On the number of unlabeled vertices in edge-friendly labelings of graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, and $f$ be a 0-1 labeling of $E(G)$ so that the absolute difference in the number of edges labeled 1 and 0 is no more than one. Call such a labeling $f$ \emph{edge-friendly}. We say an edge-friendly labeling induces a \emph{partial vertex labeling} if vertices which are incident to more edges labeled 1 than 0, are labeled 1, and vertices which are incident to more edges labeled 0 than 1, are labeled 0. Vertices that are incident to an equal number of edges of both labels we call \emph{unlabeled}. Call a procedure on a labeled graph a \emph{label switching algorithm} if it consists of pairwise switches of labels. Given an edge-friendly labeling of $K_n$, we show a label switching algorithm producing an edge-friendly relabeling of $K_n$ such that all the vertices are labeled. We call such a labeling \textit{opinionated}.Comment: 7 pages, accepted to Discrete Mathematics, special issue dedicated to Combinatorics 201

4−Equitable Tree Labelings

We assign the labels {0,1,2,3} to the vertices of a graph; each edge is assigned the absolute difference of the incident vertices’ labels. For the labeling to be 4−equitable, we require the edge labels and vertex labels to each be distributed as uniformly as possible. We study 4−equitable labelings of different trees and prove all cater-pillars, symmetric generalized n−stars (or symmetric spiders), and complete n −ary trees for all n ∈ N are 4−equitable

Spectral correlations in systems undergoing a transition from periodicity to disorder

We study the spectral statistics for extended yet finite quasi 1-d systems which undergo a transition from periodicity to disorder. In particular we compute the spectral two-point form factor, and the resulting expression depends on the degree of disorder. It interpolates smoothly between the two extreme limits -- the approach to Poissonian statistics in the (weakly) disordered case, and the universal expressions derived for the periodic case. The theoretical results agree very well with the spectral statistics obtained numerically for chains of chaotic billiards and graphs.Comment: 16 pages, Late

Some Graph Laplacians and Variational Methods Applied to Partial Differential Equations on Graphs

In this dissertation we will be examining partial differential equations on graphs. We start by presenting some basic graph theory topics and graph Laplacians with some minor original results. We move on to computing original Jost graph Laplacians of friendly labelings of various finite graphs. We then continue on to a host of original variational problems on a finite graph. The first variational problem is an original basic minimization problem. Next, we use the Lagrange multiplier approach to the Kazdan-Warner equation on a finite graph, our original results generalize those of Dr. Grigor’yan, Dr. Yang, and Dr. Lin. Then we do an original saddle point approach to the Ahmad, Lazer, and Paul resonant problem on a finite graph. Finally, we tackle an original Schrödinger operator variational problem on a locally finite graph inspired by some papers written by Dr. Zhang and Dr. Pankov. The main keys to handling this difficult breakthrough Schrödinger problem on a locally finite graph are Dr. Costa’s definition of uniformly locally finite graph and the locally finite graph analog Dr. Zhang and Dr. Pankov’s compact embedding theorem when a coercive potential function is used in the energy functional. It should also be noted that Dr. Zhang and Dr. Pankov’s deeply insightful Palais-Smale and linking arguments are used to inspire the bulk of our original linking proof

The Swallowing Characteristics of Thickeners, Jellies and Yoghurt Observed Using an In Vitro Model

© The Author(s) 2019 Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.Drinks and foods may be thickened to improve swallowing safety for dysphagia patients, but the resultant consistencies are not always palatable. Characterising alternative appetising foods is an important task. The study aims to characterise the in vitro swallowing behaviour of specifically formulated thickened dysphagia fluids containing xanthan gum and/or starch with standard jellies and yoghurt using a validated mechanical model, the “Cambridge Throat”. Observing from the side, the model throat can follow an experimental oral transit time (in vitro-OTT) and a bolus length (BL) at the juncture of the pharynx and larynx, to assess the velocity and cohesion of bolus flow. Our results showed that higher thickener concentration produced longer in vitro-OTT and shorter BL. At high concentration (spoon-thick), fluids thickened with starch-based thickener showed significantly longer in vitro-OTT than when xanthan gum-based thickener was used (84.5 s ± 34.5 s and 5.5 s ± 1.6 s, respectively, p < 0.05). In contrast, at low concentration (nectar-like), fluids containing xanthan gum-based thickener demonstrated shorter BL than those of starch-based thickener (6.4 mm ± 0.5 mm and 8.2 mm ± 0.8 mm, respectively, p < 0.05). The jellies and yoghurt had comparable in vitro-OTT and BL to thickeners at high concentrations (honey-like and spoon-thick), indicating similar swallowing characteristics. The in vitro results showed correlation with published in vivo data though the limitations of applying the in vitro swallowing test for dysphagia studies were noted. These findings contribute useful information for designing new thickening agents and selecting alternative and palatable safe-to-swallow foods.Peer reviewe

7. The 1970s

From View from the Dean’s Office by Robert McKersie. “I had been on the job just a week when Keith Kennedy, vice provost, called and said we needed to make a trip to Albany to meet the chancellor of SUNY, Ernest Boyer. This was late August 1971. After a few pleasantries, it became clear that this was not just the courtesy call of a new dean reporting in to the top leader of the state university. Chancellor Boyer went right to the point: a new Labor College was going to open on the premises of Local 3 IBEW’s training facility on Lexington Avenue in Manhattan, and the ILR School had to be there as a partner. It was not clear what unit of SUNY would take over the Labor College, but it was clear that given its broad mandate for labor education, the ILR School was going to play a key role.” Includes: View from the Dean’s Office; From Eric Himself; Another Perspective; Labor College Graduation: VanArsdale’s Dream Fulfilled; The View of a Visiting Faculty Member; Another Perspective; and The Student’s View

Color Preference and Personality Structure

The retina of the human eye is made up of ten layers. One of these layers, the bacillary layer, is composed of 130 million rods and 7 million cones.(1) It would appear, in view of the unusually large proportion of rods to cones, that the rods were of much greater importance to the process of seeing than the cones,-— but this is not the case at all. The cones, it is true, are very thinly scattered throughout the peripheral area of the retina; so thinly scattered, in fact, as to appear almost entirely lacking. The proportion of cones increases, however, as the visual axis is approached. They become the exclusive element of the macula, that localized area directly in line with the visual axis.(2) Thus, the focal point of vision, plus a reasonable area of the retina surrounding it, is almost exclusively made up of cones. Are the cones, i.e., the agents of color vision, important to visual perception? They certainly are. They are, by far, the most influential factor in the process of seeing

4−Equitable Tree Labelings

We assign the labels {0,1,2,3} to the vertices of a graph; each edge is assigned the absolute difference of the incident vertices’ labels. For the labeling to be 4−equitable, we require the edge labels and vertex labels to each be distributed as uniformly as possible. We study 4−equitable labelings of different trees and prove all cater-pillars, symmetric generalized n−stars (or symmetric spiders), and complete n −ary trees for all n ∈ N are 4−equitable

Normal Editions Workshop Newsletter, 1999

Annual newsletter for the Normal Editions Workshop, School of Art, Illinois State University.https://ir.library.illinoisstate.edu/new/1016/thumbnail.jp