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The multi-dimensional torus is one of the most popular underlying topologies for massively parallel systems. In this study, we consider a non-bipartiten-dimensional torus wheren≥2 and prove that for 1≤m≤2n, mvertex disjoint paths exist that cover all vertices between any two distinct vertices. In other words, we construct the one-to-onem-disjoint path cover of a non-bipartite torus for anymwhere 1≤m≤2n. [ABSTRACT FROM PUBLISHER]

The pessimistic strategy, also called the t 1/t 1-diagnosis strategy, allows to contain all faulty vertices and at most one fault-free vertex. However, the degree of the diagnosability of a system increases quickly using the pessimistic strategy. In this paper, we consider the k-ary n-cube, which is an important hypercube variant, and show that the 3-ary n-cube and k-ary n-cube with k≥4 are (4n−3)/(4n−3)-diagnosable and (4n−2)/(4n−2)-diagnosable, respectively. Finally, we obtain the degree of the diagnosability of the tori using the pessimistic strategy. [ABSTRACT FROM PUBLISHER]

A vertex set D in a graph G is called a geodetic set if each vertex of G is lying on some shortest u-v path of G, where u, v∈D. The geodetic number of a graph G is the minimum cardinality among all geodetic sets. A subset S of a geodetic set D is called a forcing subset of D if D is the unique geodetic set containing S. The forcing geodetic number of D is the minimum cardinality of a forcing subset of D, and the lower and the upper forcing geodetic numbers of a graph G are the minimum and the maximum forcing geodetic numbers, respectively, among all minimum geodetic sets of G. In this article, we find out the geodetic numbers, the lower and the upper forcing geodetic numbers of complete n-partite graphs, n-dimensional meshes and tori. [ABSTRACT FROM AUTHOR]

Generalized helicenes are defined. A theoretical analysis is presented for the intramolecular dehydrogenation of helicenes, bridging adjacent rungs of the helix with the formation of 4-, 5-, and/or 6-membered rings, yielding helical tubulenes to which Euler's formula for polyhedra no longer applies as such constructions are derived from Möbius strips and correspond to nonplanar graphs (i.e., have crossing lines when drawn in two dimensions). Intermolecular dehydrogenation of two parallel-superimposed polycyclic aromatic hydrocarbons with formation of similar rings leads to convex polyhedral carbon cages. Carbon helical nanotubes and some carbon cages that are different from fullerenes are also discussed. [ABSTRACT FROM AUTHOR]

A 56 year old African-American female with history of well-controlled hypertension and hyperlipidemia presented to the office for an annual physical examination. She did not have any complaints. She reported being compliant with her medications, exercised daily at her local gym, and maintained a low salt diet. She visits her dentist every 6 months and has had a few fillings in her premolars in the past. On physical examination, her vital signs were normal and the entirety of her physical examination was normal with the exception of her oropharynx. Throat examination revealed a 2 × 1 cm midline hard palate swelling that was bony hard in consistency and covered by normally appearing oral mucosa. It was diagnosed as a torus palatinus. The patient was unaware of its presence and denied being informed about it by her dentist on any visit. She was also educated about the diagnosis and safety-netted by being informed about red-flags that would prompt investigation such as change in size or shape. [ABSTRACT FROM AUTHOR]