Total Labeling Research Articles

How to Construct Super Edge-Magic Total Labeling of Theta Graph θ(2,b,c)

This research study and provide the property of super edge-magic total labelings of theta graph. Edge magic labeling on a graph is an injective function from to a subset of integers { } with the property that there is a positive integer such as for each . An edge-magic labeling is called super edge-magic total labeling if it satisfies . A graph is called (super) edge-magic total if it admits some (super) edge-magic total labeling. A theta graph is constructed by embedding the endpoints of three paths of length consecutive such that there are two vertices of degree three and the other of degree two. This study gave some conditions for such a super edge-magic total of theta graph. Based on this condition, this paper introduce some algorithms to apply and develop super edge-magic total labeling from some previous theta graphs.

Understanding subcortical projections to the lateral posterior thalamic nucleus and its subregions using retrograde neural tracing.

The rat lateral posterior thalamic nucleus (LP) is composed of the rostromedial (LPrm), lateral (LPl), and caudomedial parts, with LPrm and LPl being areas involved in information processing within the visual cortex. Nevertheless, the specific differences in the subcortical projections to the LPrm and LPl remain elusive. In this study, we aimed to reveal the subcortical regions that project axon fibers to the LPl and LPrm using a retrograde neural tracer, Fluorogold (FG). After FG injection into the LPrm or LPl, the area was visualized immunohistochemically. Retrogradely labeled neurons from the LPrm were distributed in the retina and the region from the diencephalon to the medulla oblongata. Diencephalic labeling was found in the reticular thalamic nucleus (Rt), zona incerta (ZI), ventral lateral geniculate nucleus (LGv), intergeniculate leaflet (IGL), and hypothalamus. In the midbrain, prominent labeling was found in the periaqueductal gray (PAG) and deep layers of the superior colliculus. Additionally, retrograde labeling was observed in the cerebellar and trigeminal nuclei. When injected into the LPl, several cell bodies were labeled in the visual-related regions, including the retina, LGv, IGL, and olivary pretectal nucleus (OPT), as well as in the Rt and anterior pretectal nucleus (APT). Less labeling was found in the cerebellum and medulla oblongata. When the number of retrogradely labeled neurons from the LPrm or LPl was compared as a percentage of total subcortical labeling, a larger percentage of subcortical inputs to the LPl included projections from the APT, OPT, and Rt, whereas a large proportion of subcortical inputs to the LPrm originated from the ZI, reticular formation, and PAG. These results suggest that LPrm not only has visual but also multiple sensory-and motor-related functions, whereas the LPl takes part in a more visual-specific role. This study enhances our understanding of subcortical neural circuits in the thalamus and may contribute to our exploration of the mechanisms and disorders related to sensory perception and sensory-motor integration.

Early postnatal development of the primary visual areas 17 and 18 of the cat cerebral cortex: An SMI-32 study.

Using anti-neurofilament H non-phosphorylated antibodies (SMI-32) as markers for the neuronal maturation level and Y channel responsible for motion processing, we investigated early postnatal development of the primary visual areas 17 and 18 in cats aged 0, 10, 14, and 34 days and in adults. Two analyzed parameters of SMI-32-immunolabeling were used: the total proportion of SMI-32-labeling and the density of labeled neurons. (i) The developmental time course of the total proportion of SMI-32-labeling shows the general increase in the accumulation of heavy-chain neurofilaments. This parameter showed a different time course for cortical layer development; the maximal increment in the total labeling in layer V occurred between the second and fifth postnatal weeks and in layers II-III and VI after the fifth postnatal week. In addition, the delay in accumulation of SMI-32-labeling was shown in layer V of the area 17 periphery representation during the first two postnatal weeks. (ii) The density of SMI-32-labeled neurons decreased in all layers of area 18, but was increased, decreased, or had a transient peak in layers II-III, V, and VI of area 17, respectively. The transient peak is in good correspondence with some transient neurochemical features previously revealed for different classes of cortical and thalamic neurons and reflects the time course of the early development of the thalamocortical circuitry. Some similarities between the time courses for the development of SMI-32-labeling in areas 17/18 and in A- and C-laminae of the LGNd allow us to propose heterochronous postnatal development of two Y sub-channels.

Total Labelings of Graphs with Prescribed Weights

The total labeling of a graph G = ( V , E ) is a bijection from the union of the vertex set and the edge set of G to the set { 1 , 2 , … , | V ( G ) | + | E ( G ) | } . The edge-weight of an edge under a total labeling is the sum of the label of the edge and the labels of the end vertices of that edge. The vertex-weight of a vertex under a total labeling is the sum of the label of the vertex and the labels of all the edges incident with that vertex. A total labeling is called edge-magic or vertex-magic when all the edge-weights or all the vertex-weights are the same, respectively. When all the edge-weights or all the vertex-weights are different then a total labeling is called edge-antimagic or vertex-antimagic total, respectively. In this paper we deal with the problem of finding a~total labeling of some classes of graphs that is simultaneously vertex-magic and edge-antimagic or simultaneously vertex-antimagic and edge-magic, respectively. We show several results for stars, paths and cycles.

\(4\)-Total Prime Cordial Labeling of Some Derived Graphs

Let \(G\) be a \((p, q)\) graph. Let \(f: V(G) \to \{1, 2, \ldots, k\}\) be a map where \(k \in \mathbb{N}\) is a variable and \(k > 1\). For each edge \(uv\), assign the label \(\gcd(f(u), f(v))\). \(f\) is called \(k\)-Total prime cordial labeling of \(G\) if \(\left|t_{f}(i) – t_{f}(j)\right| \leq 1\), \(i, j \in \{1, 2, \ldots, k\}\) where \(t_{f}(x)\) denotes the total number of vertices and edges labeled with \(x\). A graph with a \(k\)-total prime cordial labeling is called \(k\)-total prime cordial graph. In this paper, we investigate the 4-total prime cordial labeling of some graphs like dragon, Möbius ladder, and corona of some graphs.

Super Edge-magic Total Strength of Some Unicyclic Graphs

Let \(G\) be a finite simple undirected \((p, q)\)-graph, with vertex set \(V(G)\) and edge set \(E(G)\) such that \(p = |V(G)|\) and \(q = |E(G)|\). A super edge-magic total labeling \(f\) of \(G\) is a bijection \(f \colon V(G) \cup E(G) \longrightarrow \{1, 2, \dots, p+q\}\) such that for all edges \(uv \in E(G)\), \(f(u) + f(v) + f(uv) = c(f)\), where \(c(f)\) is called a magic constant, and \(f(V(G)) = \{1, \dots, p\}\). The minimum of all \(c(f)\), where the minimum is taken over all the super edge-magic total labelings \(f\) of \(G\), is defined to be the super edge-magic total strength of the graph \(G\). In this article, we work on certain classes of unicyclic graphs and provide evidence to conjecture that the super edge-magic total strength of a certain family of unicyclic \((p, q)\)-graphs is equal to \(2q + \frac{n+3}{2}\).

Sharp Bounds on Vertex N-magic Total Labeling Graphs

Sharp Bounds on Vertex N-magic Total Labeling Graphs

Evaluating convolutional neural network-enhanced electrocardiography for hypertrophic cardiomyopathy detection in a specialized cardiovascular setting.

The efficacy of convolutional neural network (CNN)-enhanced electrocardiography (ECG) in detecting hypertrophic cardiomyopathy (HCM) and dilated HCM (dHCM) remains uncertain in real-world applications. This retrospective study analyzed data from 19,170 patients (including 140 HCM or dHCM) in the Shinken Database (2010-2017). We evaluated the sensitivity, positive predictive rate (PPR), and F1 score of CNN-enhanced ECG in a ''basic diagnosis'' model (total disease label) and a ''comprehensive diagnosis'' model (including disease subtypes). Using all-lead ECG in the "basic diagnosis" model, we observed a sensitivity of 76%, PPR of 2.9%, and F1 score of 0.056. These metrics improved in cases with a diagnostic probability of ≥ 0.9 and left ventricular hypertrophy (LVH) on ECG: 100% sensitivity, 8.6% PPR, and 0.158 F1 score. The ''comprehensive diagnosis'' model further enhanced these figures to 100%, 13.0%, and 0.230, respectively. Performance was broadly consistent across CNN models using different lead configurations, particularly when including leads viewing the lateral walls. While the precision of CNN models in detecting HCM or dHCM in real-world settings is initially low, it improves by targeting specific patient groups and integrating disease subtype models. The use of ECGs with fewer leads, especially those involving the lateral walls, appears comparably effective.

Super (a,d)-P_2⨀P_m-Antimagic Total Labeling of Corona Product of Two Paths

Graph labeling involves mapping the elements of a graph (edges and vertices) to a set of positive integers. The concept of an anti-magic super outer labeling (a,d)-H pertains to assigning labels to the vertices and edges of a graph using natural numbers {1,2,3,...,p+q}. The weights of the outer labels H form an arithmetic sequence {a,a+d,a+2d,...,a+(k-1)d}, where 'a' represents the first term, 'd' is the common difference, and 'k' denotes the total number of outer labels, with the smallest label assigned to a vertex. This study investigates the super (a,d)-P_2⨀P_m-antimagic total labeling of the corona product P_n⨀P_m, where n and m are both greater than or equal to 3. We define the labeling functions for vertices and edges based on the partitioning of labels into three subsets. Using k-balanced and (k,δ)-anti balanced multisets, we demonstrate that for m being odd, P_n⨀P_m is super (9m^2 n+4mn+m-n+3,1)-P_2 ⨀▒P_(m ) -antimagic, and for m being even, P_n⨀P_m is super (9m^2 n+4mn+m-2n+5,3)-P_2 ⨀▒P_(m ) -antimagic. The labeling scheme is illustrated through examples. For the case when m is odd, an antimagic total labeling of P_3 ⨀▒P_3 forms a super (282,1)- P_2 ⨀▒P_(3 ) -antimagic labeling. In the case of even m, an antimagic total labeling of P_3 ⨀▒P_(4 ) results in a super (483,3)- P_2 ⨀▒P_(4 ) -antimagic labeling. Both of these examples provide insights into the antimagic properties of corona products.

ON THE TOTAL VERTEX IRREGULARITY STRENGTH OF SERIES PARALLEL GRAPH sp(m,r,4)

his study aims to determine the total vertex irregularity strength on a series parallel graph for and . Total labeling is said to be vertex irregular, if the weights for each vertices are different. Determination of the total vertex irregularity of series parallel graph is done by obtaining the largest lower bound and the smallest upper bound. The lower bound is obtained by analyzing the structure of the graph to obtain the largest minimum label of k and the upper bound is analyzed by labeling the vertices and edges of the graph, where the largest label is k and the values for each vertices weight is different. The result obtained for the total vertex irregularity strength of a series parallel graph is .

Assessment of subchondral bone microdamage quantification using contrast-enhanced imaging techniques.

Bone microdamage is common at subchondral bone (SCB) sites subjected to repeated high rate and magnitude of loading in the limbs of athletic animals and humans. Microdamage can affect the biomechanical behaviour of bone under physiological loading conditions. To understand the effects of microdamage on the mechanical properties of SCB, it is important to be able to quantify it. The extent of SCB microdamage had been previously estimated qualitatively using plain microcomputed tomography (μCT) and a radiocontrast quantification method has been used for trabecular bone but this method may not be directly applicable to SCB due to differences in bone structure. In the current study, SCB microdamage detection using lead uranyl acetate (LUA) and quantification by contrast-enhanced μCT and backscattered scanning electron microscopy (SEM) imaging techniques were assessed to determine the specificity of the labels to microdamage and the accuracy of damaged bone volume metrices. SCB specimens from the metacarpus of racehorses, with the hyaline articular cartilage (HAC) removed, were grouped into two with one group subjected to exvivo uniaxial compression loading to create experimental bone damage. The other group was not loaded to preserve the pre-existing invivo propagated bone microdamage. A subset of each group was stained with LUA using an established or a modified protocol to determine label penetration into SCB. The μCT and SEM images of stained specimens showed that penetration of LUA into the SCB was better using the modified protocol, and this protocol was repeated in SCB specimens with intact hyaline articular cartilage. The percentage of total label localised to bone microdamage was determined on SEM images, and the estimated labelled bone volume determined by μCT in SCB groups was compared. Label was present around diffuse and linear microdamage as well as oblique linear microcracks present at the articular surface, except in microcracks with high-density mineral infills. Bone surfaces lining pores with recent mineralisation were also labelled. Labelled bone volume fraction (LV/BV) estimated by μCT was higher in the absence of HAC. At least 50% of total labels were localised to bone microdamage when the bone area fraction (B.Ar/T.Ar) of the SCB was greater than 0.85 but less than 30% when B.Ar/T.Ar of the SCB was less than 0.85. To adjust for LUA labels on bone surfaces, a measure of the LV/BV corrected for bone surface area (LV/BV BS-1 ) was used to quantify damaged SCB. In conclusion, removal of HAC and using a modified labelling protocol effectively stained damaged SCB of the metacarpus of racehorses and represents a technique useful for quantifying microdamage in SCB. This method can facilitate future investigations of the effects of microdamage on joint physiology.

EDGE IRREGULAR REFLEXIVE LABELING ON LOBSTER GRAPH

Let $G$ be a lobster graph that have three layer of vertices where each layer is connected to each other. The total labeling of the graph is called an edge irregular reflexive $k$-labeling if the total weight of two incident vertices and the edge that joins it is different for all possible edges on the graph. In this paper, we will further discuss the minimum number of $k$ for this kind of labeling on lobster graph. In particular, we determine the exact value of the reflexive edge strength of lobster graph. To help illustrate it, we use Python code to generate the label for the graph.

Even vertex odd mean labeling of uniform theta graphs

Let $G$ be a graph with $p$ vertices and $q$ edges. A total graph labeling $ f:V(G)\bigcup E(G)\rightarrow \{0,1,2,3,...,2q\}$ is called even vertex odd mean labeling of a graph $G$ if the vertices of the graph $G$ label by distinct even integers from the set $\{0,2,...,2q\}$ and the labels of the edges are defined as the mean of the labels of its end vertices and these labels are $2q-1$ distinct odd integers from the set $\{1,3,5,...,2q-1\}$. In this paper we investigate the even vertex odd mean labeling of uniform theta graphs.

Graphical image of Trisomy Ultrascan related Total edge magic labelling

INTRODUCTION: The goal of this research is to investigate child syndromes at the overall level using total edge magic labelling. luckily discussed with chromosomal diseases consisting of Down's syndrome, the syndrome of Edwards, and Patau syndrome.
 OBJECTIVES: Ultrasound is used to check for Patau's, Edwards, and Down syndrome between 11 and 14 weeks of pregnancy. These syndromes can be determined before the baby is born. The name for trisomy 21 or Down syndrome. Trisomy 18 or Edwards syndrome; trisomy 13 or Patau syndrome.
 METHODS: The ultrasound screen test was converted to a graphical image, and Total edge magic labelling was implemented. A bijection from VUE to the numbers, {1, 2, 3, … p+q} with the characteristic that each everybody uv Ɛ E, Γ(u)+ Γ(uv)+ Γ(v) = Ψ for some constant Ψ, is known as Total edge magic labelling.
 RESULTS: The results of this test will determine the baby’s type of trisomy. This study's impartial was to assess the efficacy of screening for 21, 18, and 13 trisomies at the 12-week mark in pregnancy.
 CONCLUSION: The intended audience of this paper is a man or woman with a chromosomal disorder who should know about the health of their ancestors. A couple can go for genetic counselling and then plan for a baby.

On Super (a,d)-C_3- Antimagic Total Labeling of Dutch Windmill Graph D_3^m

This paper is aimed to investigate the existence of super (a,d)-C_3- antimagic total labeling of dutch windmill graph D_3^m . The methods to achieves the goal was taken in three step. First of all determine the edge and vertices notation on dutch windmill graph . At the second step, labeling the vertices and edges of several dutch windmill graphs, then obtained the pattern. Finally pattern must be proven to become theorem. Based on the study, The Dutch Windmill Graph D_3^m, with m>=2 has super (14m+9,5)-C_3- antimagic total labeling, super (13m+8,3)-C_3- antimagic total labeling, super (12m+9.,5)-C_3- antimagic total labeling, super (11m+10.,7)-C_3- antimagic total labeling, super (10m+8.,3)-C_3- antimagic total labeling.

Total Face Irregularity Strength of Certain Graphs

The edge k-labeling ψ of G is defined by a mapping from EG to a set of integers 1,2,…,k, where the integer weight assigned to the vertex x∈VG is given as wψx=∑ψxy, such that the sum is taken over every vertex of y∈VG that is adjacent to x and the integer weights of adjacent vertices must be distinct for all vertices with x≠y. An irregular assignment of G using atmost k labels which is considered to be a minimum k is defined as irregularity strength of a graph G and can be denoted as sG. There are also further works on familiar irregular assignments, such as edge irregular labelings, vertex irregular total labelings, edge irregular total labelings, and face irregular entire k-labelings of plane graphs. A plane graph can be defined as a graph that is embedded in the plane in which no two lines will be intersected. In a plane graph the number of regions present are called faces and we denote it as F. The concept of total face irregularity strength is defined by the motivation of irregular networks and entire irregular face k-labeling. In our paper, we have obtained a minimum bound for the total face irregularity strength of two-connected plane graphs like cycle-of-ladder, C-necklace graph, P-necklace graph, sibling tree, and triangular graph.

Himpunan Kritis pada Graf Bintang

Labeling is a one-to-one mapping that maps each element of a graph to Positive numbers called labels. One of its kind is edge-magic total labeling. Under special conditions, the results set of labeled graphs whose subsets are labeled and positioned, which builds the same graph as the labeling, is called the critical set. To obtain the critical set of a graph we must know the type of graph. In this study is a star graph. This study aims to determine the critical set in star graphs. The star graph used is the K1.5 star graph using center points 1, n + 1 and 2n + 1. The research results show that by labeling the total magic side of the K1.5 star graph with center point λ(c) = 1, the magic number k=14 is obtained. The possible critical set of K1.5 graphs is 120. In the K1.5 Star Graph with center point λ(c) = n+ 1, the magic number k=18 is obtained. The possible critical set of K1.5 graphs is 120. In the K1.5 Star Graph with center point λ(c) = 2n + 1, the magic number k=22 is obtained. The possible critical set of K1.5 graphs is 120.

Totally antimagic total labeling of ladders, prisms and generalised Petersen graphs

A total labeling is called edge-antimagic total (vertex-antimagic total) if all edge-weights (vertex-weights) are pairwise distinct. If a labeling is simultaneously edge-antimagic total and vertex-antimagic total, it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we prove that ladders, prisms and generalised Pertersen graphs are totally antimagic total graphs. We also show that the chain graph of totally antimagic total graphs is a totally antimagic total graphs.

On magic total labelings in combinatorial designs

Let H be a graph and ẞ be an H-decomposition of G. The H-magic total labeling of a graph G is an assignment of integers set [1, |V(G)∪E(G)|] to V(G)∪E(G) such that: any vertex and any edge of G receive distinct integers, and the sum of all vertices receive integers and all edges receive integers on B is a constant for any B ∈ ẞ. In this paper we study the H-super magic problems of balanced incomplete block designs and resolvable balanced incomplete block designs.

\((d,1)\)-Total Labelling of Generalized Petersen Graphs \(P(n,k)\)

In this paper, we investigate the ( d , 1 ) -total labelling of generalized Petersen graphs P ( n , k ) for d ≥ 3 . We find that the ( d , 1 ) -total number of P ( n , k ) with d ≥ 3 is d + 3 for even n and odd k or even n and k = n 2 , and d + 4 for all other cases.