One can define the independence polynomial of a graph G as follows. Let i(k)(G) denote the number of independent sets of size k of G, where i(0)(G) = 1. Then the independence polynomial of G is I(G,x) = Sigma(n)(k=0)(-1)(k)i(k)(G)x(k). In this paper we give a new proof of the fact that the root of I(G,x) having the smallest modulus is unique and is real

Hajiabolhassan

Levit

PÉTER CSIKVÁRI

English

One can define the independence polynomial of a graph G as follows. Let ik(G) denote the number of independent sets of size k of G, where i0(G)=1. Then the independence polynomial of G is I(G,x)=∑k=0n(−1)kik(G)xk. In this paper we give a new proof of the fact that the root of I(G,x) having the smallest modulus is unique and is real.</jats:p

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Note on the Smallest Root of the Independence Polynomial

Csíkvári, Péter

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NOTE ON THE SMALLEST ROOT OF THEINDEPENDENCE POLYNOMIALPÉTER CSIKVÁRIAbstract. One can define the independence polynomial of a graph Gas follows. Let ik(G) denote the number of independent sets of size k ofG, where i0(G) = 1. Then the independence polynomial of G isI(G, x) =n∑k=0(−1)kik(G)xk.In this paper we give a new proof for the fact that the root of I(G, x)having the smallest modulus is unique and it is real.1. IntroductionThroughout the paper G is a graph without multiple edges and loops.The independence polynomial of the graph G is defined as follows. Let ik(G)denote the number of independent sets of size k of G, where i0(G) = 1. Thenthe independence polynomial of G isI(G, x) =n∑k=0(−1)kik(G)xk.We note that in general, I(G,−x) is called the independence polynomial inthe literature. We will see that it is much more convenient to work with thisform. Since the connection between the two forms is very simple, it will notcause any confusion. We also mention that several authors consider the cliquepolynomial of the graph G, which is I(G, x) in our notation. Surprisingly,slightly more convenient to work with the independence polynomial thanwith the clique polynomial.Let β(G) be the smallest real root of the independence polynomial. Itis known that it exists and it is in the interval (0, 1] (see [5]). It is alsowell-known that if ρ is another root of the independence polynomial, then|ρ| ≥ β(G) [5]. It was only proved much later by Goldwurm and Santini[7] that in fact, the stronger statement |ρ| > β(G) holds too. On the otherhand, their proof heavily relies on the theory of formal languages. However itwas not the aim of Goldwurm and Santini, their theorem had an importantapplication, namely, it filled a gap in a proof of Fisher [3] on the minimal2000 Mathematics Subject Classification. Primary: 05C31.Key words and phrases. independence polynomial, root.The author is partially supported by the Hungarian National Foundation for Scien-tific Research (OTKA), grant no. K81310 and by grant no. CNK 77780 from the NationalDevelopment Agency of Hungary, based on a source from the Research and Technology In-novation Fund. The author is also partially supported by MTA Rényi "Lendület" Groupsand Graphs Research Group.12 PÉTER CSIKVÁRInumber of triangles of a graph with prescribed number of edges, see Remark3.3.In this short note we give a considerably simpler proof for this theorem.Theorem 1.1. Let ρ be a root of the independence polynomial I(G, z) dif-ferent from β(G) then |ρ| > β(G).The paper is organized as follows. We end this Introduction by present-ing our notations. In Section 2 we collect the required tools from complexfunction theory. In Section 3 we prove Theorem 1.1. We end the paper withsome concluding remarks on the weighted version of Theorem 1.1.Notation 1.2. Throughout the paper we will consider only simple graphs.As usual G = (V,E) will denote a graph with vertex set V (G) and edge setE(G). Let e(G) be the number of edges, i.e., |E(G)| = e(G). The set ofneighbors of the vertex v will be denoted by NG(v); if there is no confusionwe will simply write N(v) instead of NG(v). The closed neighborhood of thevertex v is NG[v] = NG(v)∪ {v}. The degree of the vertex v will be denotedby d(v) = |NG(v)|.For S ⊂ V (G) the graph G−S denotes the subgraph of G induced by thevertices V (G)\S. If e ∈ E(G) then G− e denotes the graph with vertex setV (G) and edge set E(G)\{e}.2. Lemmas from complex analysisIn this part we collect the required background from complex analysis. Weuse only two theorems from complex analysis, both theorems are well-knownand they can be found in [6].Lemma 2.1 (Pringsheim’s theorem). [6] If f(z) is representable at the ori-gin by a series expansion that has non-negative coefficients and radius ofconvergence R, then the point z = R is a singularity of f(z).Letf(z) =∞∑n=0fnznbe analytic in |z| < R, and assume that fn ≥ 0 for n ≥ 0. Then for any|s| < R we have|f(s)| =∣∣∣∣∣∞∑n=0fnsn∣∣∣∣∣ ≤∞∑n=0|fn||s|n =∞∑n=0fn|s|n = f(|s|).By investigating the case of equality we obtain Lemma 2.3. To prepare it,we need some terminology.Definition 2.2. Letf(z) =∞∑n=0fnzn.NOTE ON THE SMALLEST ROOT OF THE INDEPENDENCE POLYNOMIAL 3Let Supp(f) = {k| fk 6= 0}. The sequence (fn) as well as f(z) admits a spand if for some r, there holdsSupp(f) ⊆ r + dZ≥0 = {r, r + d, r + 2d, . . . }.The largest span, p, is the period, all other spans being divisors of p. If theperiod is equal to 1, then the sequence (fn) and f(z) are said to be aperiodic.Lemma 2.3. [6] Let f(z) be analytic in |z| < R and have non-negativecoefficients at 0. Assume that f does not reduce to a monomial and that forsome non-zero s with |s| < R, one has|f(s)| = f(|s|).Then the following holds, (i) the argument of s must be commensurate to 2pi,i.e., s = |s|eiθ with θ/2pi = rp∈ Q (an irreducible fraction) and 0 < r < p:(ii) f admits p as a span.In fact, we will only need that if the sequence (fn) is aperiodic, then|f(s)| 6= f(|s|) if s is non-zero, non-positive.3. Generator functions of the independence polynomialThroughout this section we will use the following simple recursive formulasfor the the independence polynomial.Lemma 3.1. [9] (a) If G is non-connected graph with connected componentsH1, . . . , Hk thenI(G, z) =k∏j=1I(Hj, z).(b) Let v be an arbitrary vertex of the graph G. ThenI(G, z) = I(G− v, z)− zI(G−N [v], z).(c) If e = (u, v) ∈ E(G), then we haveI(G, z) = I(G− e, z)− z2I(G−N [u]−N [v], z).The next lemma is the main tool to prove Theorem 1.1.Lemma 3.2. Let G and H be graphs and setI(H, z)I(G, z)=∞∑k=0rk(H,G)zk.Then(a) If H is a proper induced subgraph of G, then rk(H,G) > 0 for k ≥ 0.(b) If H is a proper subgraph of G, then rk(H,G) > 0 for k ≥ 2.Proof. (a) We prove the statement by induction on the number of verticesof G. If |V (G)| = 1 then |V (H)| = 0 andI(H, z)I(G, z)=11− z=∞∑j=0zj.4 PÉTER CSIKVÁRIIt is enough to prove the statement for H = G− u, where u is an arbitraryvertex of G. Indeed, if H = G− S for some S = {u1, . . . , uk} ⊆ V (G), thenI(H, z)I(G, z)=I(G− u1, z)I(G, z)I(G− {u1, u2}, z)I(G− u1, z). . .I(G− {u1, . . . uk}, z)I(G− {u1, . . . , uk−1}, z).By induction we already know that all terms except the first one have apower series expansion of positive coefficients. Hence it is enough to provethat rk(G− u,G) > 0 for all k ≥ 0.Now we use the recursive formulaI(G, z) = I(G− u, z)− zI(G−N [u], z).We haveI(G− u, z)I(G, z)=I(G− u, z)I(G− u, z)− zI(G−N [u], z)==11− z I(G−N [u],z)I(G−u,z)=∞∑k=0(zI(G−N [u], z)I(G− u, z))k.If G−N [u] is a proper subgraph of G−u then by induction all the coefficientsof the power series I(G−N [u],z)I(G−u,z)are positive and so all the coefficients of I(G−u,z)I(G,z)are positive. If G−N [u] = G−u then I(G−u,z)I(G,z)= 11−zand we are done again.(b) We prove the claim by induction on the number of edges of G. Ife(G) = 0 and |V (G)| = n, |V (H)| = k, where k < n thenI(H, z)I(G, z)=(1− z)k(1− z)n=∞∑k=0(n− k − 1 + jn− k − 1)zj.Thus the statement is true in this case. Just like in the case of the proof ofpart (a) it is enough to prove that rk(G− e,G) > 0 for all k ≥ 2. Indeed, ifE(H) = E(G)− {e1, . . . , ek} and |V (G)| − |V (H)| = s thenI(H, z)I(G, z)=I(G− e1, z)I(G, z)I(G− {e1, e2}, z)I(G− e1, z). . .I(G− {e1, . . . ek}, z)I(G− {e1, . . . , ek−1}, z)1(1− z)s.By induction we already know that all terms except the first one have apower series expansion of the required form. Hence it is enough to provethat rk(G − e,G) > 0 for all k ≥ 2. Now we use the second recursiveformula. If e = (u, v) ∈ E(G), then we haveI(G, z) = I(G− e, z)− z2I(G−N [u]−N [v], z).We haveI(G− e, z)I(G, z)=I(G− e, z)I(G− e, z)− z2I(G−N [u]−N [v], z)==11− z2 I(G−N [u]−N [v],z)I(G−e,z)=∞∑k=0(z2I(G−N [u]−N [v], z)I(G− e, z))k.Since G−N [u]−N [v] is a proper induced subgraph of G−e, we have by part(a) that all the coefficients of the power series I(G−N [u]−N [v],z)I(G−e,z)are positive andrk(G− e,G) ≥ 2 for k ≥ 2. NOTE ON THE SMALLEST ROOT OF THE INDEPENDENCE POLYNOMIAL 5Remark 3.3. From a theorem of Cartier and Foata [1] one can deduce thatrk(∅, G) counts the number of words of length k in a certain ’semi-Abelian’monoid. This deduction is explained in [2], where Fisher also gave a directproof of this combinatorial meaning. So it was known that rk(∅, G) ≥ 0.Probably, a similar interpretation of rk(H,G) can be given too.We mention that Fisher [3] used the connection between the independencepolynomial and this counting problem to give a sharp lower bound for thenumber of triangles in a graph G with n vertices and e edges if n2/4 ≤ e ≤n2/3. His proof contained a small gap: he used |ρ| > β(G) for ρ 6= β(G),but he only proved |ρ| ≥ β(G). We also note that the problem of giving(asymptoticaly) sharp lower bounds for the number of cliques of size k in agraph with fix number of vertices and edges was resolved by Razborov [11](k = 3) , V. Nikiforov [10] (k = 3, 4) and C. Reiher [12] (k ≥ 3). They usedcompletely different machinery than Fisher.Corollary 3.4. [4, 5] Let β(G) be the convergence radius of 1I(G,z). Thenβ(G) is a root of the independence polynomial I(G, z), and it has the smallestmodulus among the roots of I(G, z). Let H be a subgraph of G. Then β(G) ≤β(H).Proof. Let1I(G, z)=∞∑k=0rk(G)zk.Clearly, rk(G) = rk(∅, G) > 0 for all k ≥ 1. Hence Pringsheim’s theorem(Lemma 2.1) already gives the first part of the claim.Since1I(G, z)=I(H, z)I(G, z)1I(H, z)we obtain that rk(G) ≥ rk(H) if H is a subgraph of G. Henceβ(G) = lim infk→∞rk(G)−1/k ≤ lim infk→∞rk(H)−1/k = β(H).Remark 3.5. A very simple proof of Corollary 3.4 can also be found in [8].At this moment we do not exclude the possibility that some other root ofthe independence polynomial has modulus β(G). We will exclude it later.Our next aim is to prove that if H is a proper subgraph of the connectedgraph G then β(G) < β(H) holds. This is the last step to prove the unique-ness of the root β(G).Lemma 3.6. Let G be a connected graph and let H be a proper subgraph ofG. Then β(G) < β(H) and the multiplicity of the root β(G) is 1.Proof. We prove the claim by induction on the number of edges of G. Ife(G) = 0 or 1 then the claim is trivial. Clearly, it is enough to prove thatβ(G) < β(G − e) for any edge e = (u, v). Since β(G) ≤ β(G − e) weonly need to prove that β(G) 6= β(G − e). Assume for a contradiction thatβ(G) = β(G− e). SinceI(G, z) = I(G− e, z)− z2I(G−N [u]−N [v], z)6 PÉTER CSIKVÁRIwe obtain that β(G) = β(G − e) is also a root of I(G − N [u] − N [v], z).Hence β(G − N [u] − N [v]) = β(G − e). If G − e is connected, then it isalready a contradiction since G−N [u]−N [v] is a proper subgraph of G− ethus β(G − N [u] − N [v]) > β(G − e). If G − e is not connected, thenG− e = H1 ∪H2, where H1 and H2 are connected graphs. If u ∈ V (H1) andv ∈ V (H2) thenG−N [u]−N [v] = (H1 −N [u]) ∪ (H2 −N [v]).Henceβ(G−e) = min(β(H1), β(H2)) < min(β(H1−N [u]), β(H2−N [v])) = β(G−N [u]−N [v]).This is again contradiction. Hence we have proved that β(H) > β(G).The second claim follows from the simple identity [9]−I ′(G, z) =∑u∈V (G)I(G−N [u], z).Since all I(G−N [u], z) are positive in the interval [0, β(G)], we haveI ′(G, β(G)) < 0,thus β(G) is a simple root of I(G, z). Now we are ready to prove Theorem 1.1.Proof of Theorem 1.1. Clearly, it is enough to prove the claim for a con-nected graph G since the roots of I(G, z) are the union of the roots I(Hj, z)where Hj are the connected components of the graph G. We can also assumethat |V (G)| ≥ 2, since the claim is trivial for K1.Let u be an arbitrary vertex of the graph G. Note that G − N [u] is aproper induced subgraph of G− u since G is a connected graph on at leasttwo vertices.Once again we use the identityg(z) =I(G− u, z)I(G, z)=11− z I(G−N [u],z)I(G−u,z).Let us examinef(z) = zI(G−N [u], z)I(G− u, z).Since G−N [u] is a proper induced subgraph of G−u, the convergence radiusof f(z) is β(G − u), and all the coefficients of f(z) are positive except thecoefficient of z0. If ρ is a root of I(G, z) of modulus β(G), then it must bea singularity of g(z) since I(G − u, z) has no root of smaller modulus thanβ(G− u), and β(G− u) > β(G). Hence f(ρ) = 1. Hence |f(ρ)| = f(|ρ|) sowe can use the Lemma 2.3 to obtain that ρ = β(G)e2piir/p, where p is someperiod of f(z). But f(z) is aperiodic: all the coefficients are positive. Thusp = 1 and ρ = β(G). NOTE ON THE SMALLEST ROOT OF THE INDEPENDENCE POLYNOMIAL 74. Concluding remarksIn this last section we mention that we could have proved the same resultsfor the following weighted version of the independence polynomial.Let w : V (G) → R+ be a positive function and letI((G,w); t) =∑I∈I(∏u∈Iw(u))(−t)|I|the weighted independence polynomial of the graph G, where the summationgoes over all independent sets of the graph G including the empty set.The weighted independence polynomial satisfies the recursionsI((G,w); t) = I((G− v, w); t)− wvtI((G−N [v], w); t)andI((G,w); t) = I((G− e, w); t)− wuwvt2I((G−N [u]−N [v], w); t),where v is a vertex and e = (u, v) is an arbitrary edge. From these formulasone can see that the proof of Lemma 3.2 easily generalizes and one can provethat the weighted independence polynomial has a unique root of smallestmodulus, which is real.The importance of this weighted independence polynomial (or independence-set polynomial) is that it is related to the Lovász local lemma through thefollowing theorem of Scott and Sokal [13, 14].We say that a graph G with vertex set V is a dependency graph for theevents (Av)v∈V if, for each v ∈ V , the event Av is independent from thecollection {Au | u ∈ N [v]}. Let us say that a sequence p = (pv)v∈V ∈ [0, 1]Vis good for the dependency graph G if for every collection (Av)v∈V of eventswith dependency graph G such that P(Av) ≤ pv for all v ∈ V , we haveP(∩v∈VAv) > 0.Now combining Theorem 1.3 (The equivalence theorem) with Theorem 2.2(The fundamental theorem) of [14] we get the following result.Theorem 4.1. Let G be a finite graph with vertex set V , and let p =(pv)v∈V ∈ [0, 1]V . Then the following two statements are equivalent:(a) p is good for the dependency graph G,(b) I((G,p), t) > 0 for t ∈ [0, 1].It is not clear that the weighted version of Theorem 1.1 has any relevancein the context of Scott-Sokal theorem.Acknowledgment. I am very grateful to Vladimir Nikiforov for proposingme this problem and for lots of remarks and suggestions.References[1] P. Cartier and D. Foata: Problèmes combinatoires de commutation et rèarrangements,Lecture notes in Mathematics 85 (1969), Springer-Verlag[2] D. C. Fisher: The number of words of length n in a free ’semi-Abelian’ monoid, Amer.Math. Monthly 96 (1989), pp. 610-6148 PÉTER CSIKVÁRI[3] D. C. Fisher: Lower bounds on the number of triangles in a graph, Journal of GraphTheory, 13 (4) (1989), pp. 505-512[4] D. C. Fisher and J. Ryan: Bounds on the largest root of the matching polynomial,Discrete Mathematics 110 (1992) No. 1-3, pp. 275-278[5] D. C. Fisher and A. E. Solow: Dependence polynomials, Discrete Mathematics 82(1990), No. 3 , pp. 251-258[6] P. Flajolet and R. Sedgewick: Analytic Combinatorics, Cambridge University Press,2009[7] M. Goldwurm and M. Santini: Clique polynomials have a unique root of smallestmodulus, Information Processing Letters, 75 (3) (2000), pp. 127-132[8] H. Hajiabolhassan and M. L. Mehrabadi: On clique polynomials, Australasian J.Combin. 18 (1998), pp. 313-316[9] V. E. Levit and E. Mandrescu: The independence polynomial of a graph- a surveyIn: Proceedings of the 1st International Conference on Algebraic Informatics. Heldin Thessaloniki, October 20-23, 2005 (ED. S. Bozapalidis, A. Kalampakas and G.Rahonis), Thessaloniki, Greece: Aristotle Univ. pp. 233-254, 2005[10] V. Nikiforov: The number of cliques in graphs of given order and size, Trans. Amer.Math. Soc. 363 (3) (2011), pp. 1599-1618[11] A. Razborov: On the minimal density of triangles in graphs, Combin. Probab. Com-put. 17 (4) (2008), pp. 603-618[12] C. Reiher: The clique density theorem, preprint[13] A. D. Scott and A. D. Sokal: The repulsive lattice gas, the independent-set polynomial,and the Lovász local lemma. J. Stat. Phys. 118 (2005), No. 5-6, pp. 1151-1261[14] A. D. Scott and A. D. Sokal: On dependency graphs and the lattice gas, Combin.Probab. Comput. 15 (2006), pp. 253-279Eötvös Loránd University, Department of Computer Science, H-1117 Bu-dapest, Pázmány Péter sétány 1/C, Hungary & Alfréd Rényi Institute ofMathematics, H-1053 Budapest, Reáltanoda u. 13-15, HungaryE-mail address: csiki@cs.elte.hu

Analytic Combinatorics,

Bounds on the largest root of the matching polynomial,

Clique polynomials have a unique root of smallest modulus,

Lower bounds on the number of triangles in a graph,

On clique polynomials,

On dependency graphs and the lattice gas,

On the minimal density of triangles in graphs,

Problèmes combinatoires de commutation et rèarrangements,

Solow: Dependence polynomials,

The clique density theorem,

The independence polynomial of a graph- a survey In:

The number of cliques in graphs of given order and size,

The number of words of length n in a free ’semi-Abelian’ monoid,

The repulsive lattice gas, the independent-set polynomial, and the Lovász local lemma.

Note on the smallest root of the independence polynomial

Note on the smallest root of the independence polynomial

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