scholarly journals Extending of Edge Even Graceful Labeling of Graphs to Strong r -Edge Even Graceful Labeling

2021 ◽  
Vol 2021 ◽  
pp. 1-19
Author(s):  
Mohamed R. Zeen El Deen ◽  
Nora A. Omar

Edge even graceful labeling of a graph G with p vertices and q edges is a bijective f from the set of edge E G to the set of positive integers 2,4 , … , 2 q such that all the vertex labels f ∗ V G , given by f ∗ u = ∑ u v ∈ E G f u v mod 2 k , where k = max p , q , are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to r -edge even graceful labeling and strong r -edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an r -edge even graceful graph. Furthermore, the minimum number r for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an r -edge even graceful labeling was found. Finally, we proved that the even cycle C 2 n has a strong 2 -edge even graceful labeling when n is even.

Author(s):  
Robert D. Enright ◽  
Jacqueline Y. Song

The psychology of forgiveness originated from the creative and important work on the development of justice initiated by Piaget in 1932 and extended by Kohlberg in 1969. The scientific study of forgiveness is quite new, having emerged in print in 1989, with an examination of the developmental progression in children’s, adolescents’, and adults’ thinking about the necessary conditions for them to offer forgiveness to another person. In this chapter, the authors first review the definition of forgiveness, followed by this early cognitive work. They then turn to a discussion of the measurement of forgiveness correlates of forgiveness. The practical application of this construct is seen in the development of forgiveness therapy and forgiveness education, which the authors discuss in light of the empirical findings. Future directions for forgiveness studies are considered.


2017 ◽  
Vol 15 (19) ◽  
Author(s):  
Marek Volt

Artiklis käsitlen Lev Tolstoi kunstiteooria retseptsiooni anglo-ameerika esteetikas. Esiteks formuleerin Tolstoi kunstidefinitsiooni ja selle põhimõistete kanoonilise tõlgenduse. Seejärel analüüsin määratlust ekstensionaalse adekvaatsuse alusel, keskendudes nii tavapäraste kui võimalike uute etteheidete paikapidavuse uurimisele. Kolmandaks püstitan küsimuse, kas Tolstoi kunstidefinitsiooni kriitika ekstensionaalse adekvaatsuse alusel on üldse õigustatud. Väidan, et kuigi senistel Tolstoi meta-esteetilise rehabiliteerimise katsetel esineb puuduseid, paljastab Tolstoi kunstiteooria immanentne kriitika – teooria vaagimine eeldustelt, millelt see kritiseerib oponeerivaid teooriaid –, et ekstensionaalsest adekvaatsusest lähtuv kriitika on õigustatud. My article discusses Tolstoy’s theory of art in the context of Anglo-American aesthetics. Although Tolstoy’s What is Art touches upon a very wide spectrum of subjects (the place of art in the world, justification of sacrifices made for completing art works, criticism of previous theories of aesthetics, especially of the theory of beauty, defining of art as the expression of feelings, judging of art as such based on the religious knowledge of the era, action mechanisms of beauty/pleasure-centred art, consequences, conditions of the value of art, the relations between art and science, etc.), it has mainly been examined from the aspects of judging and defining of art.The article focuses on Tolstoy’s definition of art and consists of three notional parts. First, I present the canonical formulation of Tolstoy’s definition of art – something is a work of art if and only if the person, who lives through the feeling(s), causes by external signs that the recipients live through the same feelings. I also present the canonical interpretation of its main concepts – the conditions for creation, transmitting and reception.Second, I have an analytical insight into the criticism of the canonical treatment, displaying and commenting on, but also responding and complementing the presented arguments. The extensional adequacy-based analysis of Tolstoy’s definition of art shows that although it is possible to eliminate some of the typical criticisms, none of the three necessary conditions was necessary by itself, nor were all three of them sufficient when taken together.As Tolstoy’s definition of art has sometimes earned quite serious criticism, then, as my third point, I also examine some possibilities for rehabilitating Tolstoy’s theory of art: whether and in what sense can the extensional adequacy-based analysis of Tolstoy’s definition of art be justified at all? So far, the attempts of meta-aesthetic rehabilitation of Tolstoy (e.g., Mounce centrism) have not achieved the expected result. Furthermore, the immanent criticism of Tolstoy’s theory of art (criticism of the theory, based on the prerequisites it uses to criticize its opposing theories) reveals that the extensional adequacy-based criticism of Tolstoy’s definition of art is justified, but it is not necessarily the only yardstick for the theory.


2021 ◽  
Vol 3 (2) ◽  
pp. 103-114
Author(s):  
Meliana Pasaribu ◽  
Yundari Yundari ◽  
Muhammad Ilyas

Graceful Labeling on graph G=(V, E) is an injective function f from the set of the vertex V(G) to the set of numbers {0,1,2,...,|E(G)|} which induces bijective function f from the set of edges E(G) to the set of numbers {1,2,...,|E(G)|} such that for each edge uv e E(G) with u,v e V(G) in effect f(uv)=|f(u)-f(v)|. Meanwhile, the Skolem graceful labeling is a modification of the Graceful labeling. The graph has graceful labeling or Skolem graceful labeling is called graceful graph or Skolem graceful labeling graph. The graph used in this study is the U-star graph, which is denoted by U(Sn). The purpose of this research is to determine the pattern of the graceful labeling and Skolem graceful labeling on graph U(Sn) apply it to cryptography polyalphabetic cipher. The research begins by forming a graph U(Sn) and they are labeling it with graceful labeling and Skolem graceful labeling. Then, the labeling results are applied to the cryptographic polyalphabetic cipher. In this study, it is found that the U(Sn) graph is a graceful graph and a Skolem graceful graph, and the labeling pattern is obtained. Besides, the labeling results on a graph it U(Sn) can be used to form a table U(Sn) polyalphabetic cipher. The table is used as a key to encrypt messages.


10.37236/1171 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Ping Zhao ◽  
Kefeng Diao ◽  
Kaishun Wang

For any set $S$ of positive integers, a mixed hypergraph ${\cal H}$ is a realization of $S$ if its feasible set is $S$, furthermore, ${\cal H}$ is a one-realization of $S$ if it is a realization of $S$ and each entry of its chromatic spectrum is either 0 or 1. Jiang et al. showed that the minimum number of vertices of a realization of $\{s,t\}$ with $2\leq s\leq t-2$ is $2t-s$. Král proved that there exists a one-realization of $S$ with at most $|S|+2\max{S}-\min{S}$ vertices. In this paper, we  determine the number  of vertices of the smallest one-realization of a given set. As a result, we partially solve an open problem proposed by Jiang et al. in 2002 and by Král  in 2004.


1966 ◽  
Vol 18 ◽  
pp. 9-17
Author(s):  
Kulendra N. Majindar

In this paper, we give a connection between incidence matrices of affine resolvable balanced incomplete block designs and rectangular integer matrices subject to certain arithmetical conditions. The definition of these terms can be found in paper II of this series or in (2). For some necessary conditions on the parameters of affine resolvable balanced incomplete block designs and their properties see (2).


2019 ◽  
Vol 19 (02) ◽  
pp. 2050040 ◽  
Author(s):  
Sriparna Chattopadhyay ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo

The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity [Formula: see text] of [Formula: see text] is the minimum number of vertices which need to be removed from [Formula: see text] so that the induced subgraph of [Formula: see text] on the remaining vertices is disconnected or has only one vertex. For a positive integer [Formula: see text], let [Formula: see text] be the cyclic group of order [Formula: see text]. Suppose that the prime power decomposition of [Formula: see text] is given by [Formula: see text], where [Formula: see text], [Formula: see text] are positive integers and [Formula: see text] are prime numbers with [Formula: see text]. The vertex connectivity [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for [Formula: see text], we give a new upper bound for [Formula: see text] and determine [Formula: see text] when [Formula: see text]. We also determine [Formula: see text] when [Formula: see text] is a product of distinct prime numbers.


2020 ◽  
Vol 12 (04) ◽  
pp. 2050035
Author(s):  
Danjun Huang ◽  
Xiaoxiu Zhang ◽  
Weifan Wang ◽  
Stephen Finbow

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.


2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Luis Manuel Sánchez Ruiz ◽  
Sanjib Kumar Datta ◽  
Tanmay Biswas ◽  
Golok Kumar Mondal

The relative order of growth gives a quantitative assessment of how different functions scale each other and to what extent they are self-similar in growth. In this paper for any two positive integerspandq, we wish to introduce an alternative definition of relative(p,q)th order which improves the earlier definition of relative(p,q)th order as introduced by Lahiri and Banerjee (2005). Also in this paper we discuss some growth rates of entire functions on the basis of the improved definition of relative(p,q)th order with respect to another entire function and extend some earlier concepts as given by Lahiri and Banerjee (2005), providing some examples of entire functions whose growth rate can accordingly be studied.


2020 ◽  
Author(s):  
Fabrizio Piana ◽  
Chiara Avataneo ◽  
Luca Barale ◽  
Serena Botta ◽  
Roberto Compagnoni ◽  
...  

<p>For a reliable evaluation of the geo-environmental risk related to the presence of naturally occurring asbestos (NOA) in rocks excavated for large infrastructural projects, a proper procedure has to be followed in order to achieve:</p><p>1) the definition of a detailed geological model, tailored on NOA-related issues;</p><p>2) a representative sampling;</p><p>3) a reliable quantitative determination of asbestos content in rock samples.</p><p>Here we describe the approach followed for the evaluation of the NOA content for the excavation of a complex highway tunnel system (“Gronda di Genova” NW Italy), in NOA-bearing meta-ophiolite rocks. The NOA-oriented geological model has been constrained by the individuation of the main “NOA-related petrofacies” —i.e., classes of rocks with common lithological, structural and NOA content features—, and by the identification of “homogeneous zones” – i.e. geological units into which the NOA petrofacies are distributed. Implementation of Gy’s theory on sampling was used and here described to maintain statistical validity during sample processing from the primary rock sample to the analytical sample. SEM-EDS procedure for the quantitative determination of NOA content was improved with an error analysis delivering the minimum number of fibers to be measured to achieve the best analytical results.</p><p>The obtained results allowed the prediction of the NOA hazard in terms of risk zonation along the tunnel section and for the evaluation of the amount of asbestos-bearing spoil to be excavated and managed.</p>


Blood ◽  
2012 ◽  
Vol 120 (21) ◽  
pp. 4415-4415
Author(s):  
Massimiliano Postorino ◽  
Alessandro Lanti ◽  
Eleonora Fiorelli ◽  
Angelo Salvatore Ferraro ◽  
Oana Marilena Chiru ◽  
...  

Abstract Abstract 4415 BACKGROUND. Autologous stem cell transplantation (ASCT) of PBSCs has become a widely applied treatment for Multiple Mieloma (MM), non- Hodgking's lymphoma (NHL) and Hodgking's lymphoma (HL). Successful engraftment correlates with the number of CD34 hemopoietic progenitors cells infused. However, a part of MM or lymphoma patients (5% to 40%) fail to mobilize adequate numbers of PBSCs and thus cannot undergo to ASCT. The success of PBSCs mobilization is usually assessed by the total number of CD34+ stem cells collected, with a cutoff of 2.0–2.5 ×106 CD34+ cells/kg recipient body weight being considered as a minimum requirement for transplant. Poor mobilization of PBSCs is a major limitation to ASCT. Recently GITMO Working Group worked to define operational criteria for the identification/prediction of the poor mobilizer (PM) patients (Olivieri et al. 2011). Plerixafor, a CXCR4 chemochine antagonist, has been showed to improve significantly PBSC mobilization in PM patients. We present our experience using Plerixafor in PM patients classified according to GITMO criteria. METHODS. Between September 2009 and June 2012, a total of 17 patients (9F-8M) were enrolled. The diagnosis were: 10 MM (5F-5M), 1HL (1M), 6 NHL (4F-2M). The median age was 57 (range 15–66). 7 patients (3MM, 4NHL) were defined “Proven PM” and 10 patients (7MM, 2NHL, 1HL) “Predicted PM” according to GITMO criteria. The mobilization protocol included G-CSF, administered at a dose of 10μg/kg daily on 4 consecutive days. In the evening of the fourth day, patients received subcutaneous plerixafor at a dose of 0,24 mg/kg. Apheresis was initiated on the fifth day, 10–12 h after plerixafor and 1 h after G-CSF administration. Apheresis and daily administration of G-CSF and plerixafor continued until the patient collected enough CD34+ cells for auto- HSCT (> 2 ×106/kg; max 7 plerixafor injections if required). PBSC collection was initiated if peripheral CD34+ cells count was >10μl. A successful mobilization was defined as a total yeld of > 2×106/kg. RESULTS. 13 patients (76,5%) collected the minimum number of CD34 cells > 2×106/kg. The diagnosis were: 8MM, 1HL,1 NHL. 7 patients (2NHL; 4 MM; 1 LH; 7 predicted) were able to collect > 5×106/Kg. Only 4 patients (3 MM; 1 LNH; 4 proven) failed the mobilization because the numbers of cells CD34 were < 10μL and these patients did not undergo to apheresis procedures. The collection target of 2×106/Kg was reached in a median of 2 apheresis session (range 1–3). The technical characteristics of the procedures were (median value): blood volume processed 12 L (range 9–14), total CD34+/Kg collected 3,06 × 106(range 2,21-8,62), procedure efficiency 47,5% (range 35,3–79), duration of the procedure 261 minutes (range 210–309). Plerixafor was well tolerated and mild side effects were: reactions in the injection site, gastrointestinal disturbs, muscle pain. During administration of plerixafor we did not observe any significant laboratory abnormalities of liver or renal function. CONCLUSION. Unsuccessful mobilization represents an important limitation to ASCT in lymphoma and MM. In our experience plerixafor allowed to collect an appropriate amount of CD34 also in patients defined “proven PM” significantly reducing the percentage of patients that could not undergo ASCT (target value obtained in 43% of “proven PM”). Confirming the recent literature plerixafor is well tolerated with minimal side effects. We retrospectively applied GITMO criteria for PM patients and our experience, although limited, confirm that the use of a correct definition of PM allows the appropriate use of new mobilizing agents like plerixafor increasing significantly the therapeutic options also in patients who had no possibilities to receive an ASCT with the traditional mobilizing therapy. Disclosures: No relevant conflicts of interest to declare.


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