Extensions of Barrier Sets to Nonzero Roots of the Matching Polynomials

Some families of graphs with no nonzero real domination roots

Let G be a simple graph of order n. The domination polynomial is the generating polynomial for the number of dominating sets of G of each cardinality. A root of this polynomial is called a domination root of G. Obviously 0 is a domination root of every graph G. In the study of the domination roots of graphs, this naturally raises the question: which graphs have no nonzero real domination roots? In this paper we present some families of graphs whose have this property.

An Analogue of the Gallai-Edmonds Structure Theorem for Nonzero Roots of the Matching Polynomial

Godsil observed the simple fact that the multiplicity of 0 as a root of the matching polynomial of a graph coincides with the classical notion of deficiency. From this fact he asked to what extent classical results in matching theory generalize, replacing "deficiency" with multiplicity of $θ$ as a root of the matching polynomial. We prove an analogue of the Stability Lemma for any given root, which describes how the matching structure of a graph changes upon deletion of a single vertex. An analogue of Gallai's Lemma follows. Together these two results imply an analogue of the Gallai-Edmonds St…

Generalized $D$-graphs for Nonzero Roots of the Matching Polynomial

Recently, Bauer et al. (J Graph Theory 55(4) (2007), 343--358) introduced a graph operator $D(G)$, called the $D$-graph of $G$, which has been useful in investigating the structural aspects of maximal Tutte sets in $G$ with a perfect matching. Among other results, they proved a characterization of maximal Tutte sets in terms of maximal independent sets in the graph $D(G)$ and maximal extreme sets in $G$. This was later extended to graphs without perfect matchings by Busch et al. (Discrete Appl. Math. 155 (2007), 2487--2495). Let $θ$ be a real number and $μ(G,x)$ be the matching polynomial of a…

双 barrier 游走的核根展开中的块与残差排除

Block and Residue Exclusion in Kernel-Root Expansions for Two-Barrier Walks

针对整数上步长为 +y 和 -b 的双 barrier 游走，提出两阶段修剪方法构造核根展开，在示例中有效排除大量非必要项，提升计算效率。

Some new results on domination roots of a graph

Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G,λ)=\sum_{i=0}^{n} d(G,i) λ^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$. Every root of $D(G,λ)$ is called the domination root of $G$. We present families of graphs whose their domination polynomial have no nonzero real roots. We observe that these graphs have complex domination roots with positive real part. Then, we consider the lexicographic product of two graphs and obtain a formula for domination polynomial of this product. Using this product, we construct a family of…

On the Reliability Roots of Simplicial Complexes and Matroids

Assume that the vertices of a graph $G$ are always operational, but the edges of $G$ fail independently with probability $q \in[0,1]$. The \emph{all-terminal reliability} of $G$ is the probability that the resulting subgraph is connected. The all-terminal reliability can be formulated into a polynomial in $q$, and it was conjectured \cite{BC1} that all the roots of (nonzero) reliability polynomials fall inside the closed unit disk. It has since been shown that there exist some connected graphs which have their reliability roots outside the closed unit disk, but these examples seem to be few an…