Independence Polynomial Research Articles

On the Fibonacci numbers of the composition of graphs

Let G=(V,E) be a graph. A subset S of V is said to be independent if for every two vertices u,v∈S there is no edge between them. The Fibonacci number of a graph G is the total number of independent vertex sets of G. The problem of finding the Fibonacci number of a graph is an NP-complete Problem. A closely related concept is the independence polynomial of a graph, whose kth coefficient equals the number of independent sets of cardinality k in the graph. The composition of two graphs G and H is the graph obtained by replacing every vertex of G with a copy of H and all vertices of two copies are adjacent if the corresponding vertices in G are adjacent. Previous results on Fibonacci numbers of graphs, allow us to calculate the Fibonacci number of very few particular classes of graphs. In this paper, we show some new nice qualitative results, which allow us to calculate the Fibonacci numbers of infinitely many classes of graphs, obtained by the composition of graphs. In particular, we calculate the Fibonacci number of the composition of G and H in terms of the independence polynomial of G and the Fibonacci number of H.

The average size of independent sets of graphs

We study the average size of independent (vertex) sets of a graph. This invariant can be regarded as the logarithmic derivative of the independence polynomial evaluated at 1. We are specifically concerned with extremal questions. The maximum and minimum for general graphs are attained by the empty and complete graph respectively, while for trees we prove that the path minimises the average size of independent sets and the star maximises it. Although removing a vertex does not always decrease the average size of independent sets, we prove that there always exists a vertex for which this is the case.

On a Conjecture of Sokal Concerning Roots of the Independence Polynomial

A conjecture of Sokal [24], regarding the domain of nonvanishing for independence polynomials of graphs, states that given any natural number Δ≥3, there exists a neighborhood in C of the interval [0,(Δ−1)Δ−1/(Δ−2)Δ) on which the independence polynomial of any graph with maximum degree at most Δ does not vanish. We show here that Sokal’s conjecture holds, as well as a multivariate version, and prove the optimality for the domain of nonvanishing. An important step is to translate the setting to the language of complex dynamical systems.

Generalizations of the Matching Polynomial to the Multivariate Independence Polynomial

We generalize two main theorems of matching polynomials of undirected simple graphs, namely, real-rootedness and the Heilmann-Lieb root bound. Viewing the matching polynomial of a graph $G$ as the independence polynomial of the line graph of $G$, we determine conditions for the extension of these theorems to the independence polynomial of any graph. In particular, we show that a stability-like property of the multivariate independence polynomial characterizes claw-freeness. Finally, we give and extend multivariate versions of Godsil's theorems on the divisibility of matching polynomials of trees related to $G$.

The independence and clique polynomial of the conjugacy class graph of dihedral group

The independence and clique polynomial are two types of graph polynomial that store combinatorial information of a graph. The independence polynomial of a graph is the polynomial in which its coefficients are the number of independent sets in the graph. The independent set of a graph is a set of vertices that are not adjacent. The clique polynomial of a graph is the polynomial in which its coefficients are the number of cliques in the graph. The clique of a graph is a set of vertices that are adjacent. Meanwhile, a graph of group G is called conjugacy class graph if the vertices are non-central conjugacy classes of G and two distinct vertices are connected if and only if their class cardinalities are not coprime. The independence and clique polynomial of the conjugacy class graph of a group G can be obtained by considering the polynomials of complete graph or polynomials of union of some graphs. In this research, the independence and clique polynomials of the conjugacy class graph of dihedral groups of order 2n are determined based on three cases namely when n is odd, when n and n/2 are even, and when n is even and n/2 is odd. For each case, the results of the independence polynomials are of degree two, one and two, and the results of the clique polynomials are of degree (n-1)/2, (n+2)/2 and (n-2)/2, respectively.

Spectrahedrality of hyperbolicity cones of multivariate matching polynomials

The generalized Lax conjecture asserts that each hyperbolicity cone is a linear slice of the cone of positive semidefinite matrices. We prove the conjecture for a multivariate generalization of the matching polynomial. This is further extended (albeit in a weaker sense) to a multivariate version of the independence polynomial for simplicial graphs. As an application, we give a new proof of the conjecture for elementary symmetric polynomials (originally due to Brändén). Finally, we consider a hyperbolic convolution of determinant polynomials generalizing an identity of Godsil and Gutman.

Log-concavity of some independence polynomials via a partial ordering

Schwenk proved in 1981 that the edge independence polynomial of a graph is unimodal. It has been known since 1987 that the (vertex) independence polynomial of a graph need not be unimodal. Alavi et al. have asked whether the independence polynomial of a tree is unimodal. We apply some results on the log-concavity of combinations of log-concave sequences toward establishing the log-concavity (and, thus, the unimodality) of the independence numbers of some families of trees and related graphs.

Log-concavity of independence polynomials of some kinds of trees

An independent set in a graph G is a set of pairwise non-adjacent vertices. Let ik(G) denote the number of independent sets of cardinality k in G. Then, its generating functionI(G;x)=∑k=0α(G)ik(G)xkis called the independence polynomial of G (Gutman and Harary, 1983).Alavi et al. (1987) conjectured that the independence polynomial of any tree or forest is unimodal. This conjecture is still open. In this paper, after obtaining recurrence relations and giving factorizations of independence polynomials for certain classes of trees, we prove the log-concavity of their independence polynomials. Thus, our results confirm the conjecture of Alavi et al. for some special cases.

Counting independent sets in cubic graphs of given girth

We prove a tight upper bound on the independence polynomial (and total number of independent sets) of cubic graphs of girth at least 5. The bound is achieved by unions of the Heawood graph, the point/line incidence graph of the Fano plane.We also give a tight lower bound on the total number of independent sets of triangle-free cubic graphs. This bound is achieved by unions of the Petersen graph.We conjecture that in fact all Moore graphs are extremal for the scaled number of independent sets in regular graphs of a given minimum girth, maximizing this quantity if their girth is even and minimizing if odd. The Heawood and Petersen graphs are instances of this conjecture, along with complete graphs, complete bipartite graphs, and cycles.

On trees with real-rooted independence polynomial

The independence polynomial of a graph G is I(G,x)=∑k≥0ik(G)xk,where ik(G) denotes the number of independent sets of G of size k (note that i0(G)=1). In this paper we show a new method to prove real-rootedness of the independence polynomials of certain families of trees.In particular we will give a new proof of the real-rootedness of the independence polynomials of centipedes (Zhu’s theorem), caterpillars (Wang and Zhu’s theorem), and we will prove a conjecture of Galvin and Hilyard about the real-rootedness of the independence polynomial of the so-called Fibonacci trees.

Optimal Graphs for Independence and k-Independence Polynomials

The independence polynomial I(G, x) of a finite graph G is the generating function for the sequence of the number of independent sets of each cardinality. We investigate whether, given a fixed number of vertices and edges, there exists optimally-least (optimally-greatest) graphs, that are least (respectively, greatest) for all non-negative x. Moreover, we broaden our scope to k-independence polynomials, which are generating functions for the k-clique-free subsets of vertices. For $$k \ge 3$$ , the results can be quite different from the $$k = 2$$ (i.e. independence) case.

Graph polynomials for a class of DI-pathological graphs

Let G=V,E be a simple graph. A dominating set D⊆V is a set such that the closed neighborhood of D is the entire vertex set. An independence set of a graph G is a subset of vertices that are pairwise non-adjacent. A DI-pathological graph is a graph where every minimum dominating set intersects every maximal independent set. Let d(G,i) denote the number of dominating sets of G of size i. The domination polynomial of a graph G is defined by D(G,x)=∑i=γ(G)Vd(G,i)xi. Let s(G,i) denote the number of independent sets of size i in a graph G. The independence polynomial is defined by I(G,x)=∑i=0α(G)s(G,i)xi. In this paper, we will examine the domination polynomial and the independence polynomial of a family of extremal DI-pathological graphs. We will further define an independent dominating set and examine the corresponding independent domination polynomial for these graphs.

Tight bounds on the coefficients of partition functions via stability

Partition functions arise in statistical physics and probability theory as the normalizing constant of Gibbs measures and in combinatorics and graph theory as graph polynomials. For instance the partition functions of the hard-core model and monomer-dimer model are the independence and matching polynomials respectively.We show how stability results follow naturally from the recently developed occupancy method for maximizing and minimizing physical observables over classes of regular graphs, and then show these stability results can be used to obtain tight extremal bounds on the individual coefficients of the corresponding partition functions.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.

Unimodality of independence polynomials of the incidence product of graphs

Given two graphs G and H, assume that V(G)={v1,v2,…,vn} and U is a subset of V(H). We introduce a new graph operation called the incidence product, denoted by G⊙HU, as follows: insert a new vertex into each edge of G, then join with edges those pairs of new vertices on adjacent edges of G. Finally, for every vertex vi∈V(G), replace it by a copy of the graph H and join every new vertex being adjacent to vi to every vertex of U. It generalizes the line graph operation. We prove that the independence polynomial IG⊙HU;x=In(H;x)MG;xI2(H−U;x)I2(H;x),where M(G;x) is its matching polynomial. Based on this formula, we show that the incidence product of some graphs preserves symmetry, unimodality, reality of zeros of independence polynomials. As applications, we obtain some graphs so-formed having symmetric and unimodal independence polynomials. In particular, the graph Q(G) introduced by Cvetković, Doob and Sachs has a symmetric and unimodal independence polynomial.

Christoffel–Darboux Type Identities for the Independence Polynomial

In this paper we introduce some Christoffel–Darboux type identities for independence polynomials. As an application, we give a new proof of a theorem of Chudnovsky and Seymour, which states that the independence polynomial of a claw-free graph has only real roots. Another application is related to a conjecture of Merrifield and Simmons.

Counting Markov equivalence classes for DAG models on trees

DAG models are statistical models satisfying a collection of conditional independence relations encoded by the nonedges of a directed acyclic graph (DAG) G. Such models are used to model complex cause–effect systems across a variety of research fields. From observational data alone, a DAG model G is only recoverable up to Markov equivalence. Combinatorially, two DAGs are Markov equivalent if and only if they have the same underlying undirected graph (i.e., skeleton) and the same set of the induced subDAGs i→j←k, known as immoralities. Hence it is of interest to study the number and size of Markov equivalence classes (MECs). In a recent paper, we introduced a pair of generating functions that enumerate the number of MECs on a fixed skeleton by number of immoralities and by class size, and we studied the complexity of computing these functions. In this paper, we lay the foundation for studying these generating functions by analyzing their structure for trees and other closely related graphs. We describe these polynomials for some well-studied families of graphs including paths, stars, cycles, spider graphs, caterpillars, and balanced binary trees. In doing so, we recover connections to independence polynomials, and extend some classical identities that hold for Fibonacci numbers. We also provide tight lower and upper bounds for the number and size of MECs on any tree. Finally, we use computational methods to show that the number and distribution of high degree nodes in a triangle-free graph dictate the number and size of MECs.

R-Stable hypersimplices

Hypersimplices are well-studied objects in combinatorics, optimization, and representation theory. For each hypersimplex, we define a new family of subpolytopes, called r-stable hypersimplices, and show that a well-known regular unimodular triangulation of the hypersimplex restricts to a triangulation of each r-stable hypersimplex. For the case of the second hypersimplex defined by the two-element subsets of an n-set, we provide a shelling of this triangulation that sequentially shells each r-stable sub-hypersimplex. In this case, we utilize the shelling to compute the Ehrhart h⁎-polynomials of these polytopes, and the hypersimplex, via independence polynomials of graphs. For one such r-stable hypersimplex, this computation yields a connection to CR mappings of Lens spaces via Ehrhart–MacDonald reciprocity.

On the Stability of Independence Polynomials

The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size and its roots are called independence roots. We investigate the stability of such polynomials, that is, conditions under which the independence roots lie in the left half-plane. We use results from complex analysis to determine graph operations that result in a stable or nonstable independence polynomial. In particular, we prove that every graph is an induced subgraph of a graph with stable independence polynomial. We also show that the independence polynomials of graphs with independence number at most three are necessarily stable, but for larger independence number, we show that the independence polynomials can have roots arbitrarily far to the right.

Some results on the independence polynomial of unicyclic graphs

Some results on the independence polynomial of unicyclic graphs

The independence polynomial of conjugate graph and noncommuting graph of groups of small order.

An independent set of a graph is a set of pairwise non-adjacent vertices. The independence polynomial of a graph is defined as a polynomial in which the coefficient is the number of the independent set in the graph. Meanwhile, a graph of a group G is called conjugate graph if the vertices are non-central elements of G and two distinct vertices are adjacent if they are conjugate. The noncommuting graph is defined as a graph whose vertex set is non-central elements in which two vertices are adjacent if and only if they do not commute. In this research, the independence polynomial of the conjugate graph and noncommuting graph are found for three nonabelian groups of order at most eight.