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AbstractIn this paper we show that if G is a connected graph such that [Formula: see text], [Formula: see text] and [Formula: see text] then [Formula: see text] exists and [Formula: see text] if and only if G is not super-[Formula: see text]. We also obtain some other results.

The paper presents the results of numerical calculations to determine the regularities underlying the influence of the characteristics of a mode I breathing fatigue crack on the leading edge of the aircraft gas turbine engine blade airfoil - that is responsible for the nonlinearity of the vibratory system - on its flexural forced vibration behavior. Their comparison with similar data for the rod of a rectangular cross section has shown satisfactory agreement. The calculation of the vibration diagnostic parameters indicating the presence of damage is performed using the developed finite element models of the investigation objects whose forced vibrations were excited by the kinematic displacement of fixed edge elements. The breathing crack is modeled as a mathematical cross section, the non-penetration of its faces is ensured by solving the contact problem of their interaction. The problem of the system forced vibrations is solved using the Newmark method and fast Fourier transformation. The obtained amplitude-frequency characteristics of the undamaged and damaged blade airfoil are indicative of the increase in the energy dissipation in the system in the presence of a breathing crack. The change in the resonant vibration frequency of the cracked object of investigation and the amplitude ratio of the dominant harmonics of displacements and accelerations (the ratio of the former to the latter at the superharmonic resonance and the latter to the former at the main and subharmonic resonances) were chosen as the vibration diagnostic indicators of damage. The amplitude ratio of the dominant harmonics of accelerations at the superharmonic resonance and that of displacements at the subharmonic resonance is found to be the most sensitive indicator of the presence of cracks. [ABSTRACT FROM AUTHOR]

Background: Powdery mildew (PM) is the most common fungal disease of cucumber and other cucurbit crops, while breeding the PM-resistant materials is the effective way to defense this disease, and the recent development of modern genetics and genomics make us aware of that studying the resistance genes is the essential way to breed the PM high-resistance plant. With the ever increasing throughput of next-generation sequencing (NGS), the development of specific length amplified fragment sequencing (SLAF-seq) as a high-resolution strategy for large-scale de novo SNP discovery is gradually applied for functional gene mining. Here we combined the bulked segregant analysis (BSA) with SLAF-seq to identify candidate genes associated with PM resistance in cucumber. Methods: A segregating population comprising 251 F2 individuals was developed using H136 (female parent) as susceptible parent and BK2 (male parent) as resistance donor. After PMR test, total genomic DNA was prepared from each plant. Systemic genomic analysis of the GC content, repeat sequence, etc. was carried out by prediction software SLAF_Predict to establish condition to ensure the uniformity and density of the molecular markers. After samples were gel purified, SLAFs were generated at Biomarker Technologies Corporation in Beijing. Based on SLAF tags and the PMR test result, the hot region were annotated. Results: A total of 73,100 high-quality SLAF tags with an average depth of 99.11x were sequenced. Among these, 5,355 polymorphic tags were identified with a polymorphism rate of 7.34 %, including 7.09 % SNPs and other polymorphism types. Finally, 140 associated SLAFs were identified, and two main Hot Regions were detected on chromosome 1 and 6, which contained five genes invovled in defense response, toxin metabolism, cell stress response, and injury response in cucumber. Conclusions: Associated markers identified by super-BSA in this study, could not only speed up the study of the PMR genes, but also provide a feasible solution for breeding the marker-assisted PMR cucumber. Moreover, this study could also be extended to any other species with reference genome. [ABSTRACT FROM AUTHOR]

Abstract: For a connected graph , an edge set is a -restricted edge-cut if is disconnected and every component of has at least vertices. Graphs that allow -restricted edge-cuts are called -connected. The -edge-degree of a graph  is the minimum number of edges between a connected subgraph of order and its complement . A -connected graph is called -optimal if its -restricted edge-connectivity equals its minimum -edge-degree and super- if every minimum -restricted edge-cut isolates a connected subgraph of order . In this paper we consider the cases and . For triangle-free graphs that are not -optimal, we establish lower bounds for the order of components left by a minimum -restricted edge-cut in terms of the minimum -edge-degree. Sufficient conditions for a triangle-free graph to be -optimal and super- follow. [Copyright &y& Elsevier]

Abstract: Let be a -connected graph. is called -optimal, if its -restricted edge-connectivity equals its minimum -edge degree.  is called super- if every -cut isolates a connected subgraph of order . Firstly, we give a lower bound on the order of -fragments in triangle-free graphs that are not -optimal. Secondly, we present an Ore-type condition for triangle-free graphs to be -optimal. Thirdly, we prove a lower bound on the order of -fragments in triangle-free -connected graphs, and use it to show that triangle-free graphs with high minimum degree are -optimal and super-. [Copyright &y& Elsevier]

Abstract: An edge cut of a connected graph is called restricted if it separates this graph into components each having order at least 2; a graph is super restricted edge connected if contains an isolated edge for every minimum restricted edge cut of . It is proved in this paper that a connected regular edge-transitive graph of valency at least 3 is not super restricted edge connected if and only if it is either the three dimensional hypercube, or a tetravalent edge-transitive graph of girth 3 and of order at least 6. As a result, there are infinitely many -regular Hamiltonian graphs with which are not super restricted edge connected. This answers negatively a question in [J. Ou, F. Zhang, Super restricted edge connectivity of regular graphs, Graphs & Combin. 21 (2005) 459–467] regarding the relationship between restricted edge connected graphs and Hamiltonian graphs. [Copyright &y& Elsevier]

Abstract: Let be a connected graph. An edge set is a 3-restricted edge cut, if is disconnected and every component of has at least three vertices. The 3-restricted edge connectivity of is the cardinality of a minimum 3-restricted edge cut of . A graph is -connected, if 3-restricted edge cuts exist. A graph is called -optimal, if , where is the set of edges of with one end in and the other in and . Furthermore, if every minimum 3-restricted edge cut is a set of edges incident to a connected subgraph induced by three vertices, then G is said to be super 3-restricted edge connected or super- for simplicity. In this paper we show that let be a -regular connected graph of order , if , then is super-. [Copyright &y& Elsevier]

Abstract: An edge cut X of a connected graph G is a k-restricted edge cut if is disconnected and every component of has at least k vertices. Additionally, if the deletion of a minimum k-restricted edge cut isolates a connected component of k vertices, then the graph is said to be super- . In this paper, several sufficient conditions yielding super- graphs are given in terms of the girth and the diameter. [Copyright &y& Elsevier]