Exact Results on Potts/Tutte Polynomials for Families of Networks with Edge and Vertex Inflations

0.25 MB·English

by  Robert Shrock

Exact Results on Potts/Tutte Polynomials for Families of Networks with Edge and
Vertex Inflations
Robert Shrock
C. N. Yang Institute for Theoretical Physics
Stony Brook University
Stony Brook, NY 11794
We derive exact relations between the Potts model partition function, or equivalently the Tutte
1
polynomial, for a network (graph) G and a network obtained from G by (i) by replacing each
1
edge (i.e., bond) of G by two or more edges joining the same vertices, and (ii) by inserting one
0
or more degree-2 vertices on edges of G. These processes are called edge and vertex inflation,
2
respectively. The physicaleffects of these edge and vertex inflations are discussed. Wealso present

n exact calculations of these polynomials for families of networks obtained via the operation (ii) on
a a subset of the bonds of the network. Applications of these results include calculations of some
J
network reliability polynomials. In addition, we evaluate our results to calculate various quantities

0 of structural interest such as numbers of spanning trees, etc., and to determine their asymptotic
3 behaviorfor large networks.

]
PACSnumbers: 05.50.+q,64.60.Cn,75.10.Hk
h
c
e
m I. INTRODUCTION the corresponding polynomial calculated for a graph G˜
which is obtained by a specified modification of G. In
-
t this paper we will first present a general solution to this
a The Potts model has long been valuable as a system problem for two classes of G˜’s, namely those obtained
st exhibiting many-body cooperative phenomena [1]. On (i) by replacing each edge of G by two or more edges
. a lattice, or, more generally, on a network (i.e., graph)
t joining the same vertices, and (ii) by inserting one or
a G, at temperature T, the partition function for this
m more degree-2 vertices on each edge of G (where the de-
model (in zero external field) is Z = e−βH, where
d- β = 1/(kBT), the Hamiltonian H = P−J{σi}eij δσiσj, J is tghreaet,cκovnin,eocftatvoeirtt)e.xWvieisdedneofitneedthaessethoepneruamtiboenrsoafseeddggees
n the spin-spin exchange constant, i and j denote vertices and vertex inflations of G, respectively. In the litera-
P
o (sites)onG, eij istheedge(bond)connectingthem,and ture on mathematical graph theory, a vertex inflation
c σi are classical spins taking values in the set 1,...,q . of G is also called a homeomorphic inflation of G, and
[ We use the notation K βJ and v eK 1. T{hus, fo}r the reverse procedure, of removing one or more degree-2
≡ ≡ −
1 the Potts ferromagnet(FM, J >0) and antiferromagnet vertices from edges of G, is called a homeomorphic re-
v (AFM, J < 0), the physical ranges of v are v 0 and duction of G. In the remaining sections of our paper
6 1 v 0,respectively. Ingeneral,a graphG=≥(V,E) we will present results for Potts/Tutte polynomials of
2 − ≤ ≤
isdefinedbyitsvertexset,V,anditsedgeset,E. Wede- certain families of graphs obtained by vertex inflations
8
note the number of vertices of G as n n(G) V and for a subset of edges. These families include longitudi-
5 ≡ ≡| |
. the number of edges of G as e(G) E . Z is a polyno- nal vertex inflations of free, cyclic, and Mo¨bius ladder
1 mial in q and v, as will be evident≡fro|m|Eq. (2.1) below. graphs of arbitrarily great length and a family that we
0
The Potts model partition function Z is equivalent to a call hammock graphs. These results extend our previ-
1
functionofconsiderableinterestinmodernmathematical ous work in two ways: (a) as generalizations to the full
1
: graph theory, namely the Tutte polynomial [2]-[4], and Potts/Tutte polynomials of our earlier calculations with
v we shall therefore often refer to this object in a unified S.-H. Tsai of the chromatic polynomials for homeomor-
i
X manner as the Potts/Tutte polynomial. The partition phic inflations of these families of graphs in Refs. [7]-[9]
r function of the zero-temperature Potts antiferromagnet and (b) as homeomorphic inflations of our previous cal-
a is identical to another function of longstanding interest culationofthe Potts/Tuttepolynomialsforladder strips
ingraphtheory,namelythechromaticpolynomial,which in Ref. [10].
counts the number of ways of assigning q colors to the
Thereareseveralmotivationsforthiswork. Foranar-
vertices of G subject to the condition that no two adja-
bitrarygraphG,thecalculationofthePotts/Tuttepoly-
centverticeshavethesamecolor. Thesearecalledproper
nomial involves a number of computational steps and a
q-coloringsofG. AnimportantpropertyofthePottsan-
correspondingtime thatgrowexponentiallyrapidlywith
tiferromagnet is that for sufficiently large q, it exhibits
n(G) and e(G) (e.g., [11, 12]). Furthermore, there is no
nonzeroentropy per site at zerotemperature and is thus
known exact closed-form solution for Z for arbitrary q
an exception to the third law of thermodynamics [5, 6].
and temperature T on the thermodynamic limit of lat-
An interesting problem involving both statistical me- ticegraphsofdimensionalityd 2. Hence,itisoffunda-
≥
chanics and mathematical graph theory is to relate the mental value to carry out exact analytic calculations of
Potts/Tutte polynomial calculated for a graph G with Potts/Tutte polynomials on various families of graphs,
2
such as lattice strip graphs and modifications thereof. ForagraphG,letusdenoteG easthegraphobtained
−
Furthermore,specialcasesofthePotts/Tuttepolynomial bydeleting the edgee andG/eas the graphobtainedby
are of considerable interest in their own right. We have deleting the edge e and identifying the two vertices that
notedtheimportanceofthezero-temperaturePottsanti- wereconnectedbythisedgeofG. Thisoperationiscalled
ferromagnet(chromaticpolynomial). Thus, anothermo- a contraction of G on e. From Eq. (2.1), it follows that
tivation for our work, as embodied in point (a) above, is Z(G,q,v) satisfies the deletion-contractionrelation
togeneralizeourpreviouscalculationswithS.-H.Tsai[7]-
[9]tothebroadercontextofthefinite-temperaturePotts Z(G,q,v)=Z(G e,q,v)+vZ(G/e,q,v) . (2.5)
−
antiferromagnet and the Potts ferromagnet (the latter
An analogous deletion-contraction relation holds for
also at arbitrary temperature). A central motivation is
T(G,x,y).
to investigate the effects of edge and vertex inflations of
a graph on the Potts/Tutte polynomial of that graph.
This involves the generalization in point (b) above. One
III. TYPES OF INFLATIONS OF A GRAPH
particular value of our presentwork is its demonstration
of the usefulness of relating calculations in the context
of the Potts model to corresponding calculations for the Here we discuss in greater detail the edge and vertex
Tutte polynomial and vice versa. Finally, special cases inflationofa graphG. As partofourstudy,we willcon-
of our results yield various quantities of interest such as sider the infinite-length limit of lattice strip graphs G of
reliability polynomials, numbers of spanning trees, etc. finite width, and also the thermodynamic limit of lattice
for these families of graphs. graphs G of dimensionality d 2. In these cases, we
≥
will often use the notation G to indicate these limits.
{ }
With no loss of generality, we begin by assuming that G
II. SOME GENERAL BACKGROUND has no multiple edges. We define an edge inflation of G
to be a graph obtained by replacing one or more of the
edges of G by multiple edges joining the same vertices.
In this section we briefly discuss some necessary back-
Clearly,thisleavesthenumberofverticesn(G)invariant.
ground that will be used for our calculations. There is
Inparticular,itisnaturaltodefine auniformℓ-foldedge
a useful relation that expresses the Potts model parti-
inflation of G as the graph (G) obtained by replacing
tion function Z(G,q,v) as a sum of contributions from ℓ
E
each edge of G by ℓ edges joining the same two vertices.
spanning subgraphs of G. Here, a spanning subgraph
G′ = (V,E′) has the same vertex set as G and a subset Thus, the number of edges of ℓ(G) is e( ℓ(G))=ℓe(G).
E E
of the edge set E, E′ E. This relation is [13] A κ-regular graph is a graph with the property that all
⊆ vertices have the same degree, κ = κ v V. Al-
vi ∀ i ∈
Z(G,q,v)= qkc(G′)ve(G′) , (2.1) though a general graph is not κ-regular, one can define
an average or effective vertex degree κ as
G′⊆G eff
X
where kc(G′) denotes the number of connected compo- κ = vi∈V κvi . (3.1)
nentsinG′. ThisformulashowsthatZ(G,q,v)isapoly- eff n(G)
P
nomial in q and v. For the Potts ferromagnet, Eq. (2.1)
enablesonetogeneralizeqfromthenon-negativeintegers Clearly,auniform ℓ-foldinflationofallof the edges ofG
tothenon-negativerealnumberswhilekeepingZ(G,q,v) multiplies κeff by ℓ. A remark is in order here concern-
positive and hence maintaining a Gibbs measure. ing loops. A loop is defined as an edge joining a vertex
The Tutte polynomial of a graph G, denoted back to itself. This would not occur in the statistical
T(G,x,y), is defined by [2]-[4] mechanicalframework,since it would mean a spin σi in-
teracting with itself, and hence we will usually assume
T(G,x,y)= (x 1)kc(G′)−kc(G)(y 1)c(G′) , (2.2) that G is a loopless graph.
− −
G′⊆G We define a vertex (i.e., homeomorphic)inflation of G
X
tobeagraphobtainedbyinsertingoneormoredegree-2
where c(G′) is the number of linearly independent cycles
vertices on one or several edges of G [14]. This leaves
(circuits) in G′. Let us define
the numberof(linearlyindependent)circuits inG, c(G),
q invariant. Homeomorphic inflations and reductions of
x=1+ , y =eK =v+1 , (2.3)
v graphs have been of interest in the study of chromatic
andTuttepolynomials[7]-[9],[15]-[21]andhavealsobeen
sothatq =(x 1)(y 1). TherelationbetweenZ(G,q,v)
− − studied in the context of “decorated” spin models [22].
and T(G,x,y) follows directly from Eqs. (2.1) and (2.2)
As with edge inflations, it is natural to define a uniform
and is
ℓ-fold vertex inflation (G) as the graph obtained by
ℓ
V
Z(G,q,v)=(x 1)kc(G)(y 1)n(G)T(G,x,y) . (2.4) inserting ℓ degree-2 vertices on each edge of G. For the
− − specialcasewhere Gis asectionofa regularlattice, itis
With no loss of generality, we will restrict here to con- also natural to consider edge or homeomorphic inflation
nected graphs G, so k (G)=1. ofedgesformingasubsetoflatticevectors. Forexample,
c
3
forad-dimensionalEuclideanlatticeEd,onecanconsider the edges e , we note parenthetically that one can con-
ij
thecaseinwhichtheedgeorvertexinflationisperformed sider a generalization in which the J are different for
ij
on the edges along one or more lattice vectors eˆ , where each edge of G, e . From the original 1944 Onsager
j ij
j takes values in a subset of 1,...,d . solution of the two-dimensional Ising model [23], a large
{ }
numberofstudiesofspinmodelshavedealtwiththegen-
eral case with different spin-spin exchange constants for
IV. RELATIONS FOR UNIFORM EDGE AND different lattice directions. Studies of spin models with
VERTEX INFLATIONS spin-spinexchangeconstantsJ thatcanbedifferent(in
ij
magnitudeandsign)foreachedge,e weremotivatedby
ij
A. Edge Inflation early work on spin glasses [24]. In this case, where J ’s
ij
dependone ,thetransformationoftheHamiltonianbe-
ij
The effect of a uniform edge inflation on the Potts comes
modelpartitionfunctioncanbedeterminedfromananal-
= J δ ℓ J δ . (4.8)
ysis of the Potts Hamiltonian. As we edge-expand G to H − ij σiσj →− ij σiσj
(G), i.e., replaceeachedgeby ℓ edgesjoining the same Xeij Xeij
ℓ
E
vertices, we induce the change in the Hamiltonian Hence, defining Kij βJij and vij eKij 1, we have
≡ ≡ −
= J δ ℓJ δ , (4.1) v v =(v +1)ℓ 1 . (4.9)
H − σiσj →− σiσj ij → ij,e,ℓ ij −
Xeij Xeij Denoting v as the set of v ’s, we then have
ij
{ }
i.e., J ℓJ, and hence
→ Z( (G),q, v )=Z(G,q, v ) . (4.10)
ℓ e,ℓ
E { } { }
y y =yℓ , (4.2)
→ e,ℓ Since J < 0 for the Potts antiferromagnet, it follows
that, as T 0, K (and v 1). Hence, in this
wherethesubscriptse,ℓrefertotheℓ-foldedgeinflation. → →−∞ →−
limit,theonlycontributionstothepartitionfunctionare
Equivalently,
fromspinconfigurationsinwhichadjacentspinshavedif-
v v =(v+1)ℓ 1 , (4.3) ferent values. The resultant T =0 Potts AFM partition
e,ℓ
→ − function is therefore precisely the chromatic polynomial
where v y 1. This proves the following relation P(G,q) of the graph G counting the number of proper
e,ℓ e,ℓ
connecting≡Z on G−with Z on (G): q-colorings of G:
ℓ
E
Z(G,q, 1)=P(G,q) . (4.11)
Z( (G),q,v)=Z(G,q,v ) . (4.4)
ℓ e,ℓ −
E
This is equivalent, via Eq. (2.4), to the relation
We next determine the effect of this edge inflation on
the Tutte polynomial. Given the transformation (4.2) P(G,q)=( q)kc(G)( 1)n(G)T(G,1 q,0) . (4.12)
and the fact that q does not change, so that − − −
The minimum number of colors necessary for a proper
q =(x 1)(y 1)=(xe,ℓ 1)(ye,ℓ 1) , (4.5) q-coloring of G is the chromatic number χ(G). For
− − − −
q > χ(G), P(G,q) grows exponentially with n, leading
we have to a groundstate degeneracyper vertex, W( G ,q)>1,
where W( G ,q) = lim P(G,q)1/n. T{he} ground
(x 1)(y 1) x 1 { } n→∞
x =1+ − − =1+ − . (4.6) state entropy per vertex of the Potts model on G is
e,ℓ yℓ 1 ℓ−1yj { }
− (cid:18) j=0 (cid:19) S0( G ,q)= kBln[W( G ,q)]. In Refs. [6], [7]-[9], [16],
{ } { }
weappliedourexactresultsonchromaticpolynomialsto
Combining these transformations ofPvariables with Eq.
study the phenomenon of nonzero ground state entropy
(2.4), we derive the relation connecting the Tutte poly-
per site in Potts antiferromagnets. These exact results
nomials of G and of (G), namely
ℓ complement other approaches to studying S , such as
E 0
rigorous bounds, series, and Monte Carlo measurements
T( (G),x,y)=T(G,x ,y ) , (4.7)
Eℓ e,ℓ e,ℓ [25, 26].
It is clear from the definition of the chromatic poly-
where x and y were given in Eqs. (4.6) and (4.2).
e,ℓ e,ℓ
nomial that P(G,q) does not change if one replaces any
Note that the effect of the edge inflation on the Potts
edgeofGbytwoormoreedgesjoiningthesamevertices.
partitionfunction is simpler than the effect onthe Tutte
In particular, for the case of an ℓ-fold uniform edge in-
polynomial,since inthe formercase,onlyoneofits vari-
flation of G,
ables is modified, namely, v v , whereasin the latter
e,ℓ
→
case, both of its variables are modified, as x x and
e,ℓ P( (G),q)=P(G,q) . (4.13)
→ ℓ
y y . E
e,ℓ
→
Although our main focus is on the Potts model with This is also clear analytically from Eqs. (4.3) and (4.4),
spin-spin exchange constants J that are independent of since the condition that v = 1 implies that v = 1.
e,ℓ
− −
4
B. Vertex Inflation Theflowoneachedgecanthustakeonanyofq 1values
−
modulo q. The flow or current conservation condition is
ToanalyzetheeffectofavertexinflationofagraphG, that the flows into any vertex must be equal, mod q, to
it is again convenient to start with the Potts model for- the flows outward from this vertex. These are called q-
mulation. Forsimplicity,weassumeherethatGdoesnot flows on G, and the number of these is given by the flow
have any multiple edges; it is straightforward to extend polynomial, F(G,q). This is a special case of the Tutte
our calculationto the caseof multiple edges. We use the polynomial for x = 0 and y = 1 q or equivalently, in
−
fact that in the basic expression Z = e−βH, one terms of Potts model variables, v = q:
{σi} −
can perform the summations over the spins that are lo- F(G,q)=( 1)c(G)T(G,0,1 q) . (4.19)
P
catedatthesedegree-2vertices. Letusconsideranedge, − −
e , and insert a degree-2 vertex v (and its associated As is clear from the definition of the flow polynomial,
ij a
spin, σ ) on this edge. Then adding or removing a degree-2 vertex from an edge of G
a
does not change the number of allowed q-flows on G, so
(1+vδ )(1+vδ )
σiσa σaσj F( ℓ(G),q)=F(G,q) . (4.20)
σiX,σa,σj V
= 1+v(δσiσa +δσaσj)+v2δσiσj . Aimspilsiesevtihdaetntvfrom= Eqq.an(d4.1h6e)n,cet,hemcoorendgietnioenravlly=, th−aqt
v,1
σiX,σa,σjh i v = q for arbitra−ry ℓ 1. Recall that the number of
v,ℓ
(4.14) cyclesi−nGisunchangedb≥ythishomeomorphicinflation:
c( (G))=c(G).
ℓ
Now, carrying out the summation over σ , we find that V
a
if σ =σ ,then the resultis q+2v+v2,while ifσ =σ ,
i j i j
6
then the result is q+2v. Therefore,
V. PHYSICAL EFFECTS OF EDGE AND
VERTEX INFLATION
(1+vδ )(1+vδ )
σiσa σaσj
σiX,σa,σj A. General
=(q+2v) (1+v δ ) , (4.15)
v,1 σiσj
σXi,σj An importantquestionconcerns the effect ofedge and
vertex inflations on the physical properties of the Potts
where
model. First, one may investigate these effects for the
v2 q-state Potts model on (the thermodynamic limit of) a
vv,1 = (4.16) regular lattice graph of dimension d 2, where the fer-
q+2v ≥
romagneticversionofthemodelhasafinite-temperature
and the subscript v,1 refers to the insertion of one addi- order-disorder phase transition, and, depending on the
tional vertex on the edges. Performing this summation lattice type and the value of q, the Potts antiferromag-
for each edge, we derive the relation net may have a finite-temperature order-disorder phase
transition. In particular, one can study the effect on the
Z( (G),q,v)=(q+2v)e(G)Z(G,q,v ) . (4.17) phase transition (pt) temperature T for lattices where
V1 v,1 pt
the equation for this quantity is known exactly. How-
This operation can be performed iteratively ℓ times, ever, aside from the q =2 Ising case on two-dimensional
thereby giving a relation between Z on ℓ(G) and Z on lattices, there is no known exact closed-form solution
V
G. For example, for ℓ=2, we have for the free energy of the q-state Potts model at arbi-
trary temperature on these lattices with d 2. There
v2 ≥
v = v,1 is thus also some interest in studying the effect of edge
v,2
q+2vv,1 andvertexinflationforinfinite-lengthlimitsofquasi-one-
v4 dimensional lattice strips, where one can obtain exact
= , (4.18)
(q+2v)(q2+2qv+2v2) closed-form expressions for the free energy for arbitrary
q and T. Of course, a spin model with short-ranged in-
and so forth for higher values of ℓ. We can thus relate Z teractions, such as the Potts model, does not have any
on (G) to Z on G. finite-temperature phase transition on an infinite-length
ℓ
V
A basic problem in graph theory is the enumeration quasi-one-dimensionallatticestrip. Nevertheless,onein-
of discretized flows on the edges of a (connected) G that terestingapplicationofexactsolutionsforthefreeenergy
satisfy flow conservation at each vertex, i.e. for which andthermodynamicquantitiesonthesestripsisthatone
there are no sources or sinks. One arbitrarily chooses can study their dependence (and the dependence of the
a direction for each edge of G and assigns a discretized associatedTuttepolynomials)ongraphicalproperties,in
flow value to it. The value zero is excluded, since it is particular, on the edge or vertex inflation.
equivalent to the edge being absent from G; henceforth, In accordance with our notation above for the dimen-
wetakeaq-flowtomeanimplicitlyanowhere-zeroq-flow. sionless inverse temperature K J/(k T), we define
B
≡
5
K J/(k T ). We also introduce some notation transition in the presence of greater thermal fluctua-
pt B pt
≡
that we will use below. We define the shifted values tions. Let us denote T as the shifted phase tran-
pt,es,ℓ
of T due to a uniform ℓ-fold edge inflation (symbol- sition temperature after an ℓ-fold edge inflation along
pt
ized by the subscript e,ℓ) and a uniform ℓ-fold vertex a subset s of the lattice directions, and similarly de-
inflation (symbolized by the subscript v,ℓ) by T note K J/(k T ). Thenthe reasoningabove
pt,e,ℓ pt,es,ℓ B pt,es,ℓ
≡
and T . We thus denote K J/(k T ) and yieldstheinequalityT >T forℓ 2. Asanexam-
pt,v,ℓ pt,e,ℓ B pt,e,ℓ pt,es,ℓ pt
≡ ≥
K J/(k T ). ple, we again consider the (infinite) square lattice sq
pt,v,ℓ B pt,v,ℓ
≡ { }
and perform an edge inflation with ℓ = 2 for edges in
oneofthe twolattice directions,sayeˆ , sothatK =K,
2 1
B. Effect on Tpt Due to Edge Inflation K2 = 2K. The equation for the shifted inverse phase
transition temperature is given by v2(v +2) = q, with
physical solution
AgeneralresultisthatforthePottsmodelon(aninfi-
nite)regularlatticegraph G withdimensionalityd 2
1
that has a finite-temperatu{re}phase transition (of eit≥her Kpt,e2,2 =ln A1/3+4A−1/3+1 , (5.5)
3
ferromagnetic or antiferromagnetic type, and either first (cid:20) n o(cid:21)
or second order) at a temperature Tpt, a uniform ℓ-fold wherethe subscripte2meansinflationalongedgesalong
inflation of all edges of G has the effect of multiplying eˆ and
{ } 2
T by the factor ℓ, i.e.,
pt
1
A= 27q 16+3 3q(27q 32) . (5.6)
K 2 − −
pt
Tpt,e,ℓ =ℓTpt , Kpt,e,ℓ = ℓ . (5.1) TheresultanthK <Kp,i.e.,T >iT ,inagree-
pt,e2,2 pt pt,e2,2 pt
ment with the generalargumentgivenabove. For exam-
Analytically,thisfollowsbecausetheuniformℓ-foldedge
ple,forq =2,K =ln(1+√2) 0.88137,whileK
inflationof G changesJ toℓJ andTptisproportionalto pt ≃ pt,e2,2
{ } is given by
J. Physically, it follows because replacing J by ℓJ with
ℓ 2 strengthens the spin-spin interaction and hence
1
ma≥kes possible the onset of long-range magnetic order eKpt,e2,2 = (19+3√33 )1/3+4(19+3√33 )−1/3+1 ,
3
in the presence of greater thermal fluctuations, i.e., at a (cid:20) (cid:21)
(5.7)
higher temperature.
so that K 0.60938.
pt,e2,2
As an example, consider the Potts ferromagneton the ≃
(thermodynamic limit of the) square lattice, sq = E2.
{ }
Denotingthespin-spinexchangeconstantsinthetwolat- C. Effect on T Due to Vertex Inflation
pt
tice directions eˆ, i = 1,2, as J , with K = βJ and
i i i i
vi = eKi −1, we recall the well-known equation for the One canalsodeduce a generalresultfor the effect of a
phase transition temperature, namely [1],
uniformℓ-foldvertex(i.e., homemorphic)inflationofthe
thermodynamic limit ofa lattice graph G with dimen-
v1v2 =q . (5.2) sionalityd 2. Ageneric feature ofthe{ph}ase transition
≥
temperatureT ofaferromagneticspinmodel(aboveits
Let us initially assume K = K K and thus v = pt
1 2 1
≡ lower critical dimensionality, so that this temperature is
v v, so Eq. (5.2) becomes v2 = q. The dimensionless
2
≡ finite) is that, other things being equal, T increases as
inverse phase transition temperature K is then given pt
pt
a function of the vertex degree, i.e., coordination num-
by
ber, of the lattice. This feature is observed in approx-
imate determinations of T from high-temperature and
K =ln(1+√q ) . (5.3) pt
pt
low-temperature series expansions, Monte Carlo simula-
tions, mean-field approximations, and, where available,
Nowletuscarryoutanℓ-foldedgeinflationonthelattice.
exact solutions for T . It is understood physically as a
Denoting the resultant inverse phase transition temper- pt
consequence of the fact that increasing the coordination
ature in an obvious notation as K , we have
pt,e,ℓ
number increases the effect of the spin-spin interactions,
1 K sothat the orderingassociatedwith the phase transition
pt
Kpt,e,ℓ = ln(1+√q )= . (5.4) canoccurinthepresenceofgreaterthermalfluctuations.
ℓ ℓ
On an (infinite) line, with coordination number κ=2, a
One can also consider the effect of an edge inflation spin model with short-range interactions does not have
on all edges along a subset of lattice directions. The a finite-temperature phase transition, so the lattices of
effect is simplest for the ferromagnetic case, since this interest in this section are lattices with d 2, which
≥
does not involves competing interactions or frustration. necessarily have coordination number κ 3 (where this
≥
This edge inflation along a subset of lattice directions minimum value, κ = 3, is realized for the honeycomb
strengthens the net spin-spin interaction and therefore lattice). Ifonestartswithalatticegraph G withcoor-
{ }
makes possible the ordering associated with the phase dination number κ 3, then a uniform vertex inflation,
≥
6
which consists of the addition of ℓ degree-2 vertices on Comparing this with the inverse critical temperature for
each edge of G , reduces the effective vertex degree, theoriginal G ,Kpt =ln(1+√q),weseethatKpt,v,1 >
{ } { }
κ . K , in agreement with our general argument above.
eff pt
Let us, for technical simplicity, consider the thermo- The situation is more complicated with the Potts an-
dynamic limit G to be reached as the n(G) tiferromagnet; we will show that vertex inflation can ei-
{ } → ∞
limit of a regular lattice graph G with periodic bound- ther lower or raise a phase transition (critical) temper-
aryconditions, anduniformvertex degreeκ. Then, with ature, depending on the value of q and the lattice type.
e(G) = (κ/2)n(G), the number of vertices and edges of First, consider the q = 2 (Ising) Potts antiferromagnet
ℓ(G) are on a bipartite lattice G . The bipartite property of
V { }
G means that it can be expressed as the union of an
κℓ {eve}n and an odd sublattice, G = G G , with
n( ℓ(G))= 1+ n(G) (5.8) { } { 1}∪{ 2}
V 2 the property that each vertex in G has, as its only ad-
(cid:18) (cid:19) 1
jacent vertices, members of the vertex set of G and
2
and viceversa. Thereisawell-knownisomorphismt{hat}maps
theIsingantiferromagneton G toanIsingferromagnet
e( (G))=(1+ℓ)e(G) (5.9) { }
ℓ on G , namely the simultaneous replacement J J
V
{ } → −
andσ σ ,withσ unchanged,whereherev and
so that v1 →− v1 v2 1
v denote vertices in G and G , respectively. Thus, in
2 1 2
1+ℓ this case, the effect of an ℓ-fold vertex inflation on all
κeff(Vℓ(G))= 1+ κℓ κ . (5.10) edges is the same for the antiferromagnetic case as for
(cid:20) 2 (cid:21) the ferromagnetic case discussed above, namely that it
reduces T . However, we next prove that vertex infla-
The fact that the vertex inflation reduces κ if κ > 2 pt
eff
tion can also have the opposite effect, of increasing a
is clear analytically from this result, since
critical temperature. For this purpose, let us consider
κ ( (G))<κ if ℓ 1 and κ>2 . (5.11) the q = 2 Potts (Ising) antiferromagnet on the infinite
eff ℓ
V ≥ triangularlattice tri (withequalnegativespin-spinex-
{ }
For example, for the square lattice, with κ = 4, one has change constants in each of the three lattice directions,
κeff(V1(sq)) = 8/3 = 2.666.., κeff(V2(sq)) = 12/5 = J1 = J2 = J3 ≡ J < 0). This antiferromagnet is frus-
2.4, κ ( (sq)) = 16/7 2.286, and so forth, with an trated,andis only criticalatT =0[27]. Nowlet us per-
eff 3
V ≃
approach to 2 from above as ℓ . Indeed, in general, form a uniform ℓ-fold vertex inflation on all edges, with
→∞
foranarbitraryregularlatticegraphGwithcoordination ℓ = 2k +1 odd, thereby obtaining 2k+1(tri) . This
{V }
number κ, lattice, 2k+1(tri) , is bipartite, in contrast with the
{V }
triangularlatticeitself. Becauseofthis,theIsingantifer-
lim κ =2 (5.12) romagnetisnotfrustratedon (tri) ,andtherefore
eff 2k+1
ℓ→∞ {V }
one can apply a standard Peierls-type argument to infer
For ℓ>>1, one has the Taylor series expansion thatithasafinite-temperaturesymmetry-breakingphase
transition. Indeed, because (tri) is bipartite and
2k+1
{V }
2 1 2 1 because of the isomorphism mentioned above, the Ising
κeff =2"1+(cid:18)1− κ(cid:19)(cid:26)ℓ + κℓ2 +O(cid:16)ℓ3(cid:17)(cid:27)# (5.13) foetrhreorm, aagnndehtaavnedtahnetiriferrersopmecatgivneetpchaansebtermanaspitpioednstoateathche
same T . Thus, as this example shows, in contrast with
Hence, for a Potts ferromagnet on a lattice graph G pt
{ } the situation for the Potts ferromagnet, vertex inflation
with d 2, a uniform ℓ-fold vertex inflation of G with
≥ { } for the Potts antiferromagnet may actually raise a criti-
ℓ 1 leads to a decrease in the phase transition tem-
≥ calorphase transitiontemperature rather thanlowering
perature. The same conclusion holds if one performs a
it, depending on q and the lattice type.
vertex inflation on all edges along a subset of the lattice
directions.
We illustrate the effect of vertex inflation for the q-
state Potts ferromagnet on the square lattice. Let us D. Invariance of Universality Class Under Edge
Inflations
performauniformvertexinflationwithℓ=1onalledges
of the lattice, i.e., add one degree-2 vertex to each edge
of this lattice. Then, combining Eqs. (4.3) and (5.2), we We recall that on two-dimensional lattices the (zero-
find that the equation for the phase transition temper- field)q-state Pottsferromagnetwithq 4 hasa second-
≤
ature is vv,1 = √q, where vv,1 was given in Eq. (4.16). order phase transition with an associated q-dependent
Thesolutionfortheinversephasetransitiontemperature universality class and corresponding thermal and mag-
K is netic critical exponents that are independent of the lat-
pt,v,1
ticetype. Itisofinteresttostudywhethervertexoredge
inflationchangestheuniversalityclassofthisphasetran-
K =ln 1+√q 1+ 1+√q . (5.14)
pt,v,1 sition. Ageneralresultofrenormalization-groupanalyses
(cid:20) n q o(cid:21)
7
of second-order phase transitions is that the universal- is more complex, but the decay of the envelope curve
ity class of a second-order phase transition (in a model is described by η = 1/2. Now, just as we did before,
that is free of complications such as competing interac- let us perform a uniform ℓ-fold vertex inflation on all
tions,frustration,and/orquencheddisorder)dependson edges, with ℓ = 2k +1 odd, thereby obtaining the bi-
the lattice dimensionality, d and the symmetry group of partitelattice (tri) . Asdiscussedabove,because
2k+1
{V }
the Hamiltonian (e.g., [28]). For the Potts model, the (tri) isbipartite,theIsingantiferromagnetisnot
2k+1
{V }
symmetry group of the Hamiltonian is the symmetric frustratedonit,and,indeed,canbemappedtothe Ising
(permutation) group on q indices, S . Edge inflations of ferromagnet by the mapping given in the previous sub-
q
the lattice have no effect on the dimensionality or sym- section. Owingtothis,theIsingferromagnetandantifer-
metry group of the Hamiltonian; hence, for the range romagnet on this lattice have the same phase transition
q 4 where the two-dimensional Potts ferromagnet has temperatures, and, furthermore, the Ising antiferromag-
≤
a second-order phase transition, they do not change the net is automatically in the same universality class as the
universality class of this transition. Ising ferromagnet, with η = 1/4 [28, 29]. Thus, in this
case, the vertex inflation does change both the value of
the critical temperature and the universality class of the
E. Effects of Vertex Inflation on Universality Class phase transition.
We again consider the interval q 4 where the two-
≤ VI. LONGITUDINAL HOMEOMORPHIC
dimensional Potts ferromagnethas a second-orderphase
INFLATIONS OF FREE LADDER GRAPH
transition. As with edge inflation, we note that vertex
inflation has no effect on the lattice dimensionality or
symmetry group of the Hamiltonian, and hence, by the In the previous sections we have derived general re-
same argument as before, it does not change the univer- lations that connect Z(G,q,v) and Z(G˜,q,v), where G˜
sality class of the phase transition. is obtained from G by uniform edge or vertex inflations.
As before, the situation is more complicated for the By Eq. (2.4), these enable one to calculate the equiva-
Potts antiferromagnet, because, in contrast to the ferro- lentTuttepolynomial,T(G˜,x,y). Wehavealsodiscussed
magnet, its properties depend sensitively on the type of thecasewhereedgeorvertexinflationsareperformedon
lattice. To show this, it is convenient to continue with all edges along a subset of lattice directions of a lattice
thesameillustrativeexamplethatweusedabove,namely graph. In the rest of this paper we explore the latter
the q =2 Ising antiferromagnet. We will give two exam- type of vertex inflation further. We present exact calcu-
ples which exhibit opposite behaviors; in the first, the lations of Potts/Tutte polynomials for a class of vertex
vertex inflation lowers the phase transition temperature (i.e., homeomorphic) inflations of a subset of the edges
andmakesnochangeintheuniversalityclass. Inthesec- of ladder graphs. Our results generalize our calculations
ond, the vertex inflation raises the critical temperature ofthe chromaticpolynomialsforthese graphswithS.-H.
and does change the universality class. The first exam- Tsai in Refs. [9] and [7] (see also [8, 16]).
ple uses the Ising antiferromagnet on a bipartite lattice We consider the free strip of the ladder graph com-
graph G of dimensionality d 2, where it has a phase prised of m squares, which we denote as Sm. Now we
{ } ≥
transition at a temperature T (with antiferromagnetic perform an ℓ-fold vertex inflation on all of the longitudi-
pt
long-range order for T < T ). Now let us perform a nal edges, with ℓ=k 2 and k 3, i.e., we insert k 2
pt − ≥ −
uniformℓ-foldvertexinflationonallofthe edgesof G , degree-2 vertices on each of these edges. The resultant
{ }
thereby obtaining ℓ(G) . By an argument similar to strip graph is denoted Sk,m. The transverse edges are
{V }
theonegivenabove,thislowersT . Sincethisvertexin- notaffectedbythisoperation. Theoriginalladdergraph
pt
flation does not change either the lattice dimensionality itselfisS2,m. ThenumbersofverticesandedgesonSk,m
orthesymmetrygroupS Z ,itleavestheuniversality are
2 2
≈
class unchanged.
Toshowthatthe oppositecanalsohappen,letuscon- n(Sk,m)=2(k 1)m+2 (6.1)
−
sider the (isotropic) Ising antiferromagnet on the trian-
and
gular lattice. As mentioned before, because of the frus-
tration, this model has no finite-temperature transition,
e(S )=(2k 1)m+1 . (6.2)
butiscriticalatT =0;thespin-spincorrelationfunction k,m −
decays asymptotically like σ σ r−1/2cos(2πr/3) for
0 ~r
h i∝ Usingasystematiciterativeapplicationofthedeletion-
large r = ~r [27]. Normally, at a second-order phase
| | contractionproperty,we calculate the Tutte polynomial,
transition of a spin model on a d-dimensional lattice in
T(S ,x,y). One way to express this is in terms of a
which the (connected) spin-spin correlation function de- k,m
cays asymptotically like σ σ r−(d−2+η), one assigns generating function. We use a generating function
0 ~r
h i∝
the critical exponent η to this transition. Because of the
∞
oscillatory nature of the asymptotic decay of the T = 0 Γ(S ,x,y;z)= T(S ,x,y)zm (6.3)
k k,m+1
Ising antiferromagnet on the triangular lattice, this case
m=0
X
8
of the form a = yx2k−1 , (6.8)
1
−
a +a z
0 1
Γ(S ,x,y;z)= . (6.4)
k 1+b z+b z2
1 2
b = [1+T(C ,x,y)] , (6.9)
The denominator can be written as 0 − 2k−1
1+b z+b z2 =(1 λ z)(1 λ z) . (6.5) and
1 2 − k,0,1 − k,0,2
Recall that for the circuit graph with n vertices, C ,
n b =x2k−2y . (6.10)
1
xn+c(1) n−1
T(Cn,x,y)= =y+ xj , (6.6) Thus,
x 1
− j=1
X
1
where c(1) =q 1=xy x y. We calculate λ = b b2 4b . (6.11)
− − − k,0,j 2 − 0± 0− 1
h q i
a =T(C ,x,y) , (6.7)
0 2k
By a generalization of Eq. (2.15) in [16] from chromatic
polynomials to the full Tutte polynomial, it follows that
(a λ +a ) (a λ +a )
T(S ,x,y)= 0 k,0,1 1 (λ )m−1+ 0 k,0,2 1 (λ )m−1 . (6.12)
k,m k,0,1 k,0,2
(λ λ ) (λ λ )
k,0,1 k,0,2 k,0,2 k,0,1
− −
Note that T(S ,x,y) is symmetric under the inter- For the cyclic strip graph L we calculate
k,m k,m
change λ λ . It is straightforward, using Eq.
k,0,1 k,0,2
(2.4), to re-ex↔press these results in terms of the Potts 1 2 nT(2,d)
model partition function Z(Sk,m,q,v); for brevity, we T(Lk,m,x,y)= x 1 c(d) (λk,d,j)m (7.3)
omit the explicit results. − d=0 j=1
X X
wherec(0) =1,c(1) =q 1,c(2) =q2 3q+1,n (2,0)=
T
− −
2, n (2,1)=3, and n (2,2)=1. The λ for j =1,2
VII. LONGITUDINAL HOMEOMORPHIC T T k,0,j
INFLATIONS OF CYCLIC AND MO¨BIUS were given above in Eq. (6.11). For the others, we find
LADDER GRAPHS
λ =xk−1 , (7.4)
k,1,1
In this section we present an exact calculation of the
Tutte polynomial for longitudinal homeomorphic infla- 1
λ = (u √r ) , j =2, 3 , (7.5)
k,1,j k k
tions of cyclic and Mo¨bius ladder graphs. We start with 2 ±
a cyclic ladder strip of length m squares and add k 2
where
−
degree-2 vertices, with k 3, to each longitudinal edge.
This yields the strip graph≥with longitudinal homeomor- k−2
phic inflation that we denote Lk,m. The corresponding uk = 2 xj +xk−1+y
Mo¨biusstripMk,misobtainedbycuttingthecyclicgraph (cid:16)Xj=0 (cid:17)
at any transverse edge and reattaching the ends after a xk+xk−1 2
= − +y , (7.6)
verticaltwist. The graphsLk,m and Mk,m eachhavethe x 1
−
number of vertices
and
n(L )=n(M )=2(k 1)m (7.1)
k,m k,m − r =u2 4xk−1y , (7.7)
k k−
and the number of edges
and finally,
e(L )=e(M )=(2k 1)m . (7.2)
k,m k,m
−
λ =1 . (7.8)
k,2,1
For a given m and for k =2 these are the original cyclic
and Mo¨bius ladder strip graphs; the case k = 3 is the Although our result (7.3) (and (7.9) below) are formally
first homeomorphic inflation of these respective graphs, rational functions in x, one easily verifies that the pref-
and so forth for higher values of k. actor1/(x 1)dividestheexpressiontoitsright,sothat
−
9
T(L ,x,y) and T(M ,x,y) are polynomials in x as whereweintroducethenotationG tostandforeither
k,m k,m k,m
wellasy,asguaranteedbyEq. (2.2). Itisagainstraight- L or M .
k,m k,m
forward, using Eq. (2.4), to re-express these results in We calculate
termsofthePottsmodelpartitionfunctionZ(S ,q,v).
k,m
For the Mo¨bius strip M , we find that the λ’s are 1/2
k,m λ =2k−1 2k−1 22(k−1) 1 , (8.2)
the same as for the cyclic strip Lk,m, and the pattern k,0,j x=2 ± −
of changes in the coefficients is the same as was shown where th(cid:12)ey=1sign apphlies for j(cid:16)=1,2, resp(cid:17)ectiviely,
(cid:12)
in [10, 30, 31] for lattice strips without homeomorphic (cid:12) ±
expansion, so that λ =2k−1 , (8.3)
k,1,1
x=2
2 (cid:12)y=1
T(Mk,m,x,y)= x 1 1 (λk,0,j)m and (cid:12)(cid:12)
"
− j=1
X
3
+c(1) (λk,1,1)m+ (λk,1,j)m 1 . λk,1,j x=2 =
− − # (cid:12)y=1
(cid:16) Xj=2 (cid:17) 1 3 (cid:12)(cid:12)2k−1 1 (3 2k−1 1)2 2k+1 1/2 ,
(7.9) 2 · − ± · − −
(cid:20) h i (cid:21)
Fromthese generalexpressions,onecanspecialize to the (8.4)
chromaticpolynomialsandtheflowpolynomials. ViaEq.
(4.12), one readily checks that the results for the chro- where the sign applies for j =2,3, respectively. Next,
±
matic polynomials agree with those that we calculated define the notation
before with S.-H. Tsai in Ref. [9]. The flow polynomials
are unaffected by homeomorphic expansion, and hence +1 if G =L
η η = k,m k,m (8.5)
coincide with those that we calculated with S.-C. Chang G ≡ Gk,m ( 1 if Gk,m =Mk,m
in Ref. [32]. −
In terms of these quantities, we have, for G = L
k,m k,m
or M ,
VIII. VALUATIONS OF TUTTE POLYNOMIALS k,m
FOR HOMEOMORPHIC EXPANSIONS OF
CYCLIC AND MO¨BIUS LADDER GRAPHS NSF(Gk,m)=(λk,0,1)m+(λk,0,2)m
+η 1 (λ )m (λ )m+(λ )m
SpecialvaluationsoftheTuttepolynomialyieldseveral G − k,1,1 − k,1,2 k,1,3
quantities of graph-theoretic interest. In this section we h i h i
(8.6)
calculatetheseforthelongitudinalhomeomorphicexpan-
sions of cyclic and Mo¨bius ladder graphs. We first recall
whereinEq. (8.6)theλ ’sareevaluatedatx=2,y =
somedefinitions. Atreegraphisaconnectedgraphwith k,d,j
1asinEqs. (8.2)-(8.4). Fork=2,i.e. theoriginalcyclic
no circuits (cycles). A spanning tree of a graph G is a
and Mo¨bius strips, one checks that eq. (8.6) reduces to
spanning subgraph of G that is also a tree. A spanning
eq. (D.14) of our previous work[10]. As an example, for
forestofagraphGisaspanningsubgraphofGthatmay
thefirsthomeomorphicexpansion,k =3,eq. (8.6)yields
consist of more than one connected component but con-
tains no circuits. The special valuations of interest here
are (i) T(G,1,1) = NST(G), the number of spanning NSF(G3,m)=
trees (ST) of G; (ii) T(G,2,1)=N (G) the number of
SF
spanningforests(SF)ofG;(iii)T(G,1,2)=NCSSG(G), [4(4+√15 )]m+[4(4−√15 )]m+ηG(1−22m)
thenumberofconnectedspanningsubgraphs(CSSG)of
G; and (iv) T(G,2,2) = NSSG(G) = 2e(G), the number 11+√105 m+ 11−√105 m . (8.7)
of spanning subgraphs (SSG) of G. The last of these − 2 2
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
quantities is determined directly from Eq. (7.2) as
ForvaluationsofT(G ,x,y)withx=1andG =
k,m k,m
N (G )=2(2k−1)m , (8.1) L or M , a useful equality is
SSG k,m k,m k,m
1 1/2
λ =λ = 2k 1+y (2k 1+y)2 4y , (8.8)
k,0,j k,1,j+1
x=1 x=1 2 − ± − −
(cid:12) (cid:12) (cid:20) h i (cid:21)
(cid:12) (cid:12)
wherethe signappliesforj(cid:12)=1,2,respect(cid:12)ively. ForthenumberofspanningtreesonthesefamiliesonG =L
k,m k,m
±
10
or M , we find
k,m
1 m m
N (G )=m(k 1) η + k+ k2 1 + k k2 1 . (8.9)
ST k,m G
− − 2 − − −
(cid:20) n(cid:16) p (cid:17) (cid:16) p (cid:17) o(cid:21)
For k =2, this reduces to Eq. (D.13) of [10].
TABLEI:Valuesofw ,w ,w ,andw forthem→∞
Next, we introduce the shorthand notation SSG SF CSSG ST
limit of the Gk,m family of graphs, where Gk,m denotes Sk,m,
1 Lk,m,orMk,m.
λ = 2k+1 4k2+4k 7 (8.10)
±
2 ± −
(cid:16) p (cid:17) k w w w w
and SSG SF CSSG ST
2 2.8284271 2.7320508 2.1357792 1.9318517
k 1 (2k2+k 4)
a = − k − . (8.11) 3 2.3784142 2.3689169 1.6089554 1.5537740
±
2 ± √4k2+4k 7
(cid:16) (cid:17)(cid:20) − (cid:21) 4 2.2449241 2.2434544 1.4360948 1.4104463
In terms of these quantities, we find, for the number of 5 2.1810155 2.1807485 1.3466467 1.3318300
connected spanning subgraphs, 6 2.1435469 2.1434946 1.2908358 1.2811894
7 2.1189262 2.1189154 1.2522226 1.2454451
N (L )= 2+(λ )m+(λ )m
CSSG k,m + −
− 8 2.1015133 2.1015110 1.2236924 1.2186716
9 2.0885476 2.0885471 1.2016292 1.1977615
+ m 1 k+(λ )m−1a +(λ )m−1a (8.12)
+ + − −
− 10 2.0785185 2.0785183 1.1839857 1.1809157
(cid:20) (cid:21)
∞ 2 2 1 1
and
N (M )=N (L )+1+2(k 1)m. (8.13)
CSSG k,m CSSG k,m
−
For k =2, this reduces to Eq. (D.15) of [10]. and
These graphical quantities grow exponentially as a 1
function of the strip length m and hence also n. For w = k+ k2 1 2(k−1) . (8.20)
ST({Lk}) −
each quantity N (i.e., N , N , N , and N ),
G ST SF CSSG SSG (cid:16) p (cid:17)
one can thus define a growth constant For k = 2, i.e., the original ladder strip without home-
omorphic expansion, Eqs. (8.17), (8.18), (8.19), and
wNG(G) ≡ml→im∞[NG(G)]1/n . (8.14) (8.20), together with (8.15), reduce to Eqs. (D.24),
(D.22), (D.23), and (D.21) of our previous paper, Ref.
In Ref. [10] we used an equivalent quantity to describe [10]. Some numerical values of these growth constants
the exponential growth, viz., obtained from the analytic results given above are dis-
played in Table I.
z ln[w ] . (8.15)
NG(G) ≡ NG(G)
We find that for a given set of subgraphs (G), these
G IX. TUTTE POLYNOMIALS OF HAMMOCK
growth constants are the same for the m limits of
→ ∞ GRAPHS
the S , L , and M families of strip graphs, i.e.,
k,m k,m k,m
w =w =w , (8.16) A. General Calculation
NG({Sk}) NG({Lk}) NG({Mk)}
so we will just label them by L . For each type of
{ k} A hammock graph Hk,r is defined as follows: start
subgraph , the growth constant is determined by the
G with two vertices connected by r edges (where r denotes
dominant λ , which is λ , evaluated at the respec-
k,d,j k,0,1 “rope” or, more abstractly, “route”). Now add k 2
tive values of (x,y). We find −
degree-2 vertices to each of these edges, with k 3, so
≥
2k−1 thatonanyropethereareatotalofk vertices,including
w =22(k−1) , (8.17)
SSG({Lk}) the twoend-vertices. Thus, Hk,r with k 3 is auniform
≥
ℓ = k 2 fold vertex (homeomorphic) inflation of the
−
w =2 1+ 1 2−2(k−1) 1/2 2(k1−1) , (8.18) originalgraph,H2,r withr edgesconnectingthe twoend
vertices. The number of vertices, edges, and (linearly
SF({Lk}) −
h (cid:16) (cid:17) i independent) cycles in this graph are
2k+1+(4k2+4k 7)1/2 2(k1−1) n(Hk,r)=2+(k 2)r , (9.1)
w = − , −
CSSG({Lk}) 2
(cid:20) (cid:21)
(8.19) e(H )=(k 1)r , (9.2)
k,r
−