
from the above α(G,x) and α(G,x,y). Let G = (V,E) be a graph. A matching in G is
an edge subset A ⊆ E, such that no two edges in A have a common vertex. The matching
polynomialM(G,x,y)isdefined as
M(G,x,y) = xdef(G,A)y|A|,
A⊆Eisamatching
X
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wheredef(G,A) = |V|−| e|isthenumberofverticesnotincludedinanyoftheedges
e∈A
of A. A matching A is a perfect matching, if its edges include all vertices, that means if
S
def(G,A) = 0. A near-perfect matching A is a matching that includes all vertices except
one, that means def(G,A) = 1. For more information about matchings and the matching
polynomial,see[17, 22, 51].
For a graph G = (V,E) with a vertex v ∈ V, G is the graph arising from the deletion
−v
of v, i.e., arising by the removal of v and all the edges incident with v. The multiset of
(the isomorphism classes of) the vertex-deleted subgraphs G for v ∈ V is the deck of
−v
G. The polynomialdeck D (G) with respect to a graph polynomialP(G) is the multiset of
P
P(G ) for v ∈ V. A graph polynomial P(G) is polynomial reconstructible, if P(G) can
−v
bedeterminedfrom D (G).
P
By arguments analogous to those used in Kelly’s Lemma [38], the derivative of the
matchingpolynomialofagraphG = (V,E)equalsthesumofthepolynomialsinthecorre-
spondingpolynomialdeck.
Proposition 1 (Lemma 1 in [18]). Let G = (V,E) be a graph. The matching polynomial
M(G,x,y) satisfies
δ
M(G,x,y) = M(G ,x,y). (2)
−v
δx
v∈V
X
Inotherwords,allcoefficientsofthematchingpolynomialexcepttheonecorresponding
to the number of perfect matchings can be determined from the polynomial deck and thus
also fromthedeck:
1
m (G) = m (G ) ∀i ≥ 1, (3)
i,j i,j −v
i
v∈V
X
where m (G) isthecoefficient ofthemonomialxiyj inM(G,x,y).
i,j
Consequently, the (polynomial) reconstruction of the matching polynomial reduces to
thedeterminationofthenumberofperfect matchings.
Proposition 2. The matching polynomial M(G,x,y) of a graph G can be determined from
itspolynomialdeckD (G)anditsnumberofperfectmatchings. Inparticular,thematching
M
polynomial M(G,x,y) of a graph with an odd number of vertices is polynomial recon-
structible.
Tutte[62,Statement6.9]showedthatthenumberofperfect matchingsofasimplegraph
can be determined from its deck of vertex-deleted subgraphs and therefore gave an affirma-
tiveansweronthereconstructionproblemforthematchingpolynomial.
The matching polynomial of a simple graph can also be reconstructed from the deck of
edge-extracted and edge-deleted subgraphs [18, Theorem 4 and 6] and from the polynomial
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deck of the edge-extracted graphs [24, Corollary 2.3]. For a simple graph G on n vertices,
the matching polynomial is reconstructible from the collection of induced subgraphs of G
with⌊n⌋+1 vertices[20,Theorem 4.1].
2
Thefollowingresult isfrom [47]for simplegraphs withpendantedges.
Theorem 7. Let G = (V,E) be a simple graph with a vertex of degree 1. Then, G has a
perfect matching if and only if each vertex-deleted subgraph G for v ∈ V has a near-
−v
perfect matching.
AsprovedrecentlybyHuangandLih[36],thisstatementcanbegeneralizedtoarbitrary
simplegraphs.
Corollary 8. Let G = (V,E) bea forest. Then G has a perfect matchingif andonlyif each
vertex-deleted subgraphG forv ∈ V hasa near-perfectmatching.
−v
Forestshaveeithernoneoroneperfectmatching,becauseeverypendantedgemustbein
aperfectmatching(inordertocovertheverticesofdegree1)andthesameholdsrecursively
for the subforest arising by deleting all the vertices of the pendant edges. Therefore, from
Proposition 2 and Corollary 8 the polynomialreconstructibilityof the matching polynomial
follows.
Corollary9. ThematchingpolynomialM(G,x,y)ofaforestispolynomialreconstructible.
On the other hand, arbitrary graphs with pendant edges can have more than one perfect
matching. However,Corollary8canbeextendedtoobtainthenumberofperfectmatchings.
For a graph G = (V,E), the number of perfect matchings and of near-perfect matchings of
G is denotedbyp(G) and np(G), respectively.
Theorem 10. Let G = (V,E) be a simple graph with a pendant edge e = {u,w} where w
is avertex ofdegree1. Then we have
p(G) = np(G ) ≤ np(G ) ∀v ∈ V andparticularly (4)
−u −v
p(G) = min{np(G ) | v ∈ V}. (5)
−v
By applying this theorem, the number of perfect matchings of a simple graph with pen-
dant edges can be determined from its polynomialdeck and the following result is obtained
as acorollary.
Corollary 11. The matchingpolynomialM(G,x,y) of a simplegraphwith a pendant edge
is polynomialreconstructible.
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While it is true that the matching polynomialsof graphs with an odd number of vertices
orwithapendant edgearepolynomialreconstructible,itdoes notholdforarbitrary graphs.
There are graphs which have the same polynomial deck and yet their matching polyno-
mialsaredifferent. Althoughtherearealready counterexampleswithaslittleassixvertices,
it seems that nothing has been published before in connection with the question addressed
here.
Remark 1. The matching polynomial M(G,x,y) of an arbitrary graph is not polynomial
reconstructible. The minimalcounterexamplefor simplegraphs(with respect to thenumber
of vertices andedges)areconstructedin[47].
The question arises here: whether or not there are such counterexamples consisting of
graphs with an arbitrary even number of vertices. In [47], we gave an affirmative answer to
thisquestion.
4 Roots of beta-polynomials and independence polynomi-
als
Polynomials whose all zeros are real-valued numbers are said to be real. Several graph
polynomials have been known to be real; among them the matching polynomial α(G,x)
plays adistinguishedrole[21,34].
Polynomialswithonlyreal rootsariseinvariousapplicationsincontroltheoryandcom-
puter science [65], but also admit interestingmathematicalproperties on theirown. Newton
noted that the sequence of coefficients of such polynomials form a log-concave (and hence
unimodal) sequence. These polynomials also have strong connections to totally positive
matrices.
Thefactthatforallgraphs,allzerosofthematchingpolynomialarereal-valuedwasfirst
establishedby Heilmannand Lieb [34].
LetC beacircuitcontainedinagraphG. IfC isaHamiltoniancycle,thenα(G−C,x) ≡
1. In certain considerations in theoretical chemistry [1, 40, 54, 55], graph polynomials
β(G,C,x)are encountered,defined as
β(G,C,x) = α(G,x)−2α(G−C,x) (6)
and
β(G,C,x) = α(G,x)+2α(G−C,x) (7)
Formula (6) is used in the case of so-called Hu¨ckel-type circuits, whereas formula (7) for
the so-called Mo¨bius-type circuits. For details, see [54]. These polynomials are also called
circuit characteristicpolynomials[1].
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Alreadyinthefirstpaperdevotedtothistopic[1],Aiharamentionedthatthezerosofthe
β-polynomialsare real-valued, but gaveno argument to supporthis claim. In themeantime,
for a number of classes of graphs it was shown that β(G,C,x) is indeed a real polynomial
[25,26,30,40, 49,55,41]. In additiontothis,bymeansofextensivecomputersearchesnot
a single graph with non-real β-polynomial could be detected. The following conjecture has
been put forward byGutmanand Mizoguchiin[25, 26, 30].
Conjecture 1. For any circuit C contained in any graph G, the β-polynomials β(G,C,x),
Eqs. (6)and (7), arereal.
Manyresultshavebeen obtained. In particular,β(G,C,x)has been showntobereal for
unicyclic graphs [30], bicyclic graphs [55], graphs in which no edge belongs to more than
one circuit [55], graphs without 3-matchings (i.e., m(G,3) = 0) [40], several (but not all)
classes ofgraphswithout4-matchings(i.e., m(G,4) = 0)[41].
In [46], Li et al. showed that the conjecture is true for complete graphs. Actually, they
provedastrongerresult forcompletegraphs.
Theorem 12. ForanycircuitC inthecompletegraphK , thepolynomial
n
β(K ,C,t;x) = α(K ,x)+tα(K −C,x)
n n n
is realfor anyreal tsuchthat|t| ≤ n−1.
Theproofofferedin[46]reliesonanearlierpublishedtheorembyTura´n. In[45],Liand
Gutman presented an elementary self-contained proof for complete graphs. Finally, in [50],
Li et al. showed that the conjecture is true for all graphs, and therefore completely solved
thisconjecture.
Theorem 13. For any circuit C contained in any graph G, all roots of the polynomial
β(G,C,x)arereal.
Chudnovskyand Seymour[8]provedthefollowingresult forindependencepolynomial.
Theorem 14. If Gis clawfree, then allrootsofitsindependencepolynomialarereal.
Theorem 14 extends a theorem of [34], answering a question posed by Hamidoune [33]
and Stanley [58]. Since all line graphs are clawfree, this extends the result of [34]. Later,
LevitandMandrescustudiedtherootsofindependencepolynomialsofalmostallverywell-
covered graphs [43]. In [53], Mandrescu showed that starting from a graph G whose inde-
pendence polynomial has only real roots, one can build an infinite family of graphs, whose
independencepolynomialshaveonlyreal roots.
Real roots of other graph polynomialshave also been extensivelystudied, such as edge-
cover polynomial [2], the expected independence polynomial [7], domination polynomial
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[6], sigma-polynomial [67], chromatic polynomial [14, 37, 66], Wiener polynomial [12],
flow polynomial[37], Tuttepolynomial[16, 64], etc. Formore resultson theroots of graph
polynomials,werefer to[13, 31, 32, 52, 56, 57, 65].
Acknowledgment
Theauthorsaresupportedby NSFC andthe“973”program.
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