Irregular Labeling Research Articles

Group Irregular Labelings of Disconnected Graphs

We investigate the \textit{group irregularity strength} ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge labels at every vertex are distinct. We give the exact values and bounds on $s_g(G)$ for chosen families of disconnected graphs. In addition we present some results for the \textit{modular edge gracefulness} $k(G)$, i.e. the smallest value of $s$ such that there exists a function $f:E(G)\rightarrow \zet_s$ such that the sums of edge labels at every vertex are distinct.

The Total Edge Irregular Labeling of Network Constructed by Some Copies of Cycle on Three Vertices Corona a Vertex

Many networks have been found the total edge irregularity strength’s. The aim of this paper is determined the total edge irregularity strength of network constructed by some copies of cycle on three vertices corona a vertex. The results of this paper that the total edge irregularity strength of network constructed by some copies of cycle on three vertices corona a vertex is for where is the number copies of cycle on three vertices.Keywords: Corona Product of Graph, Irregular Labeling, Irregularity Strength

On distance irregular labeling of ladder graph and triangular ladder graph

On distance irregular labeling of ladder graph and triangular ladder graph

Irregular labelings of helm and sun graphs

A vertex irregular total k-labeling of a (p,q)-graph G=(V,E) is a labeling ϕ:V∪E→{1,2,…,k} such that the weights of the vertices wt(v)=ϕ(v)+∑uv∈Eϕ(uv) are different for all vertices. The total vertex irregularity strength tvs(G) is the minimum k for which G has a vertex irregular total k-labeling. The labeling ϕ is an edge irregular total k-labeling if for any two distinct edges e1=u1v1 and e2=u2v2, one has wt(e1)≠wt(e2) where wt(e1)=ϕ(u1)+ϕ(v1)+ϕ(u1v1). The total edge irregularity strength tes(G) is the minimum k for which G has an edge irregular total k-labeling. In this paper we determine tes(G) where G is the generalized helm and tvs(G) where G is the generalized sun graph.

Total Edge Lucas Irregular Labeling

For a graph ( ) total edge Lucas irregular labeling f :V(G) ?E (G) ? {1,2,…,K} is defined as a labeling on V(G) and E (G) in such a way that for any two different edges and , their weights ( ) ( ) ( ) and ( ) ( ) ( ) are distinct Lucas numbers.The total edge Lucas irregularity strength, tels(G), is defined as the minimum K for which G has total edge Lucas irregular labeling. In this paper we prove the graphs , Cn ,K1,n and Book (with 3 sides and 4 sides) admits total edge Lucas irregular labeling and we determine the total edge Lucas irregularity strength for those graphs.

A Survey of Irregularity Strength

This survey aims to give an overview of the modifications of the well-known irregular assignments, namely edge irregular labelings, vertex irregular and edge irregular total labelings, and face irregular entire labelings of graphs.

Total Irregularity Strength of Three Families of Graphs

We deal with the totally irregular total labeling which is required to be at the same time vertex irregular total and also edge irregular total. The minimum k for which a graph G has a totally irregular total k-labeling is called the total irregularity strength of G. In this paper, we estimate the upper bound of the total irregularity strength of graphs and determine the exact value of the total irregularity strength for three families of graphs.

Total Edge Fibonacci Irregular Labeling of some Star Graphs

A total edge Fibonacci irregular labeling f : V (G) S E(G) → {1, 2, . . . ,K} of a graph G = (V,E) is a labeling of vertices and edges of G in such a way that for any different edges xy and x 0 y 0 their weights f( x ) + f( xy ) + f( y ) and f( x 0 ) + f( x 0 y 0 ) + f( y 0 ) are distinct Fibonacci numbers. The total edge Fibonacci irregularity strength, tefs( G) is defined as the minimum K for which G has a total edge Fibonacci irregular labeling. If a graph has a total edge Fibonacci irregular labeling, then it is called a total edge Fibonacci irregular graph. In this paper, we proved K1,n , Bistar , subdivision of bistar and (1 ≤ i ≤ n) are total edge Fibonacci irregular graphs.

On the Total Irregularity Strength of Fan, Wheel, Triangular Book, and Friendship Graphs

Abstract A totally irregular total k-labeling λ: V ∪ E → {1, 2, · · ·, k} of a graph G is a total labeling such that G has a total edge irregular labeling and a total vertex irregular labeling at the same time. The minimum k for which a graph G has a totally irregular total k-labeling is called the total irregularity strength of G, denoted by ts(G). In this paper, we investigate some graphs whose total irregularity strength equals to the lower bound.

On Bounds for the Product Irregularity Strength of Graphs

For a graph $$X$$X with at most one isolated vertex and without isolated edges, a product-irregular labeling$$\omega :E(X)\rightarrow \{1,2,\ldots ,s\}$$?:E(X)?{1,2,?,s}, first defined by Anholcer in 2009, is a labeling of the edges of $$X$$X such that for any two distinct vertices $$u$$u and $$v$$v of $$X$$X the product of labels of the edges incident with $$u$$u is different from the product of labels of the edges incident with $$v$$v. The minimal $$s$$s for which there exist a product irregular labeling is called the product irregularity strength of $$X$$X and is denoted by $$ps(X)$$ps(X). In this paper it is proved that $$ps(X)\le |V(X)|-1$$ps(X)≤|V(X)|-1 for any graph $$X$$X with more than $$3$$3 vertices. Moreover, the connection between the product irregularity strength and the multidimensional multiplication table problem is given, which is especially expressed in the case of the complete multipartite graphs.

Total Edge Fibonacci-Like Sequence Irregular Labeling

In this paper, we are introducing total edge Fibonacci-like sequence irregular labeling. A total edge Fibonacci-like sequence irregular labeling f : V (G) S E(G) → {1, 2, . . . ,K} of a graph G = (V,E) is a labeling of vertices and edges of G in such a way that for any different edges xy and x 0 y 0 their weights f( x ) + f( xy ) + f( y ) and f( x 0 )+f( x 0 y 0 )+f( y 0 ) are distinct Fibonacci-like sequence numbers. The total edge Fibonacci-like sequence irregularity strength, tefls(G) is defined as the minimum K for which G has a total edge Fibonacci-like sequence irregular labeling. If a graph has a total edge Fibonacci-like sequence irregular labeling, then it is called a total edge Fibonaccilike sequence irregular graph. In this paper, we proved some standard graphs such as Pn and Cn and Book (with 3 and 4 sides) are total edge Fibonacci-like sequence irregular graphs.

Product irregularity strength of certain graphs

Consider a simple graph G with no isolated edges and at most one isolated vertex. A labeling w : E ( G ) → {1, 2, …, m } is called product - irregular , if all product degrees p d G ( v ) = ∏ e ∋ v w ( e ) are distinct. The goal is to obtain a product - irregular labeling that minimizes the maximal label. This minimal value is called the product irregularity strength and denoted p s ( G ) . We give the exact values of p s ( G ) for several families of graphs, as complete bipartite graphs K m , n , where 2 ≤ m ≤ n ≤ ( m + 2) choose 2, some families of forests, including complete d -ary trees, and other graphs with δ ( G ) = 1 .

Edge irregular total labeling of certain family of graphs

An edge irregular total k-labeling φ: V(G) ≼ E(G) → {1, 2,…, k} of a graph G = (V, E) is a labeling of vertices and edges of G in such a way that for any different edges xy and x'y' their weights φ(x) + φ(xy) + φ(y) and φ(x') + φ(x'y') + φ(y') are distinct. The total edge irregularity strength, tes(G), is defined as the minimum k for which G has an edge irregular total k-labeling.We have determined the exact value of the total edge irregularity strength of the categorical product of a cycle and a path.

Irregularity strength of digraphs

It is an elementary exercise to show that any non-trivial simple graph has two vertices with the same degree. This is not the case for digraphs and multigraphs. We consider generating irregular digraphs from arbitrary digraphs by adding multiple arcs. To this end, we define an irregular labeling of a digraph D to be an arc-labeling of the digraph such that the ordered pairs of the sums of the in-labels and out-labels at each vertex are all distinct. We define the strength s → ( D ) of D to be the smallest of the maximum labels used across all irregular labelings. Similar definitions for graphs have been studied extensively and a different formulation of digraph irregularity was given in [H. Hackett, Irregularity strength of graphs and digraphs, Masters Thesis, University of Louisville, 1995]. Here we continue the study of irregular labelings of digraphs. We give a general lower bound on s → ( D ) and determine s → ( D ) exactly for tournaments, directed paths and cycles and the orientation of the path where all vertices have either in-degree 0 or out-degree 0. We also determine the irregularity strength of a union of directed cycles and a union of directed paths, the latter which requires a new result pertaining to finding circuits of given lengths containing prescribed vertices in the complete symmetric digraph with loops.

Potential of the Mycoparasite, Verticillium lecanii, to Protect Citrus Fruit Against Penicillium digitatum, the Causal Agent of Green Mold: A Comparison with the Effect of Chitosan

ABSTRACT The potential of the mycoparasite, Verticillium lecanii, at protecting citrus fruits against green mold was explored at the cellular level. Treatmentthe fruit with V. lecanii or chitosan prior to inoculation with the causal agent of green mold, Penicillium digitatum, markedly reduced disease development compared with that of nontreated control citrus fruits in which symptoms were visible by 3 days after inoculation with the pathogen. Scanning electron microscope investigations of citrus samples, collected 5 days after inoculation with the pathogen, revealed striking differences in the extent of cell surface colonization between treated and nontreated fruits. Pathogen hyphae, which sporulated abundantly at the surface of control fruits, were collapsed and severely damaged in V. lecanii and chitosan-treated fruits. Histological observations of citrus samples confirmed that restriction of pathogen colonization at the cell surface correlated with a pronounced disorganization of the pathogen hyphae. In addition, host cell changes, mainly characterized by the deposition of a new material in the exocarp cells and the thickening of cell walls, were observed. Ultrastructural investigations of citrus samples revealed that the pathogen multiplied abundantly through much of the mesocarp and exocarp tissues in V. lecanii-free citrus fruits, whereas in V. lecanii-treated citrus, pathogen growth was restricted. Penicillium hyphae that penetrated the mesocarp tissue were markedly altered. Labeling with the wheat germ agglutinin/ovomucoid-gold complex for the localization of chitin resulted in an irregular labeling of Penicillium cell walls, even at a time when in an irregular labeling of Penicillium cell walls, even at a time when these were markedly altered. Cytochemical investigations revealed that callose and lignin-like compounds accumulated at sites of pathogen colonization in the exocarp tissue. Evidence is provided in this study that V. lecanii as well as chitosan are equally capable of inducing a striking response in P. digitatum-infected citrus fruits. The marked differences observed in the rate and extent of colonization as well as in pathogen cell viability between control and treated citrus fruits demonstrate that both treatments have the ability to induce the transcriptional activation of defense genes leading to the accumulation of structural and biochemical compounds at strategic sites.

Changes in the distribution of anionic sites in brain micro-blood vessels with and without amyloid deposits in scrapie-infected mice

Cationic colloidal gold (CCG) and scrapie-infected mouse brain samples embedded in Lowicryl K4M were used for ultrastructural localization of negatively charged microdomains (anionic sites) in the cerebral microvasculature. The distribution of anionic sites on both fronts (luminal and abluminal) of endothelial cells and in the basement membrane (BM) in the majority of micro-blood vessels (MBVs) located outside the plaque area and in the remaining cerebral cortex was similar to that which has been previously observed in non-infected animals. Some MBVs (especially capillaries), however, located inside the plaque areas and surrounded directly by amyloid fibers contained attenuated endothelium, the luminal surface of which showed a segmental lack or diminution of anionic sites. In these vessels the BM was frequently infiltrated and replaced by the amyloid fibers. In some vessels located mainly in the areas of the neuropil vacuolization deposits of homogenous material causing the thickening of the BM were noted. These changes were accompanied by irregular labeling of the BM with gold particles. At the sites of bifurcation of some MBVs, predominantly in the area of the venular estuary at the mouth of capillary (at capillary-venular connections), a discontinuity in the distribution of anionic sites was noted. The observed disturbances in the distribution of anionic sites can be associated with a previously noted increased permeability of some MBVs in the brains of scrapie-infected mice.

Protein synthesis by rat hippocampal slices maintained in vitro.

The present study evaluates protein synthesis in rat hippocampal slices maintained in vitro. Transverse slices of hippocampus were prepared from both adult rats and rat pups during postnatal development and incubated in a gassed (95% O2/5% CO2) balanced salt medium containing 5 nM 3H-leucine. The time course of 3H-leucine incorporation into TCA-precipitable protein was determined using slices removed from the media after 5, 10, 20, 30, 40, 60, and 120 min of incubation. The pattern of 3H-amino acid incorporation was evaluated by fixing slices with paraformaldehyde, embedding the slices in plastic, and sectioning the slices end on and en face for autoradiographic analysis. Biochemical analysis of 300 and 400 micron slices revealed that incorporation of leucine into protein proceeds at a constant rate. The autoradiographic analysis revealed that in adult hippocampal slices of 300-600 micron thickness there was complete penetration of 3H-leucine with no indication of a gradient in the extent of incorporation throughout the slice. The pattern of grain density within 300-600 micron slices matches that previously reported after in vivo injections of radiolabeled amino acid, where grain density is highest over neuronal cell bodies and lower over the laminae that contain dendritic processes and axons (Phillips et al: Mol Brain Res 2:251-261, 1987). Hippocampal slices of 200, 800, and 1,000 micron thickness showed irregular labeling. Slices of 200 micron were filled with pyknotic nuclei and vacuoles and exhibited patchy labeling. In 800 micron slices there were isolated areas of good preservation within the slice core, but these areas exhibited little incorporation. Relative to the 300-600 micron slices, there was a higher number of pyknotic nuclei and a much deeper layer of necrosis along the cut edges. Slices of 1,000 micron thickness showed poor preservation throughout and low levels of incorporation. Biochemical studies revealed a much higher rate of incorporation in the slices prepared from postnatal animals. Autoradiography of the slices from developing rats revealed that penetration was excellent and incorporation appeared to be greater as judged by an overall higher grain density. We believe that rat hippocampal slices provide a good in vitro model of protein metabolism that will be useful for studies of protein synthesis in isolated cell body and dendritic laminae and for the evaluation of whether protein synthesis in particular laminae is regulated by synaptic activity.

Platelet Life Span

In the past decade platelet transfusions into thrombocytopenic recipients indicated that the platelet had a curvilinear lifespan. These early studies showed that platelet transfusions could be used to demonstrate shortened lifespans of platelets in idiopathic acute and chronic thrombocytopenic purpura. In addition, progressive shortening of the platelet lifespan followed the administration of multiple transfusions.A variety of isotopes has been used to gather further information regarding platelet physiology in the non‐thrombocytopenic recipient. For the most part, irregular labeling or re‐utilization of the isotope has made these procedures unsatisfactory except for sodium chromate (Cr51).Modification of ACD anticoagulant has been helpful in eliminating the initial sequestration of platelets noted earlier with EDTA anticoagulant. These more precise measurements have confirmed the linear, senescent, lifespan of the human blood platelet. A comparison of platelet measurements in the recipient by numerical counting and radioactivity of platelets has indicated that the data may not be comparable. These differences may be related to in vivo elution of the Cr51 label. For complete understanding of the platelet lifespan investigators should confirm isotope data with numerical counting of platelets.