We use Wagner's weighted subgraph counting polynomial to prove that the partition function of anti-ferromagnetic Ising model on line graphs is real rooted and roots edge cover have absolute value at most $4$. more generally show a $k$-uniform hypergraph $2^k$, discuss applications this domination polynomials graphs. moreover how our results relate efficient algorithms for approximately computin...

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination of a graph G the generating function number dominating sets each cardinality in G, have been conjectured unimodal. In this paper we will show paths, cycles complete multipartite graphs unimodal, that almost every with mode $$ \lceil \frac{n}{2}\rceil .

Let G be a simple connected graph of order n. Dcνe (G, i) the family vertex-edge dominating sets with cardinality i. The polynomial x) = is called domination where dcνe number vertex edge G. In this paper, we study some properties polynomials Gem Gn. Also obtain (Gn, and their coefficients. Also, find recursive formula, to derive

Domination is a rapidly developing area of research in graph theory, and its various applications to ad hoc networks, distributed computing, social networks and web graphs partly explain the increased interest. This thesis focuses on domination theory, and the main aim of the study is to apply a probabilistic approach to obtain new upper bounds for various domination parameters. Chapters 2 and ...

In this paper, we present new upper bounds for the global domination and Roman domination numbers and also prove that these results are asymptotically best possible. Moreover, we give upper bounds for the restrained domination and total restrained domination numbers for large classes of graphs, and show that, for almost all graphs, the restrained domination number is equal to the domination num...

In this paper, we introduce the closed domination in graphs. Some interesting relationships are known between domination and closed domination and between closed domination and the independent domination. It is also shown that any triple m, k and n of positive integers with 3 ≤ m ≤ k ≤ n are realizable as the domination number, closed domination number and independent domination number, respect...

In this work on Polynomial Identity (PI) quantized Weyl algebras we begin with a brief survey of Poisson geometry and quantum cluster algebras, before using these as tools to classify the possible centers of such algebras in two different ways. In doing so we explicitly calculate the formulas of the discriminants of these algebras in terms of a general class of central polynomial subalgebras. F...

In this paper, we survey and supplement the complexity landscape of the domination chain parameters as a whole, including classifications according to approximability and parameterised complexity. Moreover, we provide clear pointers to yet open questions. As this posed the majority of hitherto unsettled problems, we focus on Upper Irredundance and Lower Irredundance that correspond to finding t...

We show that there are polynomial-time algorithms to compute maximum independent sets in the categorical products of two cographs and two splitgraphs. The ultimate categorical independence ratio of a graph G is defined as limk→∞ α(G )/n. The ultimate categorical independence ratio is polynomial for cographs, permutation graphs, interval graphs, graphs of bounded treewidth and splitgraphs. When ...

In a finite undirected graph G = (V,E), a vertex v ∈ V dominates itself and its neighbors. A vertex set D ⊆ V in G is an efficient dominating set (e.d. for short) of G if every vertex of G is dominated by exactly one vertex of D. The Efficient Domination (ED) problem, which asks for the existence of an e.d. in G, is known to be NP-complete for P7-free graphs but solvable in polynomial time for ...