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Optimal Domination PolynomialsApr 12 2019Let $G$ be a graph on $n$ vertices and $m$ edges and $D(G,x)$ the domination polynomial of $G$. In this paper we completely characterize the values of $n$ and $m$ for which optimal graphs exist for domination polynomials. We also show that there does ... More

Maximum Modulus of Independence Roots of Graphs and TreesDec 23 2018The 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 bound the maximum modulus, $\mbox{maxmod}(n)$, of an independence root over all graphs ... More

Graph Operations and Neighborhood PolynomialsJul 11 2018The neighborhood polynomial of graph $G$ is the generating function for the number of vertex subsets of $G$ of which the vertices have a common neighbor in $G$. In this paper, we investigate the behavior of this polynomial under several graph operations. ... More

Searching for dense subsets in a graph via the partition functionJul 05 2018For a set $S$ of vertices of a graph $G$, we define its density $0 \leq \sigma(S) \leq 1$ as the ratio of the number of edges of $G$ spanned by the vertices of $S$ to ${|S| \choose 2}$. We show that, given a graph $G$ with $n$ vertices and an integer ... More

On the Stability of Independence PolynomialsFeb 07 2018The independence polynomial of a graph is the generating polynomial for the number of independent sets of each size, and its roots are called {\em independence roots}. We investigate the stability of such polynomials, that is, conditions under which the ... More

The Domination Equivalence Classes of PathsOct 11 2017A dominating set $S$ of a graph $G$ of order $n$ is a subset of the vertices of $G$ such that every vertex is either in $S$ or adjacent to a vertex of $S$. %The domination number $G$, denoted $\gamma (G)$, is the cardinality of the smallest dominating ... More

On the Unimodality of Independence Polynomials of Very Well-Covered GraphsSep 24 2017The 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 ... More

On trees with real rooted independence polynomialMar 15 2017The independence polynomial of a graph $G$ is \[I(G,x)=\sum\limits_{k\ge 0}i_k(G)x^k,\] where $i_k(G)$ denotes the number of independent sets of $G$ of size $k$ (note that $i_0(G)=1$). In this paper we show a new method to prove real-rootedness of the ... More

The Roller-Coaster Conjecture RevisitedDec 12 2016A graph is well-covered if all its maximal independent sets are of the same cardinality (Plummer, 1970). If G is a well-covered graph, has at least two vertices, and G-v is well-covered for every vertex v, then G is a 1-well-covered graph (Staples, 1975). ... More

The Independent Domination PolynomialFeb 26 2016A vertex subset $W\subseteq V$ of the graph $G=(V,E)$ is an independent dominating set if every vertex in $V\backslash W$ is adjacent to at least one vertex in $W$ and the vertices of $W$ are pairwise non-adjacent. The independent domination polynomial ... More

A survey on recurrence relations for the independence polynomial of hypergraphsJun 11 2014The independence polynomial of a hypergraph is the generating function for its independent (vertex) sets with respect to their cardinality. This article aims to discuss several recurrence relations for the independence polynomial using some vertex and ... More

Hultman Numbers and Generalized Commuting Probability in Finite GroupsMar 16 2014Oct 02 2014Let $G$ be a finite group and $\pi$ be a permutation from $S_{n}$. We investigate the distribution of the probabilities of the equality \[ a_{1}a_{2}\cdots a_{n-1}a_{n}=a_{\pi_{1}}a_{\pi_{2}}\cdots a_{\pi_{n-1}}a_{\pi_{n}} \] when $\pi$ varies over all ... More

Domination Polynomials of Graph ProductsMay 07 2013Dec 23 2013The domination polynomials of binary graph operations, aside from union, join and corona, have not been widely studied. We compute and prove recurrence formulae and properties of the domination polynomials of families of graphs obtained by various products, ... More

On f-Symmetries of the Independence PolynomialMar 11 2013An independent set in a graph is a set of pairwise non-adjacent vertices, and a(G) is the size of a maximum independent set in the graph G. If s_{k} is the number of independent sets of cardinality k in G, then I(G;x)=s_0+s_1*x+s_2*x^2+...+s_a*x^a,a=a(G), ... More

Subset-Sum Representations of Domination PolynomialsJul 10 2012The domination polynomial D(G,x) is the ordinary generating function for the dominating sets of an undirected graph G=(V,E) with respect to their cardinality. We consider in this paper representations of D(G,x) as a sum over subsets of the edge and vertex ... More

Recurrence relations and splitting formulas for the domination polynomialJun 26 2012Sep 18 2012The domination polynomial D(G,x) of a graph G is the generating function of its dominating sets. We prove that D(G,x) satisfies a wide range of reduction formulas. We show linear recurrence relations for D(G,x) for arbitrary graphs and for various special ... More

A Counterexample to rapid mixing of the Ge-Stefankovic ProcessSep 24 2011Ge and Stefankovic have recently introduced a novel two-variable graph polynomial. When specialised to a bipartite graphs G and evaluated at the point (1/2,1) this polynomial gives the number of independent sets in the graph. Inspired by this polynomial, ... More

On Symmetry of Independence PolynomialsMay 11 2011An independent set in a graph is a set of pairwise non-adjacent vertices, and alpha(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while mu(G) is the cardinality of a maximum matching. If s_{k} is ... More

On the independence polynomial of an antiregular graphJul 06 2010A graph with at most two vertices of the same degree is called antiregular (Merris 2003), maximally nonregular (Zykov 1990) or quasiperfect (Behzad, Chartrand 1967). If s_{k} is the number of independent sets of cardinality k in a graph G, then I(G;x) ... More

A Lower Bound on the Density of Sphere Packings via Graph TheoryFeb 09 2004Using graph-theoretic methods we give a new proof that for all sufficiently large $n$, there exist sphere packings in $\R^n$ of density at least $cn2^{-n}$, exceeding the classical Minkowski bound by a factor linear in $n$. This matches up to a constant ... More