Number Of Independent Sets Research Articles

On the unimodality of independence polynomials of very well-covered graphs

The independence polynomial i(G,x) of a graph G is the generating function of the numbers of independent sets of each size. A graph of order n is very well-covered if every maximal independent set has size n∕2. Levit and Mandrescu conjectured that the independence polynomial of every very well-covered graph is unimodal (that is, the sequence of coefficients is nondecreasing, then nonincreasing). In this article we show that every graph is embeddable as an induced subgraph of a very well-covered graph whose independence polynomial is unimodal, by considering the location of the roots of such polynomials.

Upper tails and independence polynomials in random graphs

The upper tail problem in the Erdős–Rényi random graph G∼Gn,p asks to estimate the probability that the number of copies of a graph H in G exceeds its expectation by a factor 1+δ. Chatterjee and Dembo showed that in the sparse regime of p→0 as n→∞ with p≥n−α for an explicit α=αH>0, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where H is a clique.Here we extend the latter work to any fixed graph H and determine a function cH(δ) such that, for p as above and any fixed δ>0, the upper tail probability is exp⁡[−(cH(δ)+o(1))n2pΔlog⁡(1/p)], where Δ is the maximum degree of H. As it turns out, the leading order constant in the large deviation rate function, cH(δ), is governed by the independence polynomial of H, defined as PH(x)=∑iH(k)xk where iH(k) is the number of independent sets of size k in H. For instance, if H is a regular graph on m vertices, then cH(δ) is the minimum between 12δ2/m and the unique positive solution of PH(x)=1+δ.

Tight bounds on the coefficients of partition functions via stability

We show how to use the recently-developed occupancy method and stability results that follow easily from the method to obtain extremal bounds on the individual coefficients of the partition functions over various classes of bounded degree graphs.As applications, we prove new bounds on the number of independent sets and matchings of a given size in regular graphs. For large enough graphs and almost all sizes, the bounds are tight and confirm the Upper Matching Conjecture of Friedland, Krop, and Markström, and a conjecture of Kahn on independent sets for a wide range of parameters. Additionally we prove tight bounds on the number of q-colorings of cubic graphs with a given number of monochromatic edges, and tight bounds on the number of independent sets of a given size in cubic graphs of girth at least 5.

On the average size of independent sets in triangle-free graphs

We prove an asymptotically tight lower bound on the average size of independent sets in a triangle-free graph on $n$ vertices with maximum degree $d$, showing that an independent set drawn uniformly at random from such a graph has expected size at least $(1+o_d(1)) \frac {\log d}{d}n$. This gives an alternative proof of Shearer’s upper bound on the Ramsey number $R(3,k)$. We then prove that the total number of independent sets in a triangle-free graph with maximum degree $d$ is at least $\exp \left [\left (\frac {1}{2}+o_d(1) \right ) \frac {\log ^2 d}{d}n \right ]$. The constant $1/2$ in the exponent is best possible. In both cases, tightness is exhibited by a random $d$-regular graph. Both results come from considering the hard-core model from statistical physics: a random independent set $I$ drawn from a graph with probability proportional to $\lambda ^{|I|}$, for a fugacity parameter $\lambda >0$. We prove a general lower bound on the occupancy fraction (normalized expected size of the random independent set) of the hard-core model on triangle-free graphs of maximum degree $d$. The bound is asymptotically tight in $d$ for all $\lambda =O_d(1)$. We conclude by stating several conjectures on the relationship between the average and maximum size of an independent set in a triangle-free graph and give some consequences of these conjectures in Ramsey theory.

Maximizing the Number of Independent Sets of Fixed Size in Connected Graphs with Given Independence Number

The Turan connected graph $$\mathrm {TC}_{n,\alpha }$$ is obtained from $$\alpha $$ cliques of size $$\lfloor \frac{n}{\alpha } \rfloor $$ or $$\lceil \frac{n}{\alpha } \rceil $$ by joining all cliques by an edge to one central vertex in one of the larger cliques. The graph $$\mathrm {TC}_{n,\alpha }$$ was shown recently by Bruyere and Melot to maximise the number of independent sets among connected graphs of order n and independence number $$\alpha $$ . We prove a generalisation of this result by showing that $$\mathrm {TC}_{n,\alpha }$$ in fact maximises the number of independent sets of any fixed cardinality $$\beta \le \alpha $$ . Several results (both old and new) on the number of independent sets or maximum independent sets follow as corollaries.

Independent sets, matchings, and occupancy fractions

AbstractWe prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of ‐regular graphs. For independent sets, this theorem is a strengthening of the results of Kahn, Galvin and Tetali, and Zhao showing that a union of copies of maximizes the number of independent sets and the independence polynomial of a ‐regular graph.For matchings, this shows that the matching polynomial and the total number of matchings of a ‐regular graph are maximized by a union of copies of . Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markström.In probabilistic language, our main theorems state that for all ‐regular graphs and all , the occupancy fraction of the hard‐core model and the edge occupancy fraction of the monomer‐dimer model with fugacity are maximized by . Our method involves constrained optimization problems over distributions of random variables and applies to all ‐regular graphs directly, without a reduction to the bipartite case.

Comparing Graphs of Different Sizes

We consider two notions describing how one finite graph may be larger than another. Using them, we prove several theorems for such pairs that compare the number of spanning trees, the return probabilities of random walks, and the number of independent sets, among other combinatorial quantities. Our methods involve inequalities for determinants, for traces of functions of operators, and for entropy.

Independent sets on the Towers of Hanoi graphs

The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. In this article, we mainly study the number of independent sets i(Hn) on the Tower of Hanoi graph Hn at stage n, and derive the recursion relations for the numbers of independent sets. Upper and lower bounds for the asymptotic growth constant μ on the Towers of Hanoi graphs are derived in terms of the numbers at a certain stage, where μ = limv → ∞(lni(G)) / v(G) and v(G) is the number of vertices in a graph G. Furthermore, we also consider the number of independent sets on the Sierpiński graphs which contain the Towers of Hanoi graphs as a special case.

Maximal-clique partitions and the Roller Coaster Conjecture

A graph G is well-covered if every maximal independent set has the same cardinality q. Let ik(G) denote the number of independent sets of cardinality k in G. Brown, Dilcher, and Nowakowski conjectured that the independence sequence (i0(G),i1(G),…,iq(G)) was unimodal for any well-covered graph G with independence number q. Michael and Traves disproved this conjecture. Instead they posited the so-called “Roller Coaster” Conjecture: that the termsi⌈q2⌉(G),i⌈q2⌉+1(G),…,iq(G) could be in any specified order for some well-covered graph G with independence number q. Michael and Traves proved the conjecture for q<8 and Matchett extended this to q<12.In this paper, we prove the Roller Coaster Conjecture using a construction of graphs with a property related to that of having a maximal-clique partition. In particular, we show, for all pairs of integers 0≤k<q and positive integers m, that there is a well-covered graph G with independence number q for which every independent set of size k+1 is contained in a unique maximal independent set, but each independent set of size k is contained in at least m distinct maximal independent sets.

Joint Caching, Routing, and Channel Assignment for Collaborative Small-Cell Cellular Networks

We consider joint caching, routing, and channel assignment for video delivery over coordinated small-cell cellular systems of the future Internet. We formulate the problem of maximizing the throughput of the system as a linear program, in which the number of variables is very large. To address channel interference, our formulation incorporates the conflict graph that arises when wireless links interfere with each other due to simultaneous transmission. We utilize the column generation method to solve the problem by breaking it into a restricted master subproblem that involves a select subset of variables and a collection of pricing subproblems that select the new variable to be introduced into the restricted master problem, if that leads to a better objective function value. To control the complexity of the column generation optimization further, due to the exponential number of independent sets that arise from the conflict graph, we introduce an approximation algorithm that computes a solution that is within $\epsilon $ to optimality, at much lower complexity. Our framework demonstrates considerable gains in average transmission rate at which the video data can be delivered to the users, over the state-of-the-art Femtocaching system, of up to 46%. These operational gains in system performance map to analogous gains in video application quality, thereby enhancing the user experience considerably.

On the coefficients of the independence polynomial of graphs

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let $$i_k = i_k(G)$$ik=ik(G) be the number of independent sets of cardinality k of G. The independence polynomial $$I(G, x)=\sum _{k\geqslant 0}i_k(G)x^k$$I(G,x)=źkź0ik(G)xk defined first by Gutman and Harary has been the focus of considerable research recently, whereas $$i(G)=I(G, 1)$$i(G)=I(G,1) is called the Merrifield---Simmons index of G. In this paper, we first proved that among all trees of order n, the kth coefficient $$i_k$$ik is smallest when the tree is a path, and is largest for star. Moreover, the graph among all trees of order n with diameter at least d whose all coefficients of I(G, x) are largest is identified. Then we identify the graphs among the n-vertex unicyclic graphs (resp. n-vertex connected graphs with clique number $$\omega $$ź) which simultaneously minimize all coefficients of I(G, x), whereas the opposite problems of simultaneously maximizing all coefficients of I(G, x) among these two classes of graphs are also solved respectively. At last we characterize the graph among all the n-vertex connected graph with chromatic number $$\chi $$ź (resp. vertex connectivity $$\kappa $$ź) which simultaneously minimize all coefficients of I(G, x). Our results may deduce some known results on Merrifield---Simmons index of graphs.

A note on the values of independence polynomials at [formula omitted

The independence polynomialI(G;x) of a graph G is I(G;x)=∑k=0α(G)skxk, where sk is the number of independent sets in G of size k. The decycling number of a graph G, denoted ϕ(G), is the minimum size of a set S⊆V(G) such that G−S is acyclic. Engström proved that the independence polynomial satisfies |I(G;−1)|≤2ϕ(G) for any graph G, and this bound is best possible. Levit and Mandrescu provided an elementary proof of the bound, and in addition conjectured that for every positive integer k and integer q with |q|≤2k, there is a connected graph G with ϕ(G)=k and I(G;−1)=q. In this note, we prove this conjecture.

Computing the partition function for graph homomorphisms

We introduce the partition function of edge-colored graph homomorphisms, of which the usual partition function of graph homomorphisms is a specialization, and present an efficient algorithm to approximate it in a certain domain. Corollaries include effcient algorithms for computing weighted sums approximating the number of k-colorings and the number of independent sets in a graph, as well as an effcient procedure to distinguish pairs of edge-colored graphs with many color-preserving homomorphisms G → H from pairs of graphs that need to be substantially modified to acquire a color-preserving homomorphism G → H.

Birthday inequalities, repulsion, and hard spheres

We study a birthday inequality in random geometric graphs: the probability of the empty graph is upper bounded by the product of the probabilities that each edge is absent. We show the birthday inequality holds at low densities, but does not hold in general. We give three different applications of the birthday inequality in statistical physics and combinatorics: we prove lower bounds on the free energy of the hard sphere model and upper bounds on the number of independent sets and matchings of a given size in d-regular graphs. The birthday inequality is implied by a repulsion inequality: the expected volume of the union of spheres of radius r around n randomly placed centers increases if we condition on the event that the centers are at pairwise distance greater than r. Surprisingly we show that the repulsion inequality is not true in general, and in particular that it fails in 24-dimensional Euclidean space: conditioning on the pairwise repulsion of centers of 24-dimensional spheres can decrease the expected volume of their union.

Ferromagnetic Potts Model: Refined #BIS-hardness and Related Results

Recent results establish for 2-spin antiferromagnetic systems that the computational complexity of approximating the partition function on graphs of maximum degree D undergoes a phase transition that coincides with the uniqueness phase transition on the infinite D-regular tree. For the ferromagnetic Potts model we investigate whether analogous hardness results hold. Goldberg and Jerrum showed that approximating the partition function of the ferromagnetic Potts model is at least as hard as approximating the number of independent sets in bipartite graphs (#BIS-hardness). We improve this hardness result by establishing it for bipartite graphs of maximum degree D. We first present a detailed picture for the phase diagram for the infinite D-regular tree, giving a refined picture of its first-order phase transition and establishing the critical temperature for the coexistence of the disordered and ordered phases. We then prove for all temperatures below this critical temperature that it is #BIS-hard to approximate the partition function on bipartite graphs of maximum degree D. As a corollary, it is #BIS-hard to approximate the number of k-colorings on bipartite graphs of maximum degree D when k <= D/(2 ln D). The #BIS-hardness result for the ferromagnetic Potts model uses random bipartite regular graphs as a gadget in the reduction. The analysis of these random graphs relies on recent connections between the maxima of the expectation of their partition function, attractive fixpoints of the associated tree recursions, and induced matrix norms. We extend these connections to random regular graphs for all ferromagnetic models and establish the Bethe prediction for every ferromagnetic spin system on random regular graphs. We also prove for the ferromagnetic Potts model that the Swendsen-Wang algorithm is torpidly mixing on random D-regular graphs at the critical temperature for large q.

Independence polynomial and matching polynomial of the Koch network

The lattice gas model and the monomer-dimer model are two classical models in statistical mechanics. It is well known that the partition functions of these two models are associated with the independence polynomial and the matching polynomial in graph theory, respectively. Both polynomials have been shown to belong to the “[Formula: see text]-complete” class, which indicate the problems are computationally “intractable”. We consider these two polynomials of the Koch networks which are scale-free with small-world effects. Explicit recurrences are derived, and explicit formulae are presented for the number of independent sets of a certain type.

Graphs with the fewest matchings

In recent years there has been increased interest in extremal problems for “counting” parameters of graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a d-regular graph. In the same spirit, the Upper Matching Conjecture claims an upper bound on the number of k-matchings in a d-regular graph. Here we consider both matchings and matchings of fixed sizes in graphs with a given number of vertices and edges. We prove that the graph with the fewest matchings is either the lex or the colex graph. Similarly, for fixed k, the graph with the fewest k-matchings is either the lex or the colex graph. To prove these results we first prove that the lex bipartite graph has the fewest matchings of all sizes among bipartite graphs with fixed part sizes and a given number of edges.

On the independence polynomial of the corona of graphs

Let α(G) be the cardinality of a largest independent set in graph G. If sk is the number of independent sets of size k in G, then I(G;x)=s0+s1x+⋯+sαxα, α=α(G), is the independence polynomial of G (Gutman and Harary, 1983). I(G;x) is palindromic if sα−i=si for each i∈{0,1,…,⌊α/2⌋}. The corona of G and H is the graph G∘H obtained by joining each vertex of G to all the vertices of a copy of H (Frucht and Harary, 1970).In this paper, we show that I(G∘H;x) is palindromic for every graph G if and only if H=Kr−e,r≥2. In addition, we connect realrootness of I(G∘H;x) with the same property of both I(G;x) and I(H;x).

Development of wireless sensor network routing model taking into account correction of energyconsumption imbalance

Designing of a wireless sensor network with autonomous nodes arises the question how to provide the maximum duration of its life. For this purpose the paper proposes to use multipath routing mode supporting energy balancing of network nodes. The results of studies show that if the role of nodes is dynamically changed and the topology of the network is dynamically rebuilt, it is possible to bring the lifetime of the network to the terminal device lifetime. This increase of the lifetime is possible due to the fact that each of the nodes most of the time will be in the role of the terminal device. At the same time sets of concurrent routers rotate each other. Decisions on how to rebuild the topology of the network are accepted by the coordinator. The problem of multipath routing in view of redressing the imbalance of power consumption is solved. To achieve maximizing network lifetime it is necessary to find the maximum number of independent sets of routers. In solving the optimization problem of quadratic programming a uniform load between the transit nodes is obtained and hence their energy consumption will be equal.

The expected values of Hosoya index and Merrifield–Simmons index in a random polyphenylene chain

The Hosoya index $$m(G)$$m(G) and the Merrifield---Simmons index $$i(G)$$i(G) of a graph $$G$$G are the number of matchings and the number of independent sets in $$G$$G. In this paper, we establish exact formulas for the expected values of the Hosoya index and Merrifield---Simmons index of a random polyphenylene chain, and generalize the results of Doslic and Litz (MATCH Commun Math Comput Chem 67:313---330, 2012). Moreover, we obtain the average values of the Hosoya index and the Merrifield---Simmons index with respect to the set of all polyphenylene chains with $$n$$n hexagons.