Let Γ=(VΓ,EΓ) be a simple undirected graph with finite vertex set VΓ and edge EΓ. A total n-labeling α:VΓ∪EΓ→{1,2,…,n} is called irregular labeling on Γ if for any two different edges xy x′y′ in EΓ the numbers α(x)+α(xy)+α(y) α(x′)+α(x′y′)+α(y′) are distinct. The smallest positive integer n such that can labeled by irregularity strength of Γ. In this paper, we provide some asymmetric graphs sym...

We investigate a modification of well known irregularity strength of graph, namely the total edge irregularity strength. An edge irregular total k-labeling φ : V ∪E → {1, 2, . . . , k} of a graph G is a labeling of vertices and edges of G in such a way that for any two different edges uv and u′v′ their weights φ(u)+φ(uv)+φ(v) and φ(u′)+φ(u′v′)+φ(v′) are distinct. The total edge irregularity str...

OBJECTIVES The aim of this study was to assess the long-term stability of treatment with self-ligating brackets compared with conventional brackets. MATERIALS AND METHODS The long-term follow-up retrospective study sample consisted of two groups of patients: group SL (including passive and interactive self ligating braces) comprised 30 subjects treated with self-ligating brackets at a mean pr...

Let ( , ) G V E be a simple graph. For a total labeling { } : 1,2,3,..., V E k ∂ ∪ → the weight of a vertex x is defined as ( ) ( ) ( ). xy E wt x x xy ∈ = ∂ + ∂ ∑ ∂ is called a vertex irregular total k-labeling if for every pair of distinct vertices x and y, ( ) ( ). wt x wt y ≠ . The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularit...

It is proposed to use irregularity, the scale invariant index based on the ideas of fractal geometry to assess the spatial features of font drawings. The index is sensitive to the shape of characters in the font, which affects text legibility. Preliminary results have shown promising application of the proposed index for classifying fonts by reading speeds.

As a standard notation, assume that G = G(V,E) is a finite, simple and undirected graph with p vertices and q edges. A labeling of a graph is any mapping that sends some set of graph elements to a set of numbers (usually positive integers). If the domain is the vertex-set or the edge-set, the labelings are called respectively vertex-labelings or edge-labelings. If the domain is V ∪E then we cal...

A total vertex irregular k-labeling φ of a graph G is a labeling of the vertices and edges of G with labels from the set {1, 2, . . . , k} in such a way that for any two different vertices x and y their weights wt(x) and wt(y) are distinct. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x. The minimum k for which the graph G...

The correction and relapse of mandibular anterior crowding was evaluated in a population of 58 patients with Angle Class I malocclusion who were treated orthodontically without extraction of permanent teeth. The subjects were retrospectively evaluated from records taken before treatment, posttreatment, and postretention. The postretention period averaged 8 years (minimum of 4 and maximum of 20 ...

PURPOSE The aim of this study was to identify the relationship between awareness of nutrition labeling and menstrual cycle irregularity in women from a nationally representative sample of the Korean population. METHODS A cross-sectional analysis was performed using hierarchical multivariable logistic regression analysis models. A total of 4,324 women aged 19~54 years from the 2010~2012 Korean...

The aim of this study was to evaluate the relationship between the tongue volume and lower incisor irregularity, using cone-beam computed tomography (CBCT), and to identify the possible gender differences. CBCT images of 60 patients between 16 and 36 years of age were selected from 1400 sets of images in the database. Tomography was carried out using iCAT® (Imaging Sciences International, Hatfi...