On graphs with cyclic defect or excess

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $\epsilon$ are called {\it graphs with defect or excess $\epsilon$}, respectively. While {\it Moore graphs} (graphs with $\epsilon=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n-B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs \emph{graphs with cyclic defect or excess}; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the M\"obius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.Comment: 20 pages, 3 Postscript figure

On graphs with cyclic defect or excess

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree d and diameter D = k and a lower bound on the order of a graph of minimum degree d and odd girth g = 2k + 1. Graphs missing or exceeding the Moore bound by ε are called graphs with defect or excess ε, respectively. While Moore graphs (graphs with ε = 0) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation Gd,k(A) = Jn +B (Gd,k(A) = Jn - B), where A denotes the adjacency matrix of the graph in question, n its order, Jn the n × n matrix whose entries are all 1's, B the adjacency matrix of a union of vertex-disjoint cycles, and Gd,k(x) a polynomial with integer coefficients such that the matrix Gd,k(A) gives the number of paths of length at most k joining each pair of vertices in the graph. In particular, if B is the adjacency matrix of a cycle of order n we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of O(64/3 d3/2) for the number of graphs of odd degree d ≥ 3 and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree d ≥ 3 and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree ≥ 3 and cyclic defect or excess exists

Manhattan orbifolds

We investigate a class of metrics for 2-manifolds in which, except for a discrete set of singular points, the metric is locally isometric to an L_1 (or equivalently L_infinity) metric, and show that with certain additional conditions such metrics are injective. We use this construction to find the tight span of squaregraphs and related graphs, and we find an injective metric that approximates the distances in the hyperbolic plane analogously to the way the rectilinear metrics approximate the Euclidean distance.Comment: 17 pages, 15 figures. Some definitions and proofs have been revised since the previous version, and a new example has been adde

On the impact of capillarity for strength at the nanoscale

The interior of nanoscale crystals experiences stress that compensates the capillary forces and that can be large, in the order of 1 GPa. Various studies have speculated on whether and how this surface-induced stress affects the stability and plasticity of small crystals. Yet, experiments have so far failed to discriminate between the surface contribution and other, bulk-related size effects. In order to clarify the issue, we study the variation of the flow stress of a nanomaterial while distinctly different variations of the two capillary parameters surface tension and surface stress are imposed under control of an applied electric potential. Our theory qualifies the suggested impact of $\textit{surface stress}$ as not forceful and instead predicts a significant contribution of the surface energy, as measured by the $\textit{surface tension}$. The predictions for the combined potential- and size dependence of the flow stress are quantitatively supported by the experiment. Previous suggestions, favoring the surface stress as the relevant capillary parameter, are not consistent with the experiment

On graphs of defect at most 2

In this paper we consider the degree/diameter problem, namely, given natural numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of vertices in a graph of maximum degree {\Delta} and diameter D. In this context, the Moore bound M({\Delta},D) represents an upper bound for N({\Delta},D). Graphs of maximum degree {\Delta}, diameter D and order M({\Delta},D), called Moore graphs, turned out to be very rare. Therefore, it is very interesting to investigate graphs of maximum degree {\Delta} \geq 2, diameter D \geq 1 and order M({\Delta},D) - {\epsilon} with small {\epsilon} > 0, that is, ({\Delta},D,-{\epsilon})-graphs. The parameter {\epsilon} is called the defect. Graphs of defect 1 exist only for {\Delta} = 2. When {\epsilon} > 1, ({\Delta},D,-{\epsilon})-graphs represent a wide unexplored area. This paper focuses on graphs of defect 2. Building on the approaches developed in [11] we obtain several new important results on this family of graphs. First, we prove that the girth of a ({\Delta},D,-2)-graph with {\Delta} \geq 4 and D \geq 4 is 2D. Second, and most important, we prove the non-existence of ({\Delta},D,-2)-graphs with even {\Delta} \geq 4 and D \geq 4; this outcome, together with a proof on the non-existence of (4, 3,-2)-graphs (also provided in the paper), allows us to complete the catalogue of (4,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2. Such a catalogue is only the second census of ({\Delta},D,-2)-graphs known at present, the first being the one of (3,D,-{\epsilon})-graphs with D \geq 2 and 0 \leq {\epsilon} \leq 2 [14]. Other results of this paper include necessary conditions for the existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 4, and the non-existence of ({\Delta},D,-2)-graphs with odd {\Delta} \geq 5 and D \geq 5 such that {\Delta} \equiv 0, 2 (mod D).Comment: 22 pages, 11 Postscript figure

Rescue of DNA damage after constricted migration reveals a mechano-regulated threshold for cell cycle.

Migration through 3D constrictions can cause nuclear rupture and mislocalization of nuclear proteins, but damage to DNA remains uncertain, as does any effect on cell cycle. Here, myosin II inhibition rescues rupture and partially rescues the DNA damage marker γH2AX, but an apparent block in cell cycle appears unaffected. Co-overexpression of multiple DNA repair factors or antioxidant inhibition of break formation also exert partial effects, independently of rupture. Combined treatments completely rescue cell cycle suppression by DNA damage, revealing a sigmoidal dependence of cell cycle on excess DNA damage. Migration through custom-etched pores yields the same damage threshold, with ∼4-µm pores causing intermediate levels of both damage and cell cycle suppression. High curvature imposed rapidly by pores or probes or else by small micronuclei consistently associates nuclear rupture with dilution of stiff lamin-B filaments, loss of repair factors, and entry from cytoplasm of chromatin-binding cGAS (cyclic GMP-AMP synthase). The cell cycle block caused by constricted migration is nonetheless reversible, with a potential for DNA misrepair and genome variation

Subspace subcodes of Reed-Solomon codes

We introduce a class of nonlinear cyclic error-correcting codes, which we call subspace subcodes of Reed-Solomon (SSRS) codes. An SSRS code is a subset of a parent Reed-Solomon (RS) code consisting of the RS codewords whose components all lie in a fixed ν-dimensional vector subspace S of GF (2m). SSRS codes are constructed using properties of the Galois field GF(2m). They are not linear over the field GF(2ν), which does not come into play, but rather are Abelian group codes over S. However, they are linear over GF(2), and the symbol-wise cyclic shift of any codeword is also a codeword. Our main result is an explicit but complicated formula for the dimension of an SSRS code. It implies a simple lower bound, which gives the true value of the dimension for most, though not all, subspaces. We also prove several important duality properties. We present some numerical examples, which show, among other things, that (1) SSRS codes can have a higher dimension than comparable subfield subcodes of RS codes, so that even if GF(2ν) is a subfield of GF(2m), it may not be the best ν-dimensional subspace for constructing SSRS codes; and (2) many high-rate SSRS codes have a larger dimension than any previously known code with the same values of n, d, and q, including algebraic-geometry codes. These examples suggest that high-rate SSRS codes are promising candidates to replace Reed-Solomon codes in high-performance transmission and storage systems

Understanding the Mechanism of Electronic Defect Suppression Enabled by Nonidealities in Atomic Layer Deposition.

Silicon germanium (SiGe) is a multifunctional material considered for quantum computing, neuromorphic devices, and CMOS transistors. However, implementation of SiGe in nanoscale electronic devices necessitates suppression of surface states dominating the electronic properties. The absence of a stable and passive surface oxide for SiGe results in the formation of charge traps at the SiGe-oxide interface induced by GeOx. In an ideal ALD process in which oxide is grown layer by layer, the GeOx formation should be prevented with selective surface oxidation (i.e., formation of an SiOx interface) by controlling the oxidant dose in the first few ALD cycles of the oxide deposition on SiGe. However, in a real ALD process, the interface evolves during the entire ALD oxide deposition due to diffusion of reactant species through the gate oxide. In this work, this diffusion process in nonideal ALD is investigated and exploited: the diffusion through the oxide during ALD is utilized to passivate the interfacial defects by employing ozone as a secondary oxidant. Periodic ozone exposure during gate oxide ALD on SiGe is shown to reduce the integrated trap density (Dit) across the band gap by nearly 1 order of magnitude in Al2O3 (<6 × 1010 cm-2) and in HfO2 (<3.9 × 1011 cm-2) by forming a SiOx-rich interface on SiGe. Depletion of Ge from the interfacial layer (IL) by enhancement of volatile GeOx formation and consequent desorption from the SiGe with ozone insertion during the ALD growth process is confirmed by electron energy loss spectroscopy (STEM-EELS) and hypothesized to be the mechanism for reduction of the interfacial defects. In this work, the nanoscale mechanism for defect suppression at the SiGe-oxide interface is demonstrated, which is engineering of diffusion species in the ALD process due to facile diffusion of reactant species in nonideal ALD