Contents
Abstract

We define -polynomial of graph which is related to clique, (in)dependence and matching polynomials. The growth rate of partially commutative monoid is equal to the largest root of -polynomial of the corresponding graph.

The random algebra is defined in such way that its growth rate equals the largest root of -polynomial of random graph. We prove that for almost all graphs all sufficiently large real roots of -polynomial lie in neighbourhoods of roots of -polynomial of random graph. We show how to calculate the series expansions of the latter roots. The average value of over all graphs with the same number of vertices is computed.

We found the graphs on which the maximal value of with fixed numbers of vertices and edges is reached. From this, we derive the upper bound of . Modulo one assumption, we do the same for minimal value of .

We study the Nordhaus—Gaddum bounds of and .

Keywords: clique polynomial, dependence polynomial, independence polynomial, matching polynomial, random graph, planar graph, partially commutative monoid, partially commutative Lie algebra, Lovász local lemma.

-polynomial of graph

Vsevolod Gubarev

Introduction

In 1969 [46], P. Cartier and D. Foata defined a partially commutative monoid by some combinatorial reasons (reproving MacMahon Master theorem). Given a simple finite graph , a partially commutative monoid is in algebraic language , i.e., a quotient of the free monoid by the congruence relation generated by . Informally, is a set of words in alphabet with the operation of concatenation. Moreover, given a word, we can interchange neighbor letters if they are connected in . Two words are equal if could be obtained from by a finite number of such interchanges of neighbor connected letters.

Further, partially commutative groups, algebras and Lie algebras appeared. Partially commutative structures are very natural object for study. On the one hand, they are defined very simply, their properties are close to the properties of free ones. On the other hand, p.c. structures are enough reach to formulate a lot of problems with nontrivial solutions. Such objects have been investigated in combinatorics, formal languages, automata, computer science [46, 69, 70, 71, 152, 155]; in algebra [15, 73, 74, 76, 103, 122, 167]; in topology [1, 48, 124, 151]; in logics [47, 105, 168]; in robotics [48].

We should clarify that elements of partially commutative monoids sometimes are called traces, a partially commutative group is called right-angled Artin group (RAAG). There is a big area of mathematics devoted to more general groups: Artin and Coxeter groups. In the first ones, we have the relations for all pairs of connected generators and some . A Coxeter group is an Artin group with additional relations , so, it is a generalization of Weil groups which play the significant role in study of simple Lie groups and simple Lie algebras.

The natural question about the numbers of different words of given length and, more specifically, about the growth rate of partially commutative monoid leads to graph polynomials111About growth rates of partially commutative groups see [6, 8, 12].. For associative/Lie algebras we should ask about the dimensions respectively.

Dependence polynomial actually appeared in the work of P. Cartier and D. Foata [46]. This polynomial was wrtitten down by D.C. Fisher and A.E. Solow in 1990 [86]. Dependence polynomial is defined as , where denotes a number of distinct cliques in of the size and is the clique number of graph, the size of a maximal clique in . The reciprocal of dependence polynomial is the generating function of partially commutative monoid .

By analogy, in 1994 C. Hoede and X. Li defined [116] the clique polynomial as .

Let denote the number of anticliques of size  in a graph , i.e., , where is a complement graph. The polynomial is called the independence polynomial of a graph . Here is the size of the maximal independent set in . Independence polynomial firstly appeared in the work of A. Motoyama and H. Hosoya [157] in 1977 for lattice graphs. A systematic study of the subject began in 1983, with the work of I. Gutman and F. Harary [109]. In 2005, V. Levit and E. Mandrescu wrote the interesting survey about independence polynomial [134].

In 1971, H. Hosoya defined [117] matching polynomial, actually it is independence polynomial of the line graph of (see also the work of H. Kunz [126]). Let us refer to the important works [82, 101, 114] and also monographs [100, 144] devoted to matching polynomial. Note that independence and matching polynomials have connections with chemistry and physics [13, 114, 157, 177].

In the work, we define -polynomial (short for partially commutative polynomial) of a graph  as , where . Actually -polynomial have already appeared in the works [20, 39, 86] without any special name. The growth rate of partially commutative monoid/associative algebra/Lie algebra equals the largest real root of  [59, 73, 86, 102]. That is why in [86] was called as the growth factor.

Let us call (in)dependence polynomials, clique polynomial and -polynomial as clique-type polynomials. They are connected in the following way:

Clique-type polynomials have different applications in counting of (anti)cliques, matchings, perfect matchings, homomorphisms and colorings in graphs with different constraints [85, 93, 96, 174, 198, 199], in the Ramsey theory and sphere packings [67, 68, 164]. See also [14, 24, 52, 60].

In 2005, A. Scott and A. Sokal showed [177] the deep connection between and Lovász local lemma.

The main goals of the current paper are the following:

0) give a survey on clique-type polynomials,

1) present new results about -polynomials of random and planar graphs,

2) state asymptotically tight lower and upper bounds on in terms of and ,

3) find graphs on which reaches the minimum or maximum among all graphs with fixed values of and ,

4) study Nordhaus—Gadddum [7, 160] inequalities for the expressions and ,

5) find the average value of among graphs of the same size.

Let us give an exposition of the work.

In , required preliminaries on polynomials and sequences are written down.

The main goal of is to prove

Theorem 1.1 [46, 84]. The numbers , , satisfy the reccurence relation

with initial data , .

We state this result in different from the proofs of P. Cartier with D. Foata and D.C. Fisher way. As a corollary, we get

Lemma 1.4 [46, 86]. The function is a generating function for the sequence , i.e.,

At the end of the section, we state the very useful equalities of clique-type polynomials related to deletion of a vertex or an edge, to graphs join and union and to derivative.

Lemma 1.5 [108, 116]. Given a graph , we have

a) , ;

b) , ;

c) ;

d) ;

e) .

Here by for we denote the subgraph of induced by the set of vertices . By we mean an open neighbourhood of .

Exposition of  in general follows the work of P. Csikvári of 2013 [59]. Let denote a root of with the largest modulus and . D.C. Fisher and A.E. Solow in 1990 stated

Lemma 2.1 [86]. The number is a root of .

D.C. Fisher and A.E. Solow also pretended that they had proved the following theorem

Theorem 2.1 [59, 102]. The number is the only complex root of with modulus greater or equal to .

In 2000, M. Goldwurm and M. Santini finally proved Theorem 2.1 [102] via the Perron—Frobenius theory. In , we consider the excellent proof of P. Csikvári [59].

Now we are ready to connect the growth rate of p.c. structures with -polynomial:

Corollary 2.1 [102]. a) The growth rate of partially commutative monoid equals .

b) The growth rate of partially commutative associative algebra equals .

In Lemmas 2.3 and 2.4 we prove the inequalities  [59, 110] for any induced subgraph of and any spanning subgraph  of .

In , some applications of clique-type polynomials in graph theory are considered.

In 2009, D. Galvin [93] proved the result based on the work of V. Alekseev [3].

Theorem 3.1 [3, 68, 93]. Given a graph with , , we have for all with equality if and only if is a complete multipartite graph with equal parts.

Theorem 3.1 allows to prove the following corollaries:

Corollary 3.1 [3, 80]. Let be a graph with , and denotes the number of all cliques in . Then . We have equality if and only if is a complete multipartite graph with equal parts.

Corollary 3.3 (Turán’s Theorem). Given a graph with , , we have .

We write down the proof of Y. Zhao of the next result (modulo the case of bipartite graphs stated by J. Kahn in 2001 [119]).

Theorem 3.2 [198]. Given a -regular graph with , for all we have .

It implies the solution of the conjecture of N. Alon [4]:

Corollary 3.4 [198]. For any -vertex -regular graph , .

Let us introduce clique-type polynomials from the point of view of statistical physics, see, e.g., [164]. By the occupancy fraction we mean the expected fraction of vertices that appear in the random independent set

Here is so called hard-core distribution which is simply the uniform distribution over all independent sets of at fugacity . The independence polynomial is interpreted as the partition function of the hard-core model on at fugacity .

Statement 3.7 [68]. a) For any graph , is monotone increasing in .

b) Let be a triangle-free graph on vertices with maximum degree , we have for any .

E. Davies et al [68] applied Statement 3.7 to reprove the best known upper bound on the Ramsey numbers .

Corollary 3.8 [68, 180]. For the Ramsey numbers , we have the upper bound .

In 2002, V. Nikiforov proved [158] for the spectral radius the inequality

where . We show that this inequality is equivalent to

Statement 3.8. Let be not empty graph with vertices and the spectral radius . Then for any .

The section  is devoted to partially commutative Lie algebras. By the definition, a partially commutative Lie algebra equals . Denote the dimension of the homogeneous space of all products of length  in the alphabet  in as .

Given roots of , define the numbers .

In 1992, G. Duchamp and D. Krob actually proved [73] the following result (but not in the most comfortable form).

Theorem 4.1 [73]. We have

where is the Möbius function.

Corollary 4.1. If , then the growth rate of partially commutative Lie algebra equals 0. Otherwise, it equals .

Also, we show that partially commutative Lie algebras are connected with Yang—Mills algebras [115] (Corollary 4.3) and the Lie algebras of primitive elements in the connected cocommutative Hopf algebra  [162] (Remark 4.4).

In , we study clique and -polynomial of the random graph with vertices and edge probability ,

and . The main result of is

Theorem 5.1 [42]. Let .

a) All roots of are real and simple.

b) Write roots of in ascending order . Then for all .

Theorem 5.1 was stated by J. Brown and R. Nowakowski in 2005 [39] for and by J. Brown et al in 2012 [42] for any . We show that Theorem 5.1a immediately follows from the result of E. Laguerre [127] (Remark 5.1).

Let , . Define the polynomial

A polynomial of degree  is called symmetric if for all .

Statement 5.1. a) The polynomial is symmetric.

b) For odd , is a middle root of . All other roots of for odd and all roots for even could be gathered in pairs with the roots product equal .

Denote by the largest root of . Statement 5.1 allows us to find for all (Corollary 5.1).

In , we define the random algebra. Let be a finite set. Fix an order on such that if . Consider a word of length . Let a letter occurs in exactly times, . We suppose that . Consider a new alphabet

Define an order on the set : if or and .

Given a word , let us construct a word of the same length as follows. If is the -th occurrence (counting from the left) of a letter in , then the -th letter of equals . Denote the set of all multipartite graphs with parts , …, as . Define be equal to the product .

Let . Define a weight of a word as

where n.f. means ‘‘normal form’’ (the maximal word among all words in a partially commutative monoid equal to it), ,

Actually equals a probability of the event that is in the normal form in hypothetical partially commutative monoid with commutativity graph , the random multipartite graph with fixed parts with vertices and edge probability .

Define on the free associative algebra as on the vector space a new product . Let denote the set of all words of length in the alphabet . For , ,

where denotes the normal form of , denotes the probability of an event in the probability theory model constructed by the random multipartite graph  and denotes the conditional probability of  given .

Let us call the space under the product as random algebra, notation: .

Lemma 5.3. a) For , the algebra is isomorphic to the free associative algebra . For , is isomorphic to the polynomial algebra .

b) The map is a semigroup homomorphism.

Extend a weight on by linearity and put .

Theorem 5.2. The numbers , , satisfy the reccurence relation

with initial data , .

Corollary 5.3. The polynomial is a characteristic polynomial for the sequence and equals its growth rate.

Lemma 5.4. a) The following inequalities hold

b) The number for fixed  is strictly monotonic function on decreasing from to 1.

In , we solve the following problem. Denote the set of all planar graphs with vertices and edges as . We find minimal and maximal values of for and the graphs for these extremal values (Theorem 6.1). Let , then the graph with the minimal is a triangle-free graph for . For , we construct as a supergraph of in which edges in the big part form a tree (see Picture 1). For , put .

Let , , then the graph with the maximal is constructed as follows. We start with . On each step, we add one new vertex inside of some (triangle) face of the graph and connect it with each vertex of the face. We proceed on while we have edges (see Picture 3). Sometimes, such graphs are called Apollonian networks.

Let denote the set of all planar graphs with vertices. In 2007, O. Giménez and M. Noy stated [99] that for and a planar graph in average contains edges.

Theorem 6.2. a) of almost all planar graphs is a polynomial of 4-th degree, has two complex and real roots. Complex roots lie in the right half-plane, real roots are simple.

b) Let , then for almost all planar graphs we have for any root of -polynomial.

Statement 6.1. The average value of the growth rate of partially commutative monoid with planar commutativity graph equals

In , we formulate the main problems of the paper. By analogy with the Nordhaus—Gadddum inequalities for chromatic number of a graph  and its complement, we post

Problem 1. To find the tight bounds for the expressions and .

Denote by the set of all graphs with vertices and edges. Introduce

Problem 2. To find the values and the graphs on which they are reached.

The following lower bound for (part a)) was proved by D.C. Fisher in 1989.

Theorem 7.1. a) [85] Given a graph with vertices and edges, .

b) The bound from a) is reached if and only if is an empty graph or is a complete multipartite graph with equal parts.

For a graph , define the edge -density , .

Corollary 7.1. For any graph with vertices, .

Corollary 7.3. For any graph with vertices,

a) ,

b) .

Moreover, the bounds are reached only if and only if .

Corollary 7.5 [85]. For any graph , we have .

Statement 7.1. Let , then . This value is reached for a graph if and only if is a triangle-free graph.

In 1990, D.C. Fisher and J.M. Nonis proved the strong lower bound.

Theorem 7.2 [87]. Given a graph with vertices and edges, let be such a natural number that . Then

The section  could be considered as the central section of the work. In , we find the graph with .

Let us define a relation on simple graphs as follows. We write that if on the line segment . From , we have .

Lemma 8.1 [56]. a) Given an induced subgraph of , we have ;
b) Given a spanning subgraph of , we have .

Let be a graph, , . In 1981, A. Kelmans defined [120] so called Kelmans transformation which transfers a graph into a graph . To get , we erase all edges between and and add all edges between and (see Picture 4). Note that .

In 2011, P. Csikvári stated [56] some important properties of Kelmans transformations.

Lemma 8.2 [56, 57]. Let be a graph and be a graph obtained from  by a Kelmans transformation. Then

a) and so, ,

b) for all .

It is easy to show that (see [55]) any graph by a series of Kelmans transformations can be transformed to a threshold graph.

Let be a graph such that . Moreover, let any vertex of be either connected or disconnected with all vertices of and be such hanging vertex in that is not complete. Define the isolating transformation which transforms  to a graph as follows. We obtain a by arising the only edge in incident to  and adding an edge in .

Lemma 8.3. Let be a graph such that . Moreover, let any vertex of be either connected or disconnected with all vertices of and be such hanging vertex in that is not complete. There exists an isolating transformation such that .

In the next statement, we prove Conjecture 1 from [87].

Theorem 8.1. Let be natural numbers, for . Construct a graph with vertices and edges as follows. We add a vertex of degree to the complete graph and leave all other vertices to be isolated. Then .

Corollary 8.1 [66, 196]. The constructed graph from Theorem 8.1 maximizes all numbers among graphs from . In particular, for any graph and for all .

In Remark 8.1, we discuss how the strategy of the proof of Theorem 8.1 could be applied to reprove the analogous result for the spectral radius. This problem was initially posed by Brualdi and Hoffman in 1976 [44] and solved by P. Rowlinson in 1988 [173].

In , we want to derive the upper bound on applying Theorem 8.1. Before this, we easily prove the necessary condition for real-rootedness of .

Statement 8.1. Let be a graph with vertices such that all roots of are real. Then and .

Lemma 8.4. Let be a graph with vertices and edges, then and .

Theorem 8.2. Let . For any graph with vertices and edges, we have

a) ,

b) ,
where .

Another important fact which follows from the proof of Theorem 8.2 is the following: we have the asymptotically tight upper bound provided

where is the Lambert -function, the inverse function to .

Corollary 8.3. Let be a graph with vertices and , , edges. Then .

Corollary 8.4. Let be a graph with vertices and edges.

a) For , we have and .

b) For , we have and .

Corollary 8.5. For any graph with vertices the following inequalities hold

a) ,

b) .

We conjecture that the maximal values of the expressions and are reached on graphs from the class . By Example 8.2, the following values are maximal among graphs from

In , we study the minimum value . At first, we state some results devoted to transformations of graphs. Given a graph and two distinct vertices , let us call a Kelmans transformation as a nontrivial one, if , where denotes the number of all cliques in a graph .

Lemma 9.1. Let be a graph with connected complement. If a graph is a result of a nontrivial Kelmans transformation of , then .

Lemma 9.2. Let be a graph with connected complement. Let be an edge lying in a clique of size  and such vertices that and . Consider the graph obtained by removing an edge and adding an edge . Then and .

Corollary 9.1. The graph constructed in Theorem 8.1 is a unique graph with the maximal  among all graphs with vertices and edges with one exception: when for some . In this case, the set consists of all graphs obtained from by adding one edge.

Corollary 9.2. Let and be a graph such that . Then is connected graph having diameter 2.

We will find the exact value of modulo the following conjecture.

Conjecture 9.1. Let and be a graph such that . Then is disconnected.

Let and for a natural number . If is not enough large to construct (with parts of and ), then we construct the graph as a supergraph of with parts with edges and parts with edges, where , and . Introduce

Further, we draw a triangle-free graphs with vertices inside all parts with vertices and draw remaining edges anywhere. So,

Let be enough large to construct a supergraph of with prescribed conditions on parts. Find a natural number such that

Denote and . We construct the graph as a supergraph of in which the edges form a triangle-free graph in each part with vertices and we put remaining edges anywhere. Hence,

where .

Theorem 9.2. Let . If Conjecture 9.1 holds, then for the constructed graph .

Corollary 9.4. Let , , then

In , we are interested on the asymptotics of the average growth rate of partially commutative monoid with -vertex commutativity graph:

Lemma 10.1. For all , there exists the limit .

Theorem 10.1. The average value of on graphs with vertices asymptotically equals

The constant firstly appeared in the article of R. Stanley [183] in 1973: the number of all acyclic orientations of a digraph was counted as for .

Statement 10.1. For almost all graphs with vertices, lies in a neighbourhood of and is the unique root