The average degree-distance index of graph operations

This paper proposes a deterministic method to construct 5-regular geometric graphs with short average distance under the constraint such that the set of vertices is a subset of \(\mathbb {N}\times \mathbb {N}\) and the length of each edge is at most 4. This problem is motivated by the design of efficient floor plan of parallel computers consisting of a number of computing nodes arranged on a two-dimensional array. In such systems, the degree of vertices is determined by the number of ports of the routers and the edge length is limited by a certain value determined by the cycle time. The goodness of the resulting geometric graph is evaluated by the average shortest path length (ASPL) between vertices which reflects the average communication delay between computing nodes. The idea of the proposed method is to arrange the basic component derived from (3, g)-cage in a two-dimensional manner and to connect adjacent components by parallel edges of length 4 each. The result of numerical calculations shows that the average distance in the resulting graph is close to the lower bound so that the gap to the lower bound is less than 0.98 when the number of vertices is 432000.

Variable neighborhood search for extremal graphs. 21. Conjectures and results about the independence number

Antibodies and immune complexes in sera of breast cancer patients

Several authors have defined a variety of so-called safe zones for deltoid-splitting incisions. The first aim of the present study was to investigate the distance of the axillary nerve from the acromion and its relation to arm length. The second aim was to identify a safe area for the axillary nerve during surgical dissection of the deltoid muscle. Twenty-four shoulders of embalmed adult cadavers were included in the study. The distance from the anterior edge of the acromion to the course of the axillary nerve was measured and was recorded as the anterior distance. The same measurement from the posterior edge of the acromion to the course of the axillary nerve was made and was recorded as the posterior distance for each limb. Correlation analysis was performed between the arm length and the anterior distance and the posterior distance for each limb. The ratios between arm length and the anterior and posterior distances were calculated for each case and were recorded as an anterior index and a posterior index. The average arm length was 30.40 cm. The average anterior distance was 6.08 cm, and the average posterior distance was 4.87 cm. There was a significant correlation between arm length and both anterior distance (r = 0.79, p < 0.001) and posterior distance (r = 0.61, p = 0.001). The axillary nerve was not found to lie at a constant distance from the acromion at every point along its course. The average anterior index was 0.20, and the average posterior index was 0.16. The present study describes a safe area above the axillary nerve that is quadrangular in shape, with the length of the lateral edges being dependent on the individual's arm length. Using this safe area should provide a safe exposure for the axillary nerve during shoulder operations.

The positive role of root decomposition on the bioremediation of organic pollutants contaminated soil: A case study using PCB-9 as a model compound

For an optically transparent, UV‐reflective film, hollow silica nanospheres smaller than the visible wavelength (&lt;λvis) are prepared and assembled into colloidal glasses, of which interstices are then backfilled with a polymer. The polymer refractive index is matched with the silica shell to minimize backscattering in the visible range, and the average distance between the hollow silica particles is adjusted by tuning the shell thickness to satisfy the interference resonance condition for a UV selective reflection. The resulting composite film shows a strong UV reflection as expected, but it is translucent in visible light due to non‐negligible backscattering, which may be caused by large defects or fluctuation of the particle concentration. In order to avoid such backscattering, another polymer is introduced of which the refractive index is matched with the average refractive index of the hollow nanospheres. This allows an optically transparent film that selectively reflects the UV light. Furthermore, spherical aggregates of hollow silica nanospheres called “supraballs” are prepared and their average refractive index is matched with a solvent by adjusting the mixture ratio of water and ethylene glycol, which yields an optically transparent solution, selectively reflecting UV.

In the study of double-loop computer networks, the diagrams known as L-shapes arise as a graphical representation of an optimal routing for every graph’s node. The description of these diagrams provides an efficient method for computing the diameter and the average minimum distance of the corresponding graphs. We extend these diagrams to multiloop computer networks. For each Cayley digraph with a finite abelian group as vertex set, we define a monomial ideal and consider its representations via its minimal system of generators or its irredundant irreducible decomposition. From this last piece of information, we can compute the graph’s diameter and average minimum distance. That monomial ideal is the initial ideal of a certain lattice with respect to a graded monomial ordering. This result permits the use of Gr¨obner bases for computing the ideal and finding an optimal routing. Finally, we present a family of Cayley digraphs parametrized by their diameter d, all of them associated to irreducible monomial ideals.

Let Γ be a finite group with a nonempty subset A. The Cayley graph Cay (Γ, A) of Γ generated by A is defined as the digraph with vertex set Γ and edge set {(x,y) | x-1 y ∈ A}. Cay (Γ, A) can be regarded as an undirected graph if x-1 ∈ A for all x ∈ A. Let [Formula: see text] denote the largest integer M so that there exists a set of integers A = {±1, ±a2;…, ±ak} such that the average distance between all pairs of vertices of Cay (ℤM,A) is at most r, where ℤM is the additive group of residue classes modulo M. It is proved in this paper that [Formula: see text] It is also proved that [Formula: see text]

Weighted multi-connected loop networks

On domination number and distance in graphs

The “requisition-compensation balance policy” is a basic arable land protection policy promulgated by the Chinese government to maintain the dynamic balance of the total arable land. Since the implementation of the “requisition-compensation balance policy”, its important role in the conservation of arable land has been widely noted. Multiple studies indicate the effectiveness of the policy by estimating the quantity and quality change of arable land. In addition, the defects and negative externalities of the policy are disputed. The overall goal of this study is to estimate the provincial-level effectiveness of the “requisition-compensation balance policy” in mainland China in two periods (i.e., 2000—2010; 2010—2020) from four perspectives: arable land quantity dynamic balance; arable land productivity balance; farming distance; and sustainability of arable land use. The results showed that, first, the arable land quantity balance was achieved during 2000–2020 from a national perspective. There are still 19–23% of provinces have failed to reach the quantity balance. Second, the decrease in China’s total arable land productivity was exacerbated from 9612.1 thousand tons. to 31254.6 thousand ton. The average potential yield balance index was less than 1 for nearly all provinces during 2000–2010 and became even worse in the next decade because of occupying superior arable land while compensating for inferior arable land. The conservation of arable land productivity has become more important than the conservation of quantity. Third, due to the lack of constraints on the farming distance changes in the “requisition-compensation balance policy”, most provinces convert arable land around urban and rural areas to built-up land while replenishing land far from residential areas without providing adequate agricultural infrastructure, which leads to an increase in the cost of farming and consequently to an increase in the marginalization of arable land. During 2010–2020, the average farming distance of compensated arable land at the provincial level reached 2–7 times that of occupied arable land. However, the average farming distance of the whole arable land at the provincial level decreased by 3.82–63.88% during the same period. This contradiction is mainly due to increasing marginalization and opportunity costs resulting in arable land with high farming distance (including arable land that was compensated in the past) to be used with low intensity or even abandoned and thus identified as other land use types by remote sensing classification models. This factor outweighed that of “occupy nearby arable land while compensating farther one”, resulting in a reduction in the provincial average farming distance. Fourth, the percentage of sustainable compensated arable land in most provinces was lower than 70%. This indicated that the utilization and protection of arable land in these provinces was insufficiently implemented and monitored. Challenges along optimization of the “requisition-compensation balance policy” were discussed from two respects: data, theory and methodology and policy design and implementation. The authors argue that a more comprehensive “requisition-compensation balance policy” should be designed considering not only the quantity and productivity of arable land but also the farming distance, sustainability and ecological protection. A differentiated regulation mechanism of arable land requisition compensation in trans-provincial areas should be formulated. This study can provide guidance for optimizing the implementation of regional arable land protection and can also provide a reference for other countries to protect arable land.

Medial transfer of the tibial tubercle has become a standard procedure in cases of patella instability caused by an increased tuberositas tibae-trochlear groove (TT-TG) distance. However, the TT-TG distance has always been assessed as an absolute value without taking individual joint size into consideration. It was assumed that the pathological influence of the TT-TG distance correlates with individual joint size. Aim of the current study therefore was to develop a method to express TT-TG distance in relation to these joint variables. Two hundred knee MRI scans of healthy individuals (69 females and 131 males) were evaluated retrospectively in a control group. First, the TT-TG distance was measured as described by Schoettle et al. To determine joint size, the proximal-distal distance between the entrance of the chondral trochlear groove (TE) and the onset of the patella tendon at the tibial tubercle (TT) was selected. Subsequently, the TT-TG/TT-TE ratio expresses the relationship between the TT-TG distance and the proximal-distal distance from the entrance of the chondral trochlear groove to the height of the tibial tubercle. The TT-TG Index can also be expressed as an angle (TT-TG angle). Likewise, in another patient group, 54 knee MRTs of patients with patellofemoral instability were evaluated. The average TT-TG distance of the control group was 7.5±3.5 mm (range 0-17.4 mm) with no significant differences between genders. The mean TT-TE distance was 63.9 mm (range 49-79 mm) with there being significant differences between genders. The resulting mean TT-TG Index was 0.12±0.05 (range 0-0.25). In the patient group, the average TT-TG distance was 13.5±4.1 mm and the average TT-TE distance was 61.3±6.8 mm. The resulting average TT-TG Index of 0.22±0.07 in the patient group (PFI) approximates the threshold determined by the 95% confidence interval in the healthy control group. A direct comparison between the control group and the patient group revealed a significant difference in the TT-TG distance (p=0.0001), in the TT-TE distance (p<0.0042) and in the resulting TT-TG Index (p<0.0001). The measurement of the TT-TG Index is a reliable and differentiated approach for determining the lateral displacement of the tibial tubercle in relation to the proximal trochlear groove. The pathological influence of the TT-TG distance in case of patella instability depends on individual joint size, confirming the initial hypothesis. We currently consider a TT-TG Index>0.23 to be pathological based on our findings. Particularly, in case of a marginal TT-TG distance, the additional relative TT-TG Index facilitates a decision concerning an indication for a operative medial transfer of the tibial tubercle. II.

Laplacian spectra of regular graph transformations

Given a graph G with vertex set V(G) = V and edge set E(G) = E, let L(G) be the line graph and Ḡ the complement of G. Let G0 be the graph with V(G0) = V and with no edges, G1 the complete graph with the vertex set V, G+ = G and G– = Ḡ. Let S(G) (S*(G)) be the graph with the vertex set V ∪ E such that two vertices of S(G) (S*(G)) are adjacent if and only if one corresponds to a vertex v of G and other to an edge e of G and v is incident (resp., not incident) to e in G. Given x, y, z ∈{0, 1, +, –}, the generalized xyz–Point-Line transformation graph Txyz(G) of G is the graph with vertex set V(Txyz(G)) = V ∪ E and the edge set E(Txyz(G)) = E(Gx) ∪ E(L(G))y ∪ E(W), where W = S(G) if z = +, W = S*(G) if z = –, W is the graph with V(W) = V ∪ E and with no edges if z = 0 and W is complete bipartite graph with parts V and E if z = 1. In this paper, we obtain order, size, connectedness and diameter of generalized xyz– Point-Line transformation graphs.

Background: In Azerbaijan, cow pea is one of the fundamental legumes. This study was carried out at the Institute of Genetic Resources, Azerbaijan National Academy of Sciences to evaluate genetic diversity in 28 cowpea accessions from Azerbaijan national gene bank. The aim of this study was to evaluate the genotypic diversity, relatedness and population structure among local cow pea resources available in Azerbaijan and the cultivated VIR accessions based on ISSR markers. Methods: The studies were conducted in 2017-2019 at the Institute of Genetic Resources (IGR) of the National Academy of Sciences (NAS) of Azerbaijan. 28 samples were used as research material: 10 of them were local forms and 18 were samples obtained from VIR. Sowing of collection samples was carried out in duplicate with an area of food of one plant 10 x 45 cm at the optimum time, in the at the end of April. Result: In the study, 28 alleles were synthesized for the collection with five ISSR markers, with an average polymorphism index of 75%. The rich genetic diversity (GMO = 0.80; PIC = 0.34) was found in the collection. Statistical analyzes have shown that the efficiency of the UBC 835 and the UBC 818 premieres was high. The average genetic distance index in the cow pea collection was 0.33. 71% of the samples were genetically identifiable and 29% revealed that the genotype was identical across the studied loci. PCoA and cluster analysis identified 3 genetically distinct groups within the collection, with the first two coordinates explaining 57% of this genetic variation.