Number Of Independent Sets Research Articles

Counting the number of independent sets in chordal graphs

We study some counting and enumeration problems for chordal graphs, especially concerning independent sets. We first provide the following efficient algorithms for a chordal graph: (1) a linear-time algorithm for counting the number of independent sets; (2) a linear-time algorithm for counting the number of maximum independent sets; (3) a polynomial-time algorithm for counting the number of independent sets of a fixed size. With similar ideas, we show that enumeration (namely, listing) of the independent sets, the maximum independent sets, and the independent sets of a fixed size in a chordal graph can be done in constant time per output. On the other hand, we prove that the following problems for a chordal graph are # P-complete: (1) counting the number of maximal independent sets; (2) counting the number of minimum maximal independent sets. With similar ideas, we also show that finding a minimum weighted maximal independent set in a chordal graph is NP-hard, and even hard to approximate.

On the number of independent sets in graphs with fixed independence number

We construct a sequence of graphs of large degree with growing number of vertices for which the number of independent sets is substantially greater than the number of all subsets of the independent set of maximal cardinality.

Independent sets in quasi-regular graphs

New upper bounds for the number of independent sets in graphs are obtained.

Independent sets in quasi-regular graphs

New upper bounds for the number of independent sets in graphs are obtained. c © 2006 Published by Elsevier Ltd 1. Definitions and the statement of results All graphs under consideration are finite, undirected, and simple. The vertices are considered as numbered. Denote the degree of a vertex v by σ(v). G is called an (n, k, θ)-graph if it has n vertices and k ≤ σ(v) ≤ k + θ for any vertex v. Everywhere we assume that n and k are sufficiently large and θ = o(k) as k → ∞. A subset of vertices of a graph G is called independent if the subgraph of G induced by A does not contain an edge. The family of all independent sets of G will be denoted by I(G). We put I (G) = |I(G)|. Let G = (V ; E) be a graph with the vertex set V and the edge set E , and v ∈ V . We call the set ∂v = {u : (u, v) ∈ E} the boundary of v. It is clear that σ(v) = |∂v|. The boundary of A ⊆ V in a graph G = (V ; E) is the set ∂A = (⋃v∈A ∂v) A. Suppose 0 ≤ < 1. Suppose l ≤ k − θ ≤ k + θ ≤ m. A graph on n vertices with minimal degree l and maximal degree m, with ∆ as the fraction of vertices with degree more than k + θ and with δ as the fraction of vertices with degree less than k − θ , will be called a (n, l, k,m, δ,∆, θ)-graph. Such a graph is called quasi-regular if θ k and k → ∞ when k → ∞. Here, we give a brief survey on the topic and prove the following theorems. Theorem 1. Let G = (V , E) be a (n, l, k,m, δ,∆, θ)-graph. Then I (G) ≤ 2 2 ( 1+δ(1−l/k)+∆(m/k−1)+O((θ+√k log k)/k)) . (1) Denote by Iβ(G) the family of sets A ∈ I(G) with ||A| − n/4| ≥ βn/4 and Iβ(G) = |Iβ(G)|. 0195-6698/$ see front matter c © 2006 Published by Elsevier Ltd doi:10.1016/j.ejc.2006.06.017 A.A. Sapozhenko / European Journal of Combinatorics 27 (2006) 1206–121

The number of independent sets in unicyclic graphs

In this paper, we determine upper and lower bounds for the number of independent sets in a unicyclic graph in terms of its order. This gives an upper bound for the number of independent sets in a connected graph which contains at least one cycle. We also determine the upper bound for the number of independent sets in a unicyclic graph in terms of order and girth. In each case, we characterize the extremal graphs.

On the number of independent sets in damaged Cayley graphs

On the number of independent sets in damaged Cayley graphs

On the number of independent sets in damaged Cayley graphs

The Cayley graph generated by a set A is the graph ΓA(V) on a set of positive integers V such that a pair (u, v) ∈ V×V is an edge of the graph if and only if |u−v| ∈ A or u+v ∈ A. We denote by (n,m) the class of graphs G = (V, E) such that G is a union of chains and cycles and |V| = n, |E| = m. In this paper, we present an upper bound for the number of independent sets in Cayley graphs ΓA(V) such that A = {⌈n/2⌉ − t, ⌈n/2⌉ − ƒ}, V ⊆ [⌊n/2⌋ + 1, ⌊n/2⌋+t] ∪ [n − t + 1, n], where n, t, ƒ ∈ N and ƒ < t < n/4. We also describe the graph with the maximum number of independent sets in the family (n, m).

Average independence polynomials

The independence polynomial of a graph G is the function i ( G , x ) = ∑ k ⩾ 0 i k x k , where i k is the number of independent sets of vertices in G of cardinality k. We investigate here the average independence polynomial, where the average is taken over all independence polynomials of (labeled) graphs of order n. We prove that while almost every independence polynomial has a nonreal root, the average independence polynomials always have all real, simple roots.

A note on the number of matchings and independent sets in trees

We prove best-possible upper and lower bounds on the number of matchings in a tree in terms of the number of independent sets and the number of 2-independent sets.

Operations on Well-Covered Graphs and the Roller-Coaster Conjecture

A graph $G$ is well-covered if every maximal independent set has the same cardinality. Let $s_k$ denote the number of independent sets of cardinality $k$, and define the independence polynomial of $G$ to be $S(G,z) = \sum s_kz^k$. This paper develops a new graph theoretic operation called power magnification that preserves well-coveredness and has the effect of multiplying an independence polynomial by $z^c$ where $c$ is a positive integer. We will apply power magnification to the recent Roller-Coaster Conjecture of Michael and Traves, proving in our main theorem that for sufficiently large independence number $\alpha$, it is possible to find well-covered graphs with the last $(.17)\alpha$ terms of the independence sequence in any given linear order. Also, we will give a simple proof of a result due to Alavi, Malde, Schwenk, and Erdős on possible linear orderings of the independence sequence of not-necessarily well-covered graphs, and we will prove the Roller-Coaster Conjecture in full for independence number $\alpha \leq 11$. Finally, we will develop two new graph operations that preserve well-coveredness and have interesting effects on the independence polynomial.

On the Location of Roots of Independence Polynomials

The independence polynomial of a graph G is the function i(G, x) = ? k?0 i k x k, where i k is the number of independent sets of vertices in G of cardinality k. We prove that real roots of independence polynomials are dense in (??, 0], while complex roots are dense in ?, even when restricting to well covered or comparability graphs. Throughout, we exploit the fact that independence polynomials are essentially closed under graph composition.

Independent Sets in Regular Hypergraphs and Multidimensional Runlength-Limited Constraints

Let G be a t-uniform s-regular linear hypergraph with r vertices. It is shown that the number of independent sets $\IS(\hgraph)$ in $\hgraph$ satisfies \[ \log_2 \IS(\hgraph) \le \frac{r}{t} \left( 1 + O \biggl( \frac{\log^2(ts)}{s} \biggr) \right) . \] This leads to an improvement of a previous bound by Alon obtained for t = 2 (i.e., for regular ordinary graphs). It is also shown that for the Hamming graph $\Hamming(n,q)$ (with vertices consisting of all n-tuples over an alphabet of size q and edges connecting pairs of vertices with Hamming distance $1$), \[ \frac{\log_2 \IS(\Hamming(n,q))}{q^n} = \frac{1}{q} + O \biggl(\frac{\log^2 (q n)}{q n} \biggr). \] The latter result is then applied to show that the Shannon capacity of the n-dimensional $(d,\infty)$-runlength-limited (RLL) constraint converges to 1/(d+1) as n goes to infinity.

Closed formulas for the numbers of small independent sets and matchings and an extremal problem for trees

We derive closed formulas for the numbers of independent sets of size at most 4 and matchings of size at most 3 in graphs without small cycles that depend only on the degree sequence and the products of the degrees of adjacent vertices. As a related problem we describe an algorithm that determines a tree of given degree sequence that maximizes the sum of the products of the degrees of adjacent vertices.

Extremal k*-cycle resonant hexagonal chains

Denote by ℬ*n the set of all k*-cycle resonant hexagonal chains with n hexagons. For any Bn∈ℬ*n, let m(Bn) and i(Bn) be the numbers of matchings (= the Hosoya index) and the number of independent sets (= the Merrifield–Simmons index) of Bn, respectively. In this paper, we give a characterization of the k*-cycle resonant hexagonal chains, and show that for any Bn∈ℬ*n, m(Hn)≤m(Bn) and i(Hn)≥i(Bn), where Hn is the helicene chain. Moreover, equalities hold only if Bn=Hn.

An Entropy Approach to the Hard-Core Model on Bipartite Graphs

We use entropy ideas to study hard-core distributions on the independent sets of a finite, regular bipartite graph, specifically distributions according to which each independent set I is chosen with probability proportional to λ[mid ]I[mid ] for some fixed λ &gt; 0. Among the results obtained are rather precise bounds on occupation probabilities; a ‘phase transition’ statement for Hamming cubes; and an exact upper bound on the number of independent sets in an n-regular bipartite graph on a given number of vertices.

On the number of independent sets in expanders

On the number of independent sets in expanders

On the Achromatic Number of Hypercubes

The achromatic number of a finite graph G, ψ(G), is the maximum number of independent sets into which the vertex set may be partitioned, so that between any two parts there is at least one edge. For an m-dimensional hypercube Pm2 we prove that there exist constants 0<c1<c2, independent of m, such that c1(m2m−1)1/2⩽ψ(Pm2)⩽c2(m2m−1)1/2.

On Markov Chains for Independent Sets

Random independent sets in graphs arise, for example, in statistical physics, in the hardcore model of a gas. In 1997, Luby and Vigoda described a rapidly mixing Markov chain for independent sets, which we refer to as the Luby–Vigoda chain. A new rapidly mixing Markov chain for independent sets is defined in this paper. Using path coupling, we obtain a polynomial upper bound for the mixing time of the new chain for a certain range of values of the parameter λ. This range is wider than the range for which the mixing time of the Luby–Vigoda chain is known to be polynomially bounded. Moreover, the upper bound on the mixing time of the new chain is always smaller than the best known upper bound on the mixing time of the Luby–Vigoda chain for larger values of λ (unless the maximum degree of the graph is 4). An extension of the chain to independent sets in hypergraphs is described. This chain gives an efficient method for approximately counting the number of independent sets of hypergraphs with maximum degree two, or with maximum degree three and maximum edge size three. Finally, we describe a method which allows one, under certain circumstances, to deduce the rapid mixing of one Markov chain from the rapid mixing of another, with the same state space and stationary distribution. This method is applied to two Markov chains for independent sets, a simple insert/delete chain, and the new chain, to show that the insert/delete chain is rapidly mixing for a wider range of values of λ than was previously known.

On computing boundary functional sums

A method for solving enumeration problems is suggested. We consider the enumeration problems which are reducible to estimation of the sums of type T( X, f)=Σ A f( A) where f is so called boundary functional (BF) on X, and the summation is over all subsets of X (or over some special subfamily of 2 X ). An evolution of the n-cube, the percolation problem, the problem of computation of the matchings number and the independent sets number, the monotone Boolean functions number, the binary codes number and so on are among such problems. We show how to obtain asymptotics for T( X, f). In conclusion we give an example of application of the BF method to finding the number of independent sets in the bipartite graphs, induced by neighbouring levels of the n-cube.

Roots of Independence Polynomials of Well Covered Graphs

Let G be a well covered graph, that is, all maximal independent sets of G have the same cardinality, and let ik denote the number of independent sets of cardinality k in G. We investigate the roots of the independence polynomial i(G, x) = ∑ ikxk. In particular, we show that if G is a well covered graph with independence number β, then all the roots of i(G, x) lie in in the disk |z| ≤ β (this is far from true if the condition of being well covered is omitted). Moreover, there is a family of well covered graphs (for each β) for which the independence polynomials have a root arbitrarily close to −β.