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 i...

Introduction. The purpose of this report is to draw attention to some nontrivial connections between the ("continuous") theory of irregularities of distribution and discrete mathematics. The object of the theory of irregularities of distribution is to measure the uniformity (or nonuniformity) of sequences and point sets. For instance: how uniformly can an arbitrary set of N points in the unit c...

PURPOSE To report refractive and visual acuity outcomes using wavefront-guided optical path difference custom ablation treatments (OPDCAT; NIDEK Co Ltd) for myopia and myopic astigmatism. METHODS One hundred eyes of 50 patients with preoperative manifest refraction spherical equivalent (MRSE) of 4.62+/-1.30 diopters (D) (range: -1.38 to -7.25 D) with mean astigmatism of -0.57+/-0.37 D (range:...

The total irregularity of a simple undirected graphG is defined as irrt(G) = 1 2 ∑ u,v∈V (G) |dG(u)− dG(v)|, where dG(u) denotes the degree of a vertex u ∈ V (G). Obviously, irrt(G) = 0 if and only if G is regular. Here, we characterize the non-regular graphs with minimal total irregularity and thereby resolve the recent conjecture by Zhu, You and Yang [18] about the lower bound on the minimal ...

A new formulation of Randić index is carried out as a minimization of a quadratic form which involves the Laplacian matrix of a graph. Using this formulation it is easy to realize that Randić index is useful in defining a new index of irregularity, which is then formulated here. A new context for the study of irregularity of graphs is advanced as a necessity for studying complex (biological) ne...

PURPOSE To investigate the characteristics of the corneal topography with three-dimensional (3-D) anterior segment optical coherence tomography (AS-OCT) following Descemet's stripping automated endothelial keratoplasty (DSAEK). METHODS Thirty-four eyes of 33 patients following DSAEK were studied. In addition to conventional topographic maps, the elevation map of the intrastromal interface and...

In this paper, we study the total edge irregularity strength of some well known graphs. An edge irregular total k-labeling φ : V ∪E → {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 are distinct. The total edge irregularity strength tes(G) is defined as the minimum k for which G has an edge irre...

In modern urban construction, irregular buildings are increasingly constructed to fulfill architectural and functional requirements. However, these revealed unfavorable seismic performance during the past earthquake records. When design codes deal with issue of building irregularity, little attention is paid location irregularity. current study, a detailed structural analysis was performed inve...

purpose: to determine the sensitivity and specificity of posterior corneal elevation in discriminating keratoconus and subclinical keratoconus from normal corneas by galilei and orbscan ii diagnosed on the basis of topography. methods: three groups (keratoconus, subclinical keratoconus and normal) were evaluated with galilei and orbscan ii and results were compared to those obtained by gold sta...