Dependence Of Number Research Articles

Common domination perfect graphs

A dominating set in a graph G is a set S of vertices such that every vertex that does not belong to S is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. The common independence number αc(G) of G is the greatest integer r such that every vertex of G belongs to some independent set of cardinality at least r. The common independence number is squeezed between the independent domination number i(G) and the independence number α(G) of G, that is, γ(G)≤i(G)≤αc(G)≤α(G). A graph G is domination perfect if γ(H)=i(H) for every induced subgraph H of G. We define a graph G as common domination perfect if γ(H)=αc(H) for every induced subgraph H of G. We provide a characterization of common domination perfect graphs in terms of ten forbidden induced subgraphs.

ON PROPERTIES OF PRIME IDEAL GRAPHS OF COMMUTATIVE RINGS

The prime ideal graph of in a finite commutative ring with unity, denoted by , is a graph with elements of as its vertices and two elements in are adjacent if their product is in . In this paper, we explore some interesting properties of . We determined some properties of such as radius, diameter, degree of vertex, girth, clique number, chromatic number, independence number, and domination number. In addition to these properties, we study dimensions of prime ideal graphs, including metric dimension, local metric dimension, and partition dimension; furthermore, we examined topological indices such as atom bond connectivity index, Balaban index, Szeged index, and edge-Szeged index.

ON THE GIRTH, INDEPENDENCE NUMBER, AND WIENER INDEX OF COPRIME GRAPH OF DIHEDRAL GROUP

The coprime graph of a finite group , denoted by , is a graph with vertex set such that two distinct vertices and are adjacent if and only if their orders are coprime, i.e., where |x| is the order of x. In this paper, we complete the form of the coprime graph of a dihedral group that was given by previous research and it has been proved that if , for some and if . Moreover, we prove that if is even, then the independence number of is , where is the greatest odd divisor of and if is odd, then the independence number of is . Furthermore, the Wiener index of coprime graph of dihedral group has been stated here.

Independence numbers of polyhedral graphs

A polyhedral graph is a 3-connected planar graph. We find the least possible order p(k,a) of a polyhedral graph containing a k-independent set of size a for all positive integers k and a. In the case k=1 and a even, we prove that the extremal graphs are exactly the vertex-face (radial) graphs of maximal planar graphs.

Quantitative estimates for the size of an intersection of sparse automatic sets

A theorem of Cobham says that if k and ℓ are two multiplicatively independent natural numbers then a subset of the natural numbers that is both k- and ℓ-automatic is eventually periodic. A multidimensional extension was later given by Semenov. In this paper, we give a quantitative version of the Cobham-Semenov theorem for sparse automatic sets, showing that the intersection of a sparse k-automatic subset of Nd and a sparse ℓ-automatic subset of Nd is finite with size that can be explicitly bounded in terms of data from the automata that accept these sets.

On the α-spectral radius of the k-uniform supertrees

Let [Formula: see text] be a [Formula: see text]-uniform hypergraph with vertex set [Formula: see text] and edge set [Formula: see text]. A connected and acyclic hypergraph is called a supertree. For [Formula: see text], the [Formula: see text]-spectral radius of [Formula: see text] is the largest [Formula: see text]-eigenvalue of [Formula: see text], where [Formula: see text] and [Formula: see text] are the diagonal tensor of the degrees and the adjacency tensor of [Formula: see text], respectively. In this paper, we determine the unique supertrees with maximum [Formula: see text]-spectral radius among all [Formula: see text]-uniform supertrees with [Formula: see text] edges and independence number [Formula: see text] for [Formula: see text], among all [Formula: see text]-uniform supertrees with given degree sequences, and among all [Formula: see text]-uniform supertrees with [Formula: see text] edges and matching number [Formula: see text] for [Formula: see text], respectively.

Assistance and interdiction problems on interval graphs

We introduce a novel framework of graph modifications specific to interval graphs. We study interdiction problems with respect to these graph modifications. Given a list of original intervals, each interval has a replacement interval such that either the replacement contains the original, or the original contains the replacement. The interdictor is allowed to replace up to k original intervals with their replacements. Using this framework we also study the contrary of interdiction problems which we call assistance problems. We study these problems for the independence number, the clique number, shortest paths, and the scattering number. We obtain polynomial time algorithms for most of the studied problems. Via easy reductions, it follows that on interval graphs, the most vital nodes problem with respect to shortest path, independence number and Hamiltonicity can be solved in polynomial time.

The exponential growth of the packing chromatic number of iterated Mycielskians

For a graph G, the packing chromatic number of G, denoted by χρ(G), is the smallest integer k such that there exists a coloring f:V(G)↦{1,…,k} of G where every two distinct vertices u and v such that f(u)=f(v) are at pairwise distance at least f(u)+1. In this paper, we study graphs G for which χρ(μt(G))=2tχρ(G) for all t≥1, where μt(G) stands for the t-iterated Mycielskian of G. We show that a first natural upper bound of χρ(μt(G)) is 2t(|V(G)|−α(G)+1) for any graph G where α(G) is the independence number of G. This bound is rounded to 2tχρ(G) if the diameter of G is two. If moreover the graph G belongs to P, class of graphs whose vertex set can be partitioned into X∪Y such that Y is an independent set, |X|<|Y| and there is an |X|-matching M such that M=xy:x∈X,y∈Y, then χρ(μt(G))=2tχρ(G). Also, we propose a special construction of a family of maximal triangle-free graph (Tn)n≥5 with typical structure and show that χρ(μt(Tn))=2tχρ(Tn).

On the Inclusion Ideal Graph of Semigroups

The inclusion ideal graph [Formula: see text] of a semigroup [Formula: see text] is an undirected simple graph whose vertices are all the nontrivial left ideals of [Formula: see text] and two distinct left ideals [Formula: see text], [Formula: see text] are adjacent if and only if either [Formula: see text] or [Formula: see text]. The purpose of this paper is to study algebraic properties of the semigroup [Formula: see text] as well as graph theoretic properties of [Formula: see text]. We investigate the connectedness of [Formula: see text] and show that the diameter of [Formula: see text] is at most 3 if it is connected. We also obtain a necessary and sufficient condition of [Formula: see text] such that the clique number of [Formula: see text] is the number of minimal left ideals of [Formula: see text]. Further, various graph invariants of [Formula: see text], viz. perfectness, planarity, girth, etc., are discussed. For a completely simple semigroup [Formula: see text], we investigate properties of [Formula: see text] including its independence number and matching number. Finally, we obtain the automorphism group of [Formula: see text].

Patterns of GP and nurse independent prescriber prescriptions for antibiotics dispensed in the community in England: a retrospective analysis.

Physician-nurse task shifting, a process of delegation whereby tasks are moved to other specialized healthcare workers, is used in primary care in many countries to improve access, efficiency and quality of care. One such task is the prescription of medicines. To identify nurse independent prescriber (NIP) and GP numbers in England, the proportions and types of NIP and GP antibiotic prescriptions dispensed in the community, and the impact of COVID-19 on the volume, rate and types of antibiotic prescriptions dispensed. Descriptive population-based retrospective cohort study using routinely collected data on prescriptions for antibiotics dispensed in the community in England between January 2014 and October 2021. Between 2014 and 2021, numbers (headcount) of NIPs whose prescriptions were dispensed in the community rose by 146% to 34 997. GP numbers (headcount) rose by 10% to 44 681. Of the 25.373 million antibiotic prescriptions dispensed between 2014 and 2021, NIPs were responsible for 8.6%. The rate of dispensed antibiotic prescriptions per prescriber per calendar year decreased (by 50% for NIPs and by 21% for GPs) between 2014 and 2020. This decreasing trend continued following the onset of the COVID-19 pandemic across both groups. Narrow-spectrum antibiotics (penicillins, macrolides, tetracyclines) were the most frequently dispensed across both NIPs and GPs. NIPs are an increasing contributory influence on total antibiotic prescribing and should be included in antimicrobial stewardship efforts. Interventions for this group need to be tailored to the population and context in which they work.

A study of upper ideal relation graphs of rings

Let R be a ring with unity. The upper ideal relation graph of the ring R is the simple undirected graph whose vertex set is the set of all non-unit elements of R and two distinct vertices x, y are adjacent if and only if there exists a non-unit element such that the ideals (x) and (y) contained in the ideal (z). In this article, we obtain the girth, minimum degree and the independence number of . We obtain a necessary and sufficient condition on R, in terms of the cardinality of their principal ideals, such that the graph is planar and outerplanar, respectively. For a non-local commutative ring , where Ri is a local ring with maximal ideal and , we prove that the graph is perfect if and only if and each is a principal ideal. We also discuss all the finite rings R such that the graph is Eulerian. Moreover, we obtain the metric dimension and strong metric dimension of , when R is a reduced ring. Finally, we determine the vertex connectivity, automorphism group, Laplacian and the normalized Laplacian spectrum of . We classify all the values of n for which the graph is Hamiltonian.

Independent k-rainbow bondage number of graphs

For an integer an independent k-rainbow dominating function (IkRDF for short) on a graph G is a function g that assigns to each vertex a set of colors chosen from the subsets of satisfying the following conditions: (i) if , then , and (ii) the set is an independent set. The weight of an IkRDF g is the value . The independent k-rainbow domination number is the minimum weight of an IkRDF on G. In this paper, we initiate a study of the independent k-rainbow bondage number of a graph G having at least one component of order at least three, defined as the smallest size of set of edges for which . We begin by showing that the decision problem associated with the independent k-rainbow bondage problem is NP-hard for general graphs for . Then various upper bounds on are established as well as exact values on it for some special graphs. In particular, for trees T of order at least three, it is shown that .

Genome-wide association studies reveal novel loci associated with pyrethroid and organophosphate resistance in Anopheles gambiae and Anopheles coluzzii

Resistance to insecticides in Anopheles mosquitoes threatens the effectiveness of malaria control, but the genetics of resistance are only partially understood. We performed a large scale multi-country genome-wide association study of resistance to two widely used insecticides: deltamethrin and pirimiphos-methyl, using sequencing data from An. gambiae and An. coluzzii from ten locations in West Africa. Resistance was highly multi-genic, multi-allelic and variable between populations. While the strongest and most consistent association with deltamethrin resistance came from Cyp6aa1, this was based on several independent copy number variants (CNVs) in An. coluzzii, and on a non-CNV haplotype in An. gambiae. For pirimiphos-methyl, signals included Ace1, cytochrome P450s, glutathione S-transferases and the nAChR target site of neonicotinoid insecticides. The regions around Cyp9k1 and the Tep family of immune genes showed evidence of cross-resistance to both insecticides. These locally-varying, multi-allelic patterns highlight the challenges involved in genomic monitoring of resistance, and may form the basis for improved surveillance methods.

On the Independence Number of Cayley Digraphs of Clifford Semigroups

Let S be a Clifford semigroup and A a subset of S. We write Cay(S,A) for the Cayley digraph of a Clifford semigroup S relative to A. The (weak, path, weak path) independence number of a graph is the maximum cardinality of an (weakly, path, weakly path) independent set of vertices in the graph. In this paper, we characterize maximal connected subdigraphs of Cay(S,A) and apply these results to determine the (weak, path, weak path) independence number of Cay(S,A).

On Critical Difference, Independence Number and Matching Number of Graphs

On Critical Difference, Independence Number and Matching Number of Graphs

A Ramsey–Turán theory for tilings in graphs

AbstractFor a ‐vertex graph and an ‐vertex graph , an ‐tiling in is a collection of vertex‐disjoint copies of in . For , the ‐independence number of , denoted , is the largest size of a ‐free set of vertices in . In this article, we discuss Ramsey–Turán‐type theorems for tilings where one is interested in minimum degree and independence number conditions (and the interaction between the two) that guarantee the existence of optimal ‐tilings. Our results unify and generalise previous results of Balogh–Molla–Sharifzadeh [Random Struct. Algoritm. 49 (2016), no. 4, 669–693], Nenadov–Pehova [SIAM J. Discret. Math. 34 (2020), no. 2, 1001–1010] and Balogh–McDowell–Molla–Mycroft [Comb. Probab. Comput. 27 (2018), no. 4, 449–474] on the subject.

Connected Outer-Hop Independent Dominating Sets in Graphs Under Some Binary Operations

Let $G$ be a connected graph. A set $D\subseteq V(G)$ is called a connected outer-hop independent dominating if $D$ is a connected dominating set and $V(G)\s D$ is a hop independent set in $G$, respectively. The minimum cardinality of a connected outer-hop independent dominating set in $G$, denoted by $\gamma_{c}^{ohi}(G)$, is called the connected outer-hop independent domination number of $G$. In this paper, we introduce and investigated the concept of connected outer-hop independent domination in a graph. We show that the connected outer-hop independent domination number and connected outer-independent domination number of a graph are incomparable. In fact, we find that their absolute difference can be made arbitrarily large. In addition, we characterize connected outer-hop independent dominating sets in graphs under some binary operations. Furthermore, these results are used to give exact values or bounds of the parameter for these graphs.

Some bounds on spectral radius of signless Laplacian matrix of k-graphs

For a $k$-graph $H=(V(H), E(H))$, let $B(H)$ be its incidence matrix, and $Q(H)=B(H)B(H)^T$ be its signless Laplacian matrix, this name comes from the fact that $Q(H)$ is exactly the well-known signless Laplacian matrix for $2$-graph. Define the largest eigenvalue $\rho(H)$ of $Q(H)$ as the spectral radius of $H$. In this paper, we give some lower and upper bounds on $\rho(H)$ by some structural parameters (such as independent number, maximum degree, minimum degree, diameter, and so on) of $H$, which are extend or improve some known results.

Strong independence and its spectrum

For μ,κ infinite, say A⊆[κ]κ is a (μ,κ)-maximal independent family if whenever A0 and A1 are pairwise disjoint non-empty in [A]<μ then ⋂A0﹨⋃A1≠∅, A is maximal under inclusion among families with this property, and moreover all such Boolean combinations have size κ. We denote by spi(μ,κ) the set of all cardinalities of such families, and if non-empty, we let iμ(κ) be its minimal element. Thus, iμ(κ) (if defined) is a natural higher analogue of the independence number on ω for the higher Baire spaces. In this paper, we study spi(μ,κ) for μ,κ uncountable. Among others, we show that:(1)The property spi(μ,κ)≠∅ cannot be decided on the basis of ZFC plus large cardinals.(2)Relative to a measurable, it is consistent that:(a)(∃κ>ω)iκ(κ)<2κ.(b)(∃κ>ω)κ+<iω1(κ)<2κ. To the best knowledge of the authors, (2b) is the first example of a (μ,κ)-maximal independent family of size strictly between κ+ and 2κ, for uncountable κ.(3)spi(μ,κ) cannot be quite arbitrary.

Reducing Graph Parameters by Contractions and Deletions

We consider the following problem: for a given graph G and two integers k and d, can we apply a fixed graph operation at most k times in order to reduce a given graph parameter π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\pi $$\\end{document} by at least d? We show that this problem is NP-hard when the parameter is the independence number and the graph operation is vertex deletion or edge contraction, even for fixed d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document} and when restricted to chordal graphs. We give a polynomial time algorithm for bipartite graphs when the operation is edge contraction, the parameter is the independence number and d is fixed. Further, we complete the complexity dichotomy for H-free graphs when the parameter is the clique number and the operation is edge contraction by showing that this problem is NP-hard in (C3+P1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(C_3+P_1)$$\\end{document}-free graphs even for fixed d=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=1$$\\end{document}. When the operation is edge deletion and the parameter is the chromatic number, we determine the computational complexity of the associated problem for cographs and complete multipartite graphs. Our results answer several open questions stated in Diner et al. (Theor Comput Sci 746:49–72, 2012, https://doi.org/10.1016/j.tcs.2018.06.023).