Radio numbers for generalized prism graphs

A radio labeling is an assignment $c:V(G) \rightarrow \textbf{N}$ such that every distinct pair of vertices $u,v$ satisfies the inequality d(u,v)+|c(u)-c(v)|\geq \diam(G)+1. The span of a radio labeling is the maximum value. The radio number of $G$, $rn(G)$, is the minimum span over all radio labelings of $G$. Generalized prism graphs, denoted $Z_{n,s}$, $s \geq 1$, $n\geq s$, have vertex set $\{(i,j)\,|\, i=1,2 \text{and} j=1,...,n\}$ and edge set $\{((i,j),(i,j \pm 1))\} \cup \{((1,i),(2,i+\sigma))\,|\,\sigma=-\left\lfloor\frac{s-1}{2}\right\rfloor\,\ldots,0,\ldots,\left\lfloor\frac{s}{2}\right\rfloor\}$. In this paper we determine the radio number of $Z_{n,s}$ for $s=1,2$ and $3$. In the process we develop techniques that are likely to be of use in determining radio numbers of other families of graphs.Comment: To appear in Discussiones Mathematicae Graph Theory. 16 pages, 1 figur

Modelling of Modular Robot Configurations Using Graph Theory

Modular robots are systems that can change its geometry or configuration when connecting more modules or when rearranging them in a different manner to perform a variety of tasks. Graph theory can be used to describe modular robots configurations, hence the possibility to determine the flexibility of the robot to move from one point to another. When the robot’s configurations are represented in a mathematical way, forward kinematics can be obtained

The Complexity of Change

Many combinatorial problems can be formulated as "Can I transform configuration 1 into configuration 2, if certain transformations only are allowed?". An example of such a question is: given two k-colourings of a graph, can I transform the first k-colouring into the second one, by recolouring one vertex at a time, and always maintaining a proper k-colouring? Another example is: given two solutions of a SAT-instance, can I transform the first solution into the second one, by changing the truth value one variable at a time, and always maintaining a solution of the SAT-instance? Other examples can be found in many classical puzzles, such as the 15-Puzzle and Rubik's Cube. In this survey we shall give an overview of some older and more recent work on this type of problem. The emphasis will be on the computational complexity of the problems: how hard is it to decide if a certain transformation is possible or not?Comment: 28 pages, 6 figure

Radio Graceful Labelling of Graphs

Radio labelling problem of graphs have their roots in communication problem known as \emph{Channel Assignment Problem}. For a simple connected graph $G=(V(G), E(G))$, a radio labeling is a mapping $f \colon V(G)\rightarrow \{0,1,2,\ldots\}$ such that $|f(u)-f(v)|\geq {\rm diam}(G)+1-d(u,v)$ for each pair of distinct vertices $u,v\in V(G)$, where $\rm{diam}(G)$ is the diameter of $G$ and $d(u,v)$ is the distance between $u$ and $v$. A radio labeling $f$ of a graph $G$ is a \emph{radio graceful labeling} of $G$ if $f(V(G)) = \{0,1,\ldots, |V(G)|-1\}$. A graph for which a radio graceful labeling exists is called \emph{radio graceful}. In this article, we study radio graceful labeling for general graphs in terms of some new parameters

On Regular Graphs Optimally Labeled with a Condition at Distance Two

For positive integers $j \geq k$, the $\lambda_{j,k}$-number of graph Gis the smallest span among all integer labelings of V(G) such that vertices at distance two receive labels which differ by at least k and adjacent vertices receive labels which differ by at least j. We prove that the $\lambda_{j,k}$-number of any r-regular graph is no less than the $\lambda_{j,k}$-number of the infinite r-regular tree $T_{\infty}(r)$. Defining an r-regular graph G to be $(j,k,r)$-optimal if and only if $\lambda_{j,k}(G) = \lambda_{j,k}(T_{\infty}(r))$, we establish the equivalence between $(j,k,r)$-optimal graphs and r-regular bipartite graphs with a certain edge coloring property for the case ${j \over k} \u3e r$. The structure of $r$-regular optimal graphs for ${j \over k} \leq r$ is investigated, with special attention to ${j \over k} = 1,2$. For the latter, we establish that a (2,1,r)-optimal graph, through a series of edge transformations, has a canonical form. Finally, we apply our results on optimality to the derivation of the $\lambda_{j,k}$-numbers of prisms

Eosinophilic bronchitis, eosinophilic granuloma, and eosinophilic bronchopneumopathy in 75 dogs (2006-2016).

BackgroundEosinophilic lung disease is a poorly understood inflammatory airway disease that results in substantial morbidity.ObjectiveTo describe clinical findings in dogs with eosinophilic lung disease defined on the basis of radiographic, bronchoscopic, and bronchoalveolar lavage fluid (BAL) analysis. Categories included eosinophilic bronchitis (EB), eosinophilic granuloma (EG), and eosinophilic bronchopneumopathy (EBP).AnimalsSeventy-five client owned dogs.MethodsMedical records were retrospectively reviewed for dogs with idiopathic BAL fluid eosinophilia. Information abstracted included duration and nature of clinical signs, bronchoscopic findings, and laboratory data. Thoracic radiographs were evaluated for the pattern of infiltrate, bronchiectasis, and lymphadenomegaly.ResultsThoracic radiographs were normal or demonstrated a bronchial pattern in 31 dogs assigned a diagnosis of EB. Nine dogs had intraluminal mass lesions and were bronchoscopically diagnosed with EG. The remaining 35 dogs were categorized as having EBP based on radiographic changes, yellow green mucus in the airways, mucosal changes, and airway collapse. Age and duration of cough did not differ among groups. Dogs with EB were less likely to have bronchiectasis or peripheral eosinophilia, had lower total nucleated cell count in BAL fluid, and lower percentage of eosinophils in BAL fluid compared to dogs in the other 2 groups. In contrast to previous reports, prolonged survival (>55 months) was documented in dogs with EG.Conclusions and clinical importanceDogs with eosinophilic lung disease can be categorized based on imaging, bronchoscopic and BAL fluid cytologic findings. Further studies are needed to establish response to treatment in these groups

Roman Domination in Complementary Prism Graphs

A Roman domination function on a complementary prism graph GGc is a function f : V [ V c ! {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number R(GGc) of a graph G = (V,E) is the minimum of Px2V [V c f(x) over such functions, where the complementary prism GGc of G is graph obtained from disjoint union of G and its complement Gc by adding edges of a perfect matching between corresponding vertices of G and Gc. In this paper, we have investigated few properties of R(GGc) and its relation with other parameters are obtaine

International Journal of Mathematical Combinatorics, Vol.6A

The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences

Radio Labelings of Distance Graphs

A radio $k$-labeling of a connected graph $G$ is an assignment $c$ of non negative integers to the vertices of $G$ such that $$|c(x) - c(y)| \geq k+1 - d(x,y),$$ for any two vertices $x$ and $y$, $x\ne y$, where $d(x,y)$ is the distance between $x$ and $y$ in $G$. In this paper, we study radio labelings of distance graphs, i.e., graphs with the set $\Z$ of integers as vertex set and in which two distinct vertices $i, j \in \Z$ are adjacent if and only if $|i - j| \in D$.Comment: 14 page

SEGUE: A Spectroscopic Survey of 240,000 Stars with g = 14-20

The Sloan Extension for Galactic Understanding and Exploration (SEGUE) Survey obtained ≈240,000 moderateresolution (R ~ 1800) spectra from 3900 Å to 9000 Å of fainter Milky Way stars (14.0 < g < 20.3) of a wide variety of spectral types, both main-sequence and evolved objects, with the goal of studying the kinematics and populations of our Galaxy and its halo. The spectra are clustered in 212 regions spaced over three quarters of the sky. Radial velocity accuracies for stars are σ(RV) ~ 4 km s^(−1) at g < 18, degrading to σ(RV) ~ 15 km s^(−1) at g ~ 20. For stars with signal-to-noise ratio >10 per resolution element, stellar atmospheric parameters are estimated, including metallicity, surface gravity, and effective temperature. SEGUE obtained 3500 deg^2 of additional ugriz imaging (primarily at low Galactic latitudes) providing precise multicolor photometry (σ(g, r, i) ~ 2%), (σ(u, z) ~ 3%) and astrometry (≈0".1) for spectroscopic target selection. The stellar spectra, imaging data, and derived parameter catalogs for this survey are publicly available as part of Sloan Digital Sky Survey Data Release 7