arXiv:1101.5826v1 [cond-mat.stat-mech] 30 Jan 2011 β atc,o,mr eeal,o ewr ie,graph) (i.e., network a on generally, On more G or, [1]. lattice, phenomena a cooperative many-body exhibiting oe i eoetra ed is field) external zero (in model h pnsi xhneconstant, exchange spin-spin the σ sts on (sites) euetenotation the use We h ot oe atto function partition model Potts The ilin mial (AFM, (FM, ferromagnet Potts the − oetenme fvrie of vertices of number the note h ubro de of edges of number the sdfie yisvre set, vertex its by defined is onstenme fwy fassigning of ways of of vertices number which interest the polynomial, longstanding chromatic counts the of namely theory, function graph another in antiferromagnet to partition Potts unified identical The a zero-temperature is in the polynomial. object of Potts/Tutte this function the to and as refer [2]-[4], often manner polynomial therefore Tutte shall the we namely theory, mathematical modern graph in interest considerable of function ot/ut oyoilcluae o graph the a relate for to is calculated theory polynomial graph Potts/Tutte mathematical and 6]. chanics [5, thermodynamics of law thus is third and the temperature to zero exception at an large site per sufficiently entropy for nonzero that is tiferromagnet proper called are These q color. same the have vertices cent clrnsof -colorings i ttemperature at , 1 1 = xc eut nPtsTtePlnmasfrFmle fNe of Families for Polynomials Potts/Tutte on Results Exact h ot oe a ogbe aubea system a as valuable been long has model Potts The nitrsigpolmivligbt ttsia me- statistical both involving problem interesting An r lsia pn aigvle nteset the in values taking spins classical are ≤ / v ( q < J ≤ k B and G ,rsetvl.I eea,agraph a general, In respectively. 0, T G , ,teHamiltonian the ), ASnmes 05.50.+q,64.60.Cn,75.10.Hk et numbers: PACS trees, spanning re of these networks. numbers of large as for Applications such behavior evaluate interest network. we structural the addition, of In of net polynomials. bonds of reliability the families network of for subset polynomials these a of calculations exact epciey h hscleet fteeeg n etxi vertex and edge these of effects physical The respectively. rmr ere2vrie negsof edges on vertices degree-2 more of or bond) (i.e., edge oyoil o ewr (graph) network a for polynomial, ) h hsclrne of ranges physical the 0), e ujc otecniinta otoadja- two no that condition the to subject G v ij edrv xc eain ewe h ot oe partition model Potts the between relations exact derive We swl eeietfo q 21 below. (2.1) Eq. from evident be will as , nipratpoet ftePtsan- Potts the of property important An . .INTRODUCTION I. steeg bn)cnetn hm and them, connecting (bond) edge the is T K h atto ucinfrthis for function partition the , ≡ G βJ as V n t deset, edge its and , > J e and ( G G G H Z ) )adantiferromagnet and 0) i as ytoo oeegsjiigtesm etcs n i)b ins by (ii) and vertices, same the joining edges more or two by v | ≡ = = and ≡ n Z .N agIsiuefrTertclPhysics Theoretical for Institute Yang N. C. P − E ≡ e seuvln oa to equivalent is J j { | v K . σ n P eoevertices denote q i are − Z ( } G q e oost the to colors e ij .Tu,for Thus, 1. texhibits it , sapolyno- a is ) − G v E δ β | ≡ G tn ro,N 11794 NY Brook, Stony etxInflations Vertex σ tn ro University Brook Stony { H ( = ≥ ede- We . i 1 σ where , G n ewr bandfrom obtained network a and V .,q ..., , G j and 0 oetShrock Robert , ,E V, | hs rcse r aldeg n etxinflation, vertex and edge called are processes These . with J and } is ) . ot/ut oyoil nvrosfmle fgraphs, of of families calculations various analytic on exact polynomials out Potts/Tutte carry to value mental i yrpaigec deof edge each replacing by (i) iegah fdimensionality of graphs tice temperature and onn h aevrie,ad(i yisrigoeor one inserting of by edge each (ii) on and vertices degree-2 vertices, more same the joining nw xc lsdfr ouinfor solution closed-form exact known hspprw ilfis rsn eea ouint this to of solution classes general two a of present for modification first problem will specified we a paper by this obtained is which oilivle ubro opttoa tp n a with rapidly and exponentially n steps grow computational that time of corresponding number a involves nomial graph bitrary strips [10]. cal- ladder Ref. for previous in polynomials our [7]-[9] Potts/Tutte of the Refs. inflations of in culation homeomorphic graphs as of homeomor- (b) families for and these polynomials of full chromatic inflations the the phic with to of calculations generalizations Tsai earlier as S.-H. previ- our (a) our of ways: extend we polynomials Potts/Tutte two that results in family These work a ous and M¨obius graphs. ladder length and hammock longitudi- great call cyclic, include arbitrarily free, families of of These graphs inflations inflations edges. vertex vertex of nal by of subset obtained polynomials a graphs Potts/Tutte for of for families results certain present will we of degree-2 of duction more edges or one from removing vertices of procedure, reverse the h orsodn oyoilcluae o graph a for calculated polynomial corresponding the ueo ahmtclgahter,avre inflation vertex a theory, graph of edge mathematical as on operations of ture these inflations denote vertex We and it). to connect that gree, ( G hr r eea oiain o hswr.Fra ar- an For work. this for motivations several are There G and ) κ u eut ocluaevrosquantities various calculate to results our sas aldahmoopi nainof inflation homeomorphic a called also is ok bandvateoeain(i on (ii) operation the via obtained works v fltosaedsusd eas present also We discussed. are nflations i . n odtrieterasymptotic their determine to and c., favertex a of , e ( ucin reuvlnl h Tutte the equivalently or function, G ut nld acltoso some of calculations include sults G ntermiigscin forpaper our of sections remaining the In . G eg,[1 2) utemr,teei no is there Furthermore, 12]). [11, (e.g., ) h aclto ftePtsTtepoly- Potts/Tutte the of calculation the , T G ntetemdnmclmto lat- of limit thermodynamic the on y()b elcn each replacing by (i) by v i G sdfie stenme fedges of number the as defined is wrswt deand Edge with tworks scle oemrhcre- homeomorphic a called is , G epciey ntelitera- the In respectively. , G ˜ d s aeytoeobtained those namely ’s, ≥ G .Hne ti ffunda- of is it Hence, 2. ytoo oeedges more or two by rigone erting Z G o arbitrary for weetede- the (where G G and , In . G ˜ q 2 such as lattice strip graphs and modifications thereof. For a graph G, let us denote G e as the graph obtained Furthermore, special cases of the Potts/Tutte polynomial by deleting the edge e and G/e as− the graph obtained by are of considerable interest in their own right. We have deleting the edge e and identifying the two vertices that noted the importance of the zero-temperature Potts anti- were connected by this edge of G. This operation is called ferromagnet (chromatic polynomial). Thus, another mo- 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-contraction relation to generalize our previous calculations with S.-H. Tsai [7]- [9] to the broader context of the finite-temperature Potts Z(G, q, v)= Z(G e,q,v)+ vZ(G/e,q,v) . (2.5) − antiferromagnet and the Potts ferromagnet (the latter also at arbitrary temperature). A central motivation is An analogous deletion-contraction relation holds for to investigate the effects of edge and vertex inflations of T (G, x, y). 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 present work 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 inflation of a graph G. As part of our study, we will con- 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 In this section we briefly discuss some necessary back- edges of G by multiple edges joining the same vertices. ground that will be used for our calculations. There is Clearly, this leaves the number of vertices n(G) invariant. a useful relation that expresses the Potts model parti- In particular, it is natural to define a uniform ℓ-fold edge inflation of G as the graph (G) obtained by replacing tion function Z(G, q, v) as a sum of contributions from Eℓ spanning subgraphs of G. Here, a spanning subgraph each edge of G by ℓ edges joining the same two vertices. ′ ′ Thus, the number of edges of (G) is e( (G)) = ℓe(G). G = (V, E ) has the same vertex set as G and a subset Eℓ Eℓ 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, κvi = κ vi V . Al- ′ ′ ∀ ∈ 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 κeff as GX⊆G ′ where k (G ) denotes the number of connected compo- κvi c κ = vi∈V . (3.1) nents in G′. This formula shows that Z(G, q, v) is a poly- eff n(G) nomial in q and v. For the Potts ferromagnet, Eq. (2.1) P enables one to generalize q from the non-negative integers Clearly, a uniform ℓ-fold inflation of all of the edges of G to the non-negative real numbers while keeping Z(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] mechanical framework, 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 to be a graph obtained by inserting one or more degree-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 number of (linearly independent) circuits in G, 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 and Tutte polynomials [7]-[9], [15]-[21] and have also been so that q = (x 1)(y 1). The relation between Z(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 Vℓ 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 − − special case where G is a section of a regular lattice, it is With no loss of generality, we will restrict here to con- also natural to consider edge or homeomorphic inflation nected graphs G, so kc(G) = 1. of edges forming a subset of lattice vectors. For example, 3
d for a d-dimensional Euclidean lattice E , one can consider the edges eij , we note parenthetically that one can con- the case in which the edge or vertex inflation is performed sider a generalization in which the Jij are different for on the edges along one or more lattice vectorse ˆj, where each edge of G, eij . From the original 1944 Onsager j takes values in a subset of 1, ..., d . solution of the two-dimensional Ising model [23], a large { } number of studies of spin models have dealt with the gen- 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-spin exchange constants Jij that can be different (in magnitude and sign) for each edge, eij were motivated by A. Edge Inflation early work on spin glasses [24]. In this case, where Jij ’s depend on eij , the transformation of the Hamiltonian be- The effect of a uniform edge inflation on the Potts comes model partition function can be determined from an anal- = J δ ℓ J δ . (4.8) ysis of the Potts Hamiltonian. As we edge-expand G to H − ij σiσj →− ij σiσj e e (G), i.e., replace each edge by ℓ edges joining the same Xij Xij Eℓ vertices, we induce the change in the Hamiltonian Hence, defining K βJ and v eKij 1, we have ij ≡ ij ij ≡ − ℓ = J δσ σ ℓJ δσ σ , (4.1) v v = (v + 1) 1 . (4.9) H − i j →− i j ij → ij,e,ℓ ij − eij eij X X 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) Eℓ { } { e,ℓ} 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 where the subscripts e,ℓ refer to the ℓ-fold edge inflation. limit, the only→ contributions→ −∞ to the→− partition function are Equivalently, from spin configurations in which adjacent spins have dif- ℓ ferent values. The resultant T = 0 Potts AFM partition v ve,ℓ = (v + 1) 1 , (4.3) → − function is therefore precisely the chromatic polynomial where ve,ℓ ye,ℓ 1. This proves the following relation P (G, q) of the graph G counting the number of proper connecting≡Z on G− with Z on (G): q-colorings of G: Eℓ Z( (G),q,v)= Z(G, q, v ) . (4.4) Z(G, q, 1) = P (G, q) . (4.11) Eℓ e,ℓ − 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 ground state degeneracy per vertex, W ( G , q) > 1, where W ( G , q) = lim P (G, q)1/n. The{ } ground (x 1)(y 1) x 1 { } n→∞ xe,ℓ =1+ − − =1+ − . (4.6) state entropy per vertex of the Potts model on G is yℓ 1 ℓ−1 yj { } − j=0 S0( G , q)= kB ln[W ( G , q)]. In Refs. [6], [7]-[9], [16], { } { } P we applied our exact results on chromatic polynomials to Combining these transformations of variables 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 ℓ(G), namely E complement other approaches to studying S0, such as 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 xe,ℓ and ye,ℓ were given in Eqs. (4.6) and (4.2). Note that the effect of the edge inflation on the Potts nomial that P (G, q) does not change if one replaces any partition function is simpler than the effect on the Tutte edge of G by two or more edges joining the same vertices. polynomial, since in the former case, only one of its vari- In particular, for the case of an ℓ-fold uniform edge in- flation of G, ables is modified, namely, v ve,ℓ, whereas in the latter case, both of its variables are→ modified, as x x and → e,ℓ P ( ℓ(G), q)= P (G, q) . (4.13) y ye,ℓ. E →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 The flow on each edge can thus take on any of q 1 values modulo q. The flow or current conservation condition− is To analyze the effect of a vertex inflation of a graph G, 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. For simplicity, we assume here that G does not 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 polynomial for x = 0 and y = 1 q or equivalently, in our calculation to the case of multiple edges. We use the − 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) cated at these degree-2 vertices. Let us considerP an edge, − − eij , and insert a degree-2 vertex va (and its associated As is clear from the definition of the flow polynomial, spin, σa) on this edge. Then adding or removing a degree-2 vertex from an edge of G does not change the number of allowed q-flows on G, so
(1 + vδσiσa )(1 + vδσaσj ) F ( ℓ(G), q)= F (G, q) . (4.20) σi,σa,σj V X 2 As is evident from Eq. (4.16), the condition v = q = 1+ v(δσiσa + δσaσj )+ v δσiσj . − implies that vv,1 = q and hence, more generally, that σi,σa,σj − X h i vv,ℓ = q for arbitrary ℓ 1. Recall that the number of (4.14) cycles in− G is unchanged by≥ this homeomorphic inflation: c( ℓ(G)) = c(G). Now, carrying out the summation over σa, we find that V if σ = σ , then the result is q +2v + v2, while if σ = σ , i j i 6 j then the result is q +2v. Therefore, V. PHYSICAL EFFECTS OF EDGE AND VERTEX INFLATION
(1 + vδσiσa )(1 + vδσaσj ) σ ,σ ,σ iXa j A. General
= (q +2v) (1 + vv,1δσiσj ) , (4.15) σ ,σ Xi j An important question concerns the effect of edge 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 v = (4.16) v,1 q +2v regular lattice graph of dimension d 2, where the fer- romagnetic version of the model has a≥ finite-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 e(G) Z( 1(G),q,v) = (q +2v) Z(G, q, vv,1) . (4.17) phase transition (pt) temperature Tpt for lattices where V 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 (G) and Z on Vℓ lattices, there is no known exact closed-form solution G. For example, for ℓ = 2, we have for the free energy of the q-state Potts model at arbi-
2 trary temperature on these lattices with d 2. There vv,1 is thus also some interest in studying the effect≥ of edge vv,2 = q +2vv,1 and vertex inflation for infinite-length limits of quasi-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 ℓ(G) to Z on G. finite-temperature phase transition on an infinite-length AV basic problem in graph theory is the enumeration quasi-one-dimensional lattice strip. Nevertheless, one in- of discretized flows on the edges of a (connected) G that teresting application of exact solutions for the free energy satisfy flow conservation at each vertex, i.e. for which and thermodynamic quantities on these strips is that one 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 associated Tutte polynomials) on graphical properties, 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- we take a q-flow to mean implicitly a nowhere-zero q-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 Tpt,es,ℓ as the shifted phase tran- of Tpt due to a uniform ℓ-fold edge inflation (symbol- sition temperature after an ℓ-fold edge inflation along 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 Tpt,e,ℓ note Kpt,es,ℓ J/(kB Tpt,es,ℓ). Then the reasoning above and T . We thus denote K J/(k T ) and yields the inequality≡ T >T for ℓ 2. As an exam- pt,v,ℓ pt,e,ℓ ≡ B pt,e,ℓ pt,es,ℓ pt ≥ Kpt,v,ℓ J/(kB Tpt,v,ℓ). ple, we again consider the (infinite) square lattice sq ≡ and perform an edge inflation with ℓ = 2 for edges{ in} one of the two lattice directions, saye ˆ2, so that K1 = K, 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 A general result is that for the Potts model on (an infi- physical solution nite) regular lattice graph G with dimensionality d 2 { } ≥ 1 1/3 −1/3 that has a finite-temperature phase transition (of either Kpt,e2,2 = ln A +4A +1 , (5.5) 3 ferromagnetic or antiferromagnetic type, and either first n o or second order) at a temperature Tpt, a uniform ℓ-fold where the subscript e2 means inflation along edges along inflation of all edges of G has the effect of multiplying eˆ and { } 2 Tpt by the factor ℓ, i.e., 1 A = 27q 16+3 3q(27q 32) . (5.6) Kpt 2 − − Tpt,e,ℓ = ℓTpt ,Kpt,e,ℓ = . (5.1) h p i ℓ The resultant Kpt,e2,2
κeff ( ℓ(G)) <κ if ℓ 1 and κ> 2 . (5.11) the q = 2 Potts (Ising) antiferromagnet on the infinite V ≥ triangular lattice tri (with equal negative spin-spin ex- { } For example, for the square lattice, with κ = 4, one has change constants in each of the three lattice directions, κ ( (sq)) = 8/3=2.666.., κ ( (sq)) = 12/5 = J1 = J2 = J3 J < 0). This antiferromagnet is frus- eff V1 eff V2 ≡ 2.4, κeff ( 3(sq)) = 16/7 2.286, and so forth, with an trated, and is only critical at T = 0 [27]. Now let us per- approachV to 2 from above≃ as ℓ . Indeed, in general, form a uniform ℓ-fold vertex inflation on all edges, with → ∞ for an arbitrary regular lattice graph G with coordination ℓ = 2k + 1 odd, thereby obtaining 2k+1(tri) . This {V } number κ, lattice, 2k+1(tri) , is bipartite, in contrast with the triangular{V lattice itself.} Because of this, the Ising antifer- lim κeff = 2 (5.12) romagnet is not frustrated on 2k+1(tri) , and therefore ℓ→∞ one can apply a standard Peierls-type{V argument} to infer For ℓ >> 1, one has the Taylor series expansion that it has a finite-temperature symmetry-breaking phase transition. Indeed, because (tri) is bipartite and {V2k+1 } 2 1 2 1 because of the isomorphism mentioned above, the Ising κeff =2 1+ 1 + 2 + O 3 (5.13) ferromagnet and antiferromagnet can be mapped to each " − κ ℓ κℓ ℓ # other, and have their respective phase transitions at the 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- cal or phase transition temperature rather than lowering 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 perform a uniform vertex inflation with ℓ = 1 on all edges Inflations 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 Potts ferromagnet with q 4 has a second- ≤ ature is vv,1 = √q, where vv,1 was given in Eq. (4.16). order phase transition with an associated q-dependent The solution for the inverse phase transition temperature universality class and corresponding thermal and mag- Kpt,v,1 is netic critical exponents that are independent of the lat- tice type. It is of interest to study whether vertex or edge inflation changes the universality class of this phase tran- K = ln 1+ √q 1+ 1+ √q . (5.14) pt,v,1 sition. A general result of renormalization-group analyses n q o 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/or quenched disorder) depends on edges, with ℓ = 2k + 1 odd, thereby obtaining the bi- the lattice dimensionality, d and the symmetry group of partite lattice (tri) . As discussed above, because {V2k+1 } the Hamiltonian (e.g., [28]). For the Potts model, the 2k+1(tri) is bipartite, the Ising antiferromagnet is not symmetry group of the Hamiltonian is the symmetric {Vfrustrated on} it, and, indeed, can be mapped to the Ising (permutation) group on q indices, Sq. Edge inflations of ferromagnet by the mapping given in the previous sub- the lattice have no effect on the dimensionality or sym- section. Owing to this, the Ising ferromagnet and antifer- 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- dimensional Potts ferromagnet has a≤ second-order phase VI. LONGITUDINAL HOMEOMORPHIC 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- lent Tutte polynomial, T (G,˜ x, y). We have also discussed magnet, its properties depend sensitively on the type of the case where edge or vertex inflations are performed on lattice. To show this, it is convenient to continue with all edges along a subset of lattice directions of a lattice the same illustrative example that we used above, 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 and makes no change in the universality class. In the sec- of ladder graphs. Our results generalize our calculations ond, the vertex inflation raises the critical temperature of the chromatic polynomials for these graphs with S.-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 Tpt (with antiferromagnetic perform an ℓ-fold vertex inflation on all of the longitudi- long-range order for T
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, Cn, 2k−2 b1 = x y . (6.10) n−1 xn + c(1) T (C ,x,y)= = y + xj , (6.6) n x 1 Thus, 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 a0 = T (C2k,x,y) , (6.7) 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 (λ λ ) k,0,1 (λ λ ) k,0,2 k,0,1 − k,0,2 k,0,2 − k,0,1
Note that T (Sk,m,x,y) is symmetric under the inter- For the cyclic strip graph Lk,m we calculate change λk,0,1 λk,0,2. It is straightforward, using Eq. ↔ 2 nT (2,d) (2.4), to re-express these results in terms of the Potts 1 T (L ,x,y)= c(d) (λ )m (7.3) model partition function Z(Sk,m,q,v); for brevity, we k,m x 1 k,d,j j=1 omit the explicit results. − Xd=0 X where c(0) = 1, c(1) = q 1, c(2) = q2 3q + 1, n (2, 0) = − − T VII. LONGITUDINAL HOMEOMORPHIC 2, nT (2, 1) = 3, and nT (2, 2)= 1. The λk,0,j for j =1, 2 INFLATIONS OF CYCLIC AND MOBIUS¨ were given above in Eq. (6.11). For the others, we find LADDER GRAPHS k−1 λk,1,1 = x , (7.4) In this section we present an exact calculation of the Tutte polynomial for longitudinal homeomorphic infla- 1 λk,1,j = (uk √rk ) , j =2, 3 , (7.5) tions of cyclic and M¨obius 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 u = 2 xj + xk−1 + y phic inflation that we denote Lk,m. The corresponding k j=0 M¨obius strip Mk,m is obtained by cutting the cyclic graph X at any transverse edge and reattaching the ends after a xk + xk−1 2 = − + y , (7.6) vertical twist. The graphs Lk,m and Mk,m each have the x 1 number of vertices − and
n(Lk,m)= n(Mk,m)=2(k 1)m (7.1) 2 k−1 − rk = uk 4x y , (7.7) and the number of edges − and finally, e(L )= e(M )=(2k 1)m . (7.2) k,m k,m − λk,2,1 =1 . (7.8) For a given m and for k = 2 these are the original cyclic and M¨obius 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. actor 1/(x 1) divides the expression to its right, so that − 9
T (Lk,m,x,y) and T (Mk,m,x,y) are polynomials in x as where we introduce the notation Gk,m to stand for either well as y, as guaranteed by Eq. (2.2). It is again straight- Lk,m or Mk,m. forward, using Eq. (2.4), to re-express these results in We calculate terms of the Potts model partition function Z(Sk,m,q,v). 1/2 For the M¨obius strip Mk,m, we find that the λ’s are k−1 k−1 2(k−1) λk,0,j =2 2 2 1 , (8.2) the same as for the cyclic strip L , and the pattern x=2 k,m y=1 ± − h i of changes in the coefficients is the same as was shown where the sign applies for j =1, 2, respectively, in [10, 30, 31] for lattice strips without homeomorphic ± expansion, so that k−1 λk,1,1 x=2 =2 , (8.3) y=1 2 1 m T (Mk,m,x,y)= (λk,0,j ) x 1 and − " j=1 3 X (1) m m λk,1,j = +c (λk,1,1) + (λk,1,j ) 1 . x=2 − − y=1 j=2 # 1/2 X 1 k−1 k−1 2 k+1 3 2 1 (3 2 1) 2 , (7.9) 2 · − ± · − − h i From these general expressions, one can specialize to the (8.4) chromatic polynomials and the flow polynomials. Via Eq. (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 Gk,m = Lk,m or M , VIII. VALUATIONS OF TUTTE POLYNOMIALS k,m FOR HOMEOMORPHIC EXPANSIONS OF m m CYCLIC AND MOBIUS¨ LADDER GRAPHS NSF (Gk,m) = (λk,0,1) + (λk,0,2)
+η 1 (λ )m (λ )m + (λ )m Special valuations of the Tutte polynomial yield several G − k,1,1 − k,1,2 k,1,3 quantities of graph-theoretic interest. In this section we h i h i calculate these for the longitudinal homeomorphic expan- (8.6) sions of cyclic and M¨obius ladder graphs. We first recall some definitions. A tree graph is a connected graph with where in Eq. (8.6) the λk,d,j ’s are evaluated at x = 2, y = no circuits (cycles). A spanning tree of a graph G is a 1 as in Eqs. (8.2)-(8.4). For k = 2, i.e. the original cyclic spanning subgraph of G that is also a tree. A spanning and M¨obius strips, one checks that eq. (8.6) reduces to forest of a graph G is a spanning subgraph of G that may eq. (D.14) of our previous work [10]. As an example, for consist of more than one connected component but con- the first homeomorphic expansion, k = 3, eq. (8.6) yields 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) = NSF (G) the number of [4(4 + √15 )]m + [4(4 √15 )]m + η (1 22m) spanning forests (SF ) of G; (iii) T (G, 1, 2) = NCSSG(G), − G − the number of connected spanning subgraphs (CSSG) of m m e(G) 11 + √105 11 √105 G; and (iv) T (G, 2, 2) = NSSG(G)=2 , the number + − . (8.7) − 2 2 of spanning subgraphs (SSG) of G. The last of these quantities is determined directly from Eq. (7.2) as For valuations of T (Gk,m,x,y) with x = 1 and Gk,m = (2k−1)m NSSG(Gk,m)=2 , (8.1) Lk,m or Mk,m, a useful equality is
1/2 1 2 λk,0,j = λk,1,j+1 = 2k 1+ y (2k 1+ y) 4y , (8.8) x=1 x=1 2 − ± − − h i where the sign applies for j =1, 2, respectively. For the number of spanning trees on these families on G = L ± k,m k,m 10 or Mk,m, we find
1 m m N (G )= m(k 1) η + k + k2 1 + k k2 1 . (8.9) ST k,m − − G 2 − − − n p p o
For k = 2, this reduces to Eq. (D.13) of [10]. TABLE I: Values of w , w , w , and w for the m →∞ Next, we introduce the shorthand notation SSG SF CSSG ST limit of the Gk,m family of graphs, where Gk,m denotes Sk,m, L M 1 k,m, or k,m. λ = 2k +1 4k2 +4k 7 (8.10) ± 2 ± − k w w w w p SSG SF CSSG ST and 2 2.8284271 2.7320508 2.1357792 1.9318517 2 k 1 (2k + k 4) 3 2.3784142 2.3689169 1.6089554 1.5537740 a± = − k − . (8.11) 2 2 ± √4k +4k 7 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
m m 7 2.1189262 2.1189154 1.2522226 1.2454451 NCSSG(Lk,m)= 2 + (λ+) + (λ−) − 8 2.1015133 2.1015110 1.2236924 1.2186716
m−1 m−1 9 2.0885476 2.0885471 1.2016292 1.1977615 + m 1 k + (λ+) a+ + (λ−) a− (8.12) − 10 2.0785185 2.0785183 1.1839857 1.1809157 ∞ 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 2(k−1) function of the strip length m and hence also n. For w = k + k2 1 . (8.20) ST ({Lk}) − each quantity N (i.e., N , N , N , and N ), G ST SF CSSG SSG p 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 1/n wNG(G) lim [NG(G)] . (8.14) (8.20), together with (8.15), reduce to Eqs. (D.24), ≡ m→∞ (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 IX. TUTTE POLYNOMIALS OF HAMMOCK growth constants are the same for the m G limits of → ∞ GRAPHS the Sk,m, Lk,m, and Mk,m families of strip graphs, i.e., A. General Calculation wNG ({Sk}) = wNG ({Lk}) = wNG ({Mk)} , (8.16) so we will just label them by Lk . For each type of A hammock graph H is defined as follows: start subgraph , the growth constant{ is} determined by the k,r G with two vertices connected by r edges (where r denotes dominant λk,d,j , which is λk,0,1, evaluated at the respec- “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 that on any rope there are a total of k vertices, including w =2 2(k−1) , (8.17) SSG({Lk}) the two end-vertices. Thus, Hk,r with k 3 is a uniform ℓ = k 2 fold vertex (homeomorphic)≥ inflation of the 1 − 1/2 2(k−1) original graph, H2,r with r edges connecting the two end w =2 1+ 1 2−2(k−1) , (8.18) SF ({Lk}) − vertices. The number of vertices, edges, and (linearly h i independent) cycles in this graph are 1 2 1/2 2k +1+(4k +4k 7) 2(k−1) n(Hk,r)=2+(k 2)r , (9.1) w = − , − CSSG({Lk}) 2 (8.19) e(H ) = (k 1)r , (9.2) k,r − 11 and where T (Cn,x,y) was given in Eq. (6.6). The second special case is for k = 2. If G is a planar graph, we c(Hk,r)= r 1 . (9.3) denote its planar dual as G∗. From Eq. (2.2), it follows − that (satisfying the general relation c(G) = e(G)+ kc(G) n(G)). Hence, − T (G, x, y)= T (G∗,y,x) . (9.9) 2r(k 1) κeff (Hk,r)= − . (9.4) 2 + (k 2)r We can apply this result here, since H is a planar − k,r graph. Now H is the graph consisting of two vertices An interesting feature of the H graphs is that if r 2,r k,r → connected by r edges. This is the planar dual to the for fixed k, the combination of tne two end vertices, circuit graph; ∞each with κ = r and the (k 2)r interior vertices on the “ropes”, each with κ = 2, yields− the result ∗ H2,r = (Cr) . (9.10) k 1 lim κeff (Hk,r)=2 − . (9.5) r→∞ k 2 − From Eqs. (9.9), (9.10), and (6.6), it follows that
Although Hk,r is much simpler than the complex net- works encountered in biological and social contexts, the yr + xy x y T (H ,x,y)= T (C ,y,x)= − − . (9.11) property that for large r it has vertices of quite differ- 2,r r y 1 ent degrees is also observed in complex networks [33]. − The girth g(G) of a graph G is the number of edges in a minimum-distance circuit in G. For r 2, the girth of Proceeding to the general Hk,r graph, we define ≥ Hk,r is k−2 xk−1 1 g(Hk,r)=2k 2 . (9.6) λ = xj = − (9.12) − H,1 x 1 j=0 − Before presenting our general results for T (Hk,r,x,y), X we note two special cases. First, for r = 2, the graph H is just the circuit graph with 2k 2 vertices: and k,2 − H = C . (9.7) k,2 2k−2 λH,2 = T (Ck−1,x,y) . (9.13) Hence, Using an iterative application of the deletion-contraction T (Hk,2,x,y)= T (C2k−2,x,y) , (9.8) relation, we calculate the Tutte polynomial of Hk,r to be
(λ )r−2 (λ )r−2 T (H ,x,y) = (λ )r−2 T (C ,x,y)+ H,1 − H,2 (λ )2 . (9.14) k,r H,1 2k−2 λ λ H,2 H,1 − H,2
B. Chromatic and Flow Polynomials dual, then the flow and chromatic polynomials satisfy the relation
Using Eq. (4.12), one readily verifies that for the case −1 ∗ x =1 q, y = 0, the Tutte polynomial (9.14) yields the F (G, q)= q P (G , q) . (9.16) − chromatic polynomial for Hk,r that we calculated in (Eq. −1 (3.7) of) Ref. [7]. We observe that F (Hk,r, q) = q P (Cr, q), in accord with (4.20) and the fact that H = (C )∗ (cf. Eq. One can also discuss other special cases. For the flow 2,r r (9.10)). polynomial F (Hk,r, q), we set x = 0, y =1 q and, with eq. (4.19), we have −
F (H , q)= q−1[(q 1)r + (q 1)( 1)r] . (9.15) C. Reliability Polynomial k,r − − − This is independent of k, in accordance with the general A communication network, such as the internet, can result (4.20). If G is a planar graph and G∗ is its planar be represented by a graph, with the vertices of the graph 12
4 2 representing the nodes of the network and the edges of R(H3,3,p)= p (7p 18p + 12) , (9.22) the graph representing the communication links between − 5 2 these nodes. In realistic networks, both the nodes and R(H3,4,p)= p (4 3p)(5p 12p + 8) . (9.23) the links between them are imperfect, and fail to oper- − − ate. One common measure of the reliability of the net- and work is the probability that there is a working commu- 6 4 3 2 R(H3,5,p)= p (31p 150p +280p 240p+80) (9.24) nications route between any node and any other node. − − This is the all-terminal reliability function. This is com- Hence, for the pairs (r, r′) = (r, 2), monly modeled by a simplification in which one assumes R(H ,p) R(H ,p)= p3(1 p)2(7p 4) , (9.25) that each node and link are operating with respective 3,3 − 3,2 − − probabilities pnode and plink. As probabilities, pnode and R(H ,p) R(H ,p)= p3(1 p)2(4 3p)(5p2 2p 1) , plink lie in the interval [0,1]. The dependence of the all- 3,4 3,2 − − − − (9.26)− terminal reliability function Rtot(G, pnode,plink) on pnode n and is an overall factor of (pnode) ; i.e., Rtot(G, pnode,plink)= (p )nR(G, p ). Thus, the difficult part of the cal- 3 2 5 4 3 2 node link R(H3,5,p) R(H3,2,p)= p (1 p) (31p 88p +73p 6p 5p 4) culation of Rtot(G, pnode,plink) is the determination of − − − (9.27)− − − R(G, plink). For notational brevity, we set plink p. As is evident from the difference in Eq. (9.25), The function R(G, p) is given by ≡ 4 ˜ ˜ R(H3,3,p) > R(H3,2,p) if 1 >p> 0.571 , R(G, p)= pe(G) (1 p)e(G)−e(G) , (9.17) 7 ≃ − ˜ (9.28) GX⊆G while where G˜ is a connected spanning subgraph of G. Clearly, 4 R(H ,p) < R(H ,p) if 0
D. Some Graphical Quantities and
1 (i): w = k k−2 . (9.43) We next discuss special valuations of T (Hk,r,x,y) that CSSG yield quantities of graph-theoretic interest. For x = 1, the following result is convenient: For the limit (ii), we calculate
r−2 (ii): w = w = 2 (9.44) T (H , 1,y) = (k 1) (2k + y 3) SSG SF k,r − − (k + y 2)r−2 (k 1)r−2 and + − − − (k + y 2)2 . y 1 − − (ii): wST = wCSSG =1 . (9.45) (9.35)
In addition to X. CONCLUSIONS N (H )=2(k−1)r , (9.36) SSG k,r In conclusion, in this paper have have derived exact we compute relations between the Potts model partition function, or equivalently, the Tutte polynomial, for a graph G and r−1 ˜ NST (Hk,r)= r(k 1) , (9.37) for a graph G obtained from G by edge or vertex infla- − tion. An analysis was given of some physical effects of uniform ℓ-fold edge and vertex inflations. We have pre- k−1 r−2 2(k−1) k−1 NSF (Hk,r)=(2 1) 2 1+(r 2)(2 1) , sented exact calculations of the Tutte polynomials for − − − − free, cyclic, and M¨obius ladder families of graphs S , (9.38) k,m h i L , and M of length m, with ℓ = k 2 vertex and k,m k,m inflation on all longitudinal edges. We have− given simi- N (H )= kr (k 1)r . (9.39) lar results for the Tutte polynomial of the family Hk,r of CSSG k,r − − hammock graphs. Our present results generalize our pre- Since these are two-parameter families of graphs, one vious calculations in Refs. [7, 9, 10]. As one application, can consider the limits (i) r with fixed finite k, we have calculated reliability polynomials for the ham- and (ii) k with fixed finite→ ∞r. We calculate the mock graphs and analyzed their properties as a function growth constants→ ∞ for each of these. The cases of interest of k and r. In addition, we have used our calculations to here are those with k 3, i.e., those with homeomorphic compute the number of spanning trees, spanning forests, expansion. For these we≥ find that for the limit (i) , and connected spanning subgraphs on these families and to determine their asymptotic behavior as the number of k−1 k−2 vertices goes to infinity. (i): wSSG =2 , (9.40)
1 Acknowledgments (i): w = (k 1) k−2 , (9.41) ST − This research was partially supported by the grant k−1 1 (i): w = (2 1) k−2 , (9.42) NSF-PHY-06-53342. SF −
[1] F. Y. Wu, Rev. Mod. Phys. 54, 235 (1982). [10] R. Shrock, Physica A 283, 388 (2000). [2] W. T. Tutte, Graph Theory, in G. C. Rota, ed., En- [11] D. J. A. Welsh, Complexity: Knots, Colourings and cyclopedia of Mathematics and its Applications, vol. 21 Counting, London Math. Soc. Lecture Notes vol. 186 (Addison-Wesley, New York, 1984). (Cambridge Univ. Press, Cambridge, 1993). [3] N. Biggs, Algebraic Graph Theory (Cambridge Univ. [12] L. Beaudin, J. Ellis-Monaghan, G. Pangborn, and R. Press, Cambridge, 1993). Shrock, Discrete Math. 310, 2037 (2010). [4] B. Bollob´as, Modern Graph Theory (Springer, New York, [13] C. M. Fortuin and P. W. Kasteleyn, Physica 57, 536 1998). (1972). [5] Y. Chow and F. Y. Wu, Phys. Rev. B 36, 285 (1987). [14] Note that with our labelling conventions, an ℓ-fold edge [6] R. Shrock and S.-H. Tsai, Phys. Rev. E 55, 5165 (1997). inflation with ℓ = 1 means to replace that edge with an [7] R. Shrock and S.-H. Tsai, J. Phys. A 31, 9641 (1998). equivalent single edge, which leaves the graph unchanged; [8] R. Shrock and S.-H. Tsai, Physica A265, 186 (1999). the choices ℓ ≥ 2 change the graph. In contrast, for an [9] R. Shrock and S.-H. Tsai, J. Phys. A Letts. 32, L195 ℓ-fold vertex inflations, the choice ℓ = 0 leaves the graph (1999). unchanged, while the choices ℓ ≥ 1 change the graph. 14
[15] E. G. Whitehead and L. C. Zhao, J. Graph Theory 8, [26] R. Shrock and S.-H. Tsai, Phys. Rev. E 55, 6791 (1997); 355 (1984). ibid. 56, 2733 (1997); ibid. 56, 4111 (1997). [16] R. Shrock and S.-H. Tsai, Physica A 259, 315 (1998). [27] J. Stephenson, J. Math. Phys. 5, 1009 (1964); ibid., 11, [17] R. C. Read and E. G. Whitehead, Discrete Math. 204, 413 (1970). 337 (1999); ibid. 243, 267 (2002); ibid. 308, 1826 (2008). [28] Early reviews include M. E. Fisher, Rev. Mod. Phys. 46, [18] L. Traldi, Discrete Math. 220, 291 (2000); ibid. 248, 279 597 (1974) and C. Domb and M. S. Green, eds., Phase (2002). Transitions and Critical Phenomena (Academic Press, [19] A. Sokal, Combin. Probab. Comput. 13, 221 (2004). New York, 1976), vol. 6. [20] Dong, F. M., Koh, K. M., Teo, K. L.: Chromatic Poly- [29] L. P. Kadanoff, Nuovo Cim. 44B, 276 (1966); T. T. Wu, nomials and Chromaticity of Graphs (World Scientific, Phys. Rev. 149, 380 (1966). Singapore, 2005). [30] R. Shrock, Phys. Lett. A 261, 57 (1999). [21] R. Shrock and Y. Xu, arXiv:1101.0852. [31] S.-C. Chang and R. Shrock, Physica A 296, 131 (2001). [22] An early review is I. Syozi, in C. Domb and M. S. Green, [32] S.-C. Chang and R. Shrock, J. Stat. Phys., 112, 815 eds., Phase Transitions and Critical Phenomena (Aca- (2003). demic Press, New York, 1972), p. 269. [33] Recent reviews include M. Newman, A.-L. Barab´asi, and [23] L. Onsager, Phys. Rev. 65, 117 (1944). D. J. Watts, The Structure and Dynamics of Networks [24] D. Sherrington and S. Kirkpatrick, Phys. Rev. Lett. 35, (Princeton Univ. Press, Princeton, 2006) and S. N. Doro- 1792 (1975); D. Elderfield and D. Sherrington, J. Phys. govstev, A. V. Goltsev, and J. F. F. Mendes, Rev. Mod. C 16, L497 (1983); H. Nishimori and M. J. Stephen, Phys. 80, 1275 (2008); Phys. Rev. B 27, 5644 (1983); M. M´ezard, G. Parisi, and [34] S.-C. Chang and R. Shrock, J. Stat. Physics 112, 1019 M. A. Virasoro, Spin Glass Theory and Beyond (World (2003). Scientific, Singapore, 1987). [25] N. L. Biggs, Bull. London Math. Soc. 9, 54 (1977).