Related papers: Independence Polynomials and Hypergeometric Series

The concept and the construction of modular graph functions are generalized from genus-one to higher genus surfaces. The integrand of the four-graviton superstring amplitude at genus-two provides a generating function for a special class of…

We define an independence system associated with simple graphs. We prove that the independence system is a matroid for certain families of graphs, including trees, with bases as minimal resolving sets. Consequently, the greedy algorithm on…

Let $\alpha=\alpha(G)$ be the independence number of a simple graph $G$ with $n$ vertices and $I(G)$ be its edge ideal in $S=K[x_1,\ldots, x_n]$. If $S/I(G)$ is Gorenstein, the graph $G$ is called Gorenstein over $K$ and if $G$ is…

This paper is concerned with the question of whether geometric structures such as cell complexes can be used to simultaneously describe the minimal free resolutions of all powers of a monomial ideal. We provide a full answer in the case of…

The hard-core model can be used to understand the numbers of independent sets in graphs in extremal graph theory. The occupancy fraction, defined as the logarithmic derivative of the independence polynomial of a graph, is a key quantity in…

A $k$-independent set in a connected graph is a set of vertices such that any two vertices in the set are at distance greater than $k$ in the graph. The $k$-independence number of a graph, denoted $\alpha_k$, is the size of a largest…

Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $D$ and valency $k \ge 3$. In [Homotopy in $Q$-polynomial distance-regular graphs, Discrete Math., {\bf 223} (2000), 189-206], H. Lewis showed that the girth of…

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph $G$ is the generating function of the number of dominating sets of each cardinality in $G$. In…

If for any $k$ the $k$-th coefficient of a polynomial I(G;x) is equal to the number of stable sets of cardinality $k$ in the graph $G$, then it is called the independence polynomial of $G$ (Gutman and Harary, 1983). Let $a$ be the size of a…

Given a projective plane $\Sigma$ and a polarity $\theta$ of $\Sigma$, the corresponding polarity graph is the graph whose vertices are the points of $\Sigma$, and two distinct points $p_1$ and $p_2$ are adjacent if $p_1$ is incident to…

We prove that the tree independence number of every even-hole-free graph is at most polylogarithmic in its number of vertices. More explicitly, we prove that there exists a constant c>0 such that for every integer n>1 every n-vertex…

We study the independence complexes of graph products where at least one factor is a path. We also analyze the complexes of their induced subgraphs. We determine the homotopy type of the independence complex of the graphs $P_n\times P_m$,…

Let $\Gamma \subset \operatorname{PU}(1,n)$ be a lattice, and $S_\Gamma$ the associated ball quotient. We prove that, if $S_\Gamma$ contains infinitely many maximal totally geodesic subvarieties, then $\Gamma$ is arithmetic. We also prove…

In this paper, we give spectral upper bounds for the independence number of even uniform hypergraphs and graphs, extend the Hoffman bound to even uniform hypergraphs, and give a simple spectral condition for determining the independence…

In this paper, we introduce a unitary invariant $\Gamma$ defined on the unit ball of $B(H)^n$ in terms of the characteristic function, the noncommutative Poisson kernel, and the defect operator associated with a row contraction. We show…

We prove that, to every abstract group $G$, we can associate a sequence of graphs $\Gamma_n$ such that the automorphism group of $\Gamma_n$ is isomorphic to $G$ and the genus of $\Gamma_n$ is an unbounded function of $n$.

Let $G_\Gamma$ be a graph product over a finite simplicial graph $\Gamma$, and let $K_\Gamma$ denote the kernel of the canonical homomorphism from $G_\Gamma$ to the direct product of its vertex groups. It is known that, up to isomorphism,…

The mapping class group $\Gamma$ of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter…

The chromatic polynomial is characterized as the unique polynomial invariant of graphs, compatible with two interacting bialgebras structures: the first coproduct is given by partitions of vertices into two parts, the second one by a…

We study identities of finite dimensional algebras over a field of characteristic zero, graded by an arbitrary groupoid $\Gamma$. First we prove that its graded colength has a polynomially bounded growth. For any graded simple algebra $A$…