Let G be a finite non-abelian group and denote by Z(G) its center. The non-commuting graph of on transversal the center is whose vertices are non-central elements in two x y adjacent whenever xy=yx. In this work, we classify groups centeris double-toroidal or 1-planar.

An (L, d)∗-coloring is a mapping φ that assigns a color φ(v) ∈ L(v) to each vertex v ∈ V (G) such that at most d neighbors of v receive colore φ(v). A graph is called (m, d)∗-choosable, if G admits an (L, d)∗-coloring for every list assignment L with |L(v)| ≥ m for all v ∈ V (G). In this note, it is proved that every toroidal graph, which contains no adjacent triangles and contains no 6-cycles ...

In the final project paper we consider a graph parameter called readability. Motivation for readability comes from bioinformatics applications. Graphs arising in problems related to genome sequencing are of small readability, which motivates the study of graphs of small readability. We present an algorithm due to Braga and Meidanis, which shows that every digraph is isomorphic to the overlap gr...

a toroidalization of a dominant morphism $varphi: xto y$ of algebraic varieties over a field of characteristic zero is a toroidal lifting of $varphi$ obtained by performing sequences of blow ups of nonsingular subvarieties above $x$ and $y$. we give a proof of toroidalization of locally toroidal morphisms of 3-folds.

We present a polynomial time algorithm to find the maximum weight of an edge-cut in graphs embeddable on an arbitrary orientable surface, with integral weights bounded in the absolute value by a polynomial of the size of the graph. The algorithm has been implemented for toroidal grids using modular arithmetics and the generalized nested dissection method. The applications in statistical physics...

A Bayesian multiscale technique for detection of statistically significant features in noisy images is proposed. The prior is defined as a stationary intrinsic Gaussian Markov random field on a toroidal graph, which enables efficient computation of the relevant posterior marginals. Hence the method is applicable to large images produced by modern digital cameras. The technique is demonstrated i...

It is shown that if a graph of n vertices can be drawn on the torus without edge crossings and the maximum degree of its vertices is at most d, then its planar crossing number cannot exceed cdn, where c is a constant. This bound, conjectured by Brass, cannot be improved, apart from the value of the constant. We strengthen and generalize this result to the case when the graph has a crossing-free...

Let L be any infinite biperiodic alternating link. We show that for any sequence of finite links that Følner converges almost everywhere to L, their determinant densities converge to the Mahler measure of the 2–variable characteristic polynomial of the toroidal dimer model on an associated biperiodic graph.

For integers m; d; D with m 3; d 2; and D 2, let T(m) be a 2{dimensional quadratic toroidal grid with side length m and let B(d;D) be the base d, dimension D de Bruijn graph; assume that jT(m)j = jB(d;D)j. The starting point for our investigations is the observation that, for m; D even, embeddings f : T(m) ! B(d; D) with load 1, expansion 1, and dilation D=2 can easily be found (and have previo...