Cubicity, Degeneracy, and Crossing Number

A $k$-box $B=(R_1,...,R_k)$, where each $R_i$ is a closed interval on the real line, is defined to be the Cartesian product $R_1\times R_2\times ...\times R_k$. If each $R_i$ is a unit length interval, we call $B$ a $k$-cube. Boxicity of a graph $G$, denoted as \boxi(G), is the minimum integer $k$ such that $G$ is an intersection graph of $k$-boxes. Similarly, the cubicity of $G$, denoted as \cubi(G), is the minimum integer $k$ such that $G$ is an intersection graph of $k$-cubes. It was shown in [L. Sunil Chandran, Mathew C. Francis, and Naveen Sivadasan: Representing graphs as the intersection of axis-parallel cubes. MCDES-2008, IISc Centenary Conference, available at CoRR, abs/cs/ 0607092, 2006.] that, for a graph $G$ with maximum degree $\Delta$, \cubi(G)\leq \lceil 4(\Delta +1)\log n\rceil. In this paper, we show that, for a $k$-degenerate graph $G$, \cubi(G) \leq (k+2) \lceil 2e \log n \rceil. Since $k$ is at most $\Delta$ and can be much lower, this clearly is a stronger result. This bound is tight. We also give an efficient deterministic algorithm that runs in $O(n^2k)$ time to output a $8k(\lceil 2.42 \log n\rceil + 1)$ dimensional cube representation for $G$. An important consequence of the above result is that if the crossing number of a graph $G$ is $t$, then \boxi(G) is $O(t^{1/4}{\lceil\log t\rceil}^{3/4})$ . This bound is tight up to a factor of $O((\log t)^{1/4})$. We also show that, if $G$ has $n$ vertices, then \cubi(G) is $O(\log n + t^{1/4}\log t)$. Using our bound for the cubicity of $k$-degenerate graphs we show that cubicity of almost all graphs in $\mathcal{G}(n,m)$ model is $O(d_{av}\log n)$, where $d_{av}$ denotes the average degree of the graph under consideration.Comment: 21 page

Boxicity and topological invariants

The boxicity of a graph $G=(V,E)$ is the smallest integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap \cdots \cap E_k$. In the first part of this note, we prove that every graph on $m$ edges has boxicity $O(\sqrt{m \log m})$, which is asymptotically best possible. We use this result to study the connection between the boxicity of graphs and their Colin de Verdi\`ere invariant, which share many similarities. Known results concerning the two parameters suggest that for any graph $G$, the boxicity of $G$ is at most the Colin de Verdi\`ere invariant of $G$, denoted by $\mu(G)$. We observe that every graph $G$ has boxicity $O(\mu(G)^4(\log \mu(G))^2)$, while there are graphs $G$ with boxicity $\Omega(\mu(G)\sqrt{\log \mu(G)})$. In the second part of this note, we focus on graphs embeddable on a surface of Euler genus $g$. We prove that these graphs have boxicity $O(\sqrt{g}\log g)$, while some of these graphs have boxicity $\Omega(\sqrt{g \log g})$. This improves the previously best known upper and lower bounds. These results directly imply a nearly optimal bound on the dimension of the adjacency poset of graphs on surfaces.Comment: 6 page

Boxicity and separation dimension

A family $\mathcal{F}$ of permutations of the vertices of a hypergraph $H$ is called 'pairwise suitable' for $H$ if, for every pair of disjoint edges in $H$, there exists a permutation in $\mathcal{F}$ in which all the vertices in one edge precede those in the other. The cardinality of a smallest such family of permutations for $H$ is called the 'separation dimension' of $H$ and is denoted by $\pi(H)$. Equivalently, $\pi(H)$ is the smallest natural number $k$ so that the vertices of $H$ can be embedded in $\mathbb{R}^k$ such that any two disjoint edges of $H$ can be separated by a hyperplane normal to one of the axes. We show that the separation dimension of a hypergraph $H$ is equal to the 'boxicity' of the line graph of $H$. This connection helps us in borrowing results and techniques from the extensive literature on boxicity to study the concept of separation dimension.Comment: This is the full version of a paper by the same name submitted to WG-2014. Some results proved in this paper are also present in arXiv:1212.6756. arXiv admin note: substantial text overlap with arXiv:1212.675

Boxicity of graphs on surfaces

The boxicity of a graph $G=(V,E)$ is the least integer $k$ for which there exist $k$ interval graphs $G_i=(V,E_i)$, $1 \le i \le k$, such that $E=E_1 \cap ... \cap E_k$. Scheinerman proved in 1984 that outerplanar graphs have boxicity at most two and Thomassen proved in 1986 that planar graphs have boxicity at most three. In this note we prove that the boxicity of toroidal graphs is at most 7, and that the boxicity of graphs embeddable in a surface $\Sigma$ of genus $g$ is at most $5g+3$. This result yields improved bounds on the dimension of the adjacency poset of graphs on surfaces.Comment: 9 pages, 2 figure

Crystalline clusters in mW water:Stability, growth, and grain boundaries

With numerical simulations of the mW model of water, we investigate the energetic stability of crystalline clusters both for Ice I (cubic and hexagonal ice) and for the metastable Ice 0 phase as a function of the cluster size. Under a large variety of forming conditions, we find that the most stable cluster changes as a function of size: at small sizes, the Ice 0 phase produces the most stable clusters, while at large sizes, there is a crossover to Ice I clusters. We further investigate the growth of crystalline clusters with the seeding technique and study the growth patterns of different crystalline clusters. While energetically stable at small sizes, the growth of metastable phases (cubic and Ice 0) is hindered by the formation of coherent grain boundaries. A fivefold symmetric twin boundary for cubic ice, and a newly discovered coherent grain boundary in Ice 0, promotes cross nucleation of cubic ice. Our work reveals that different local structures can compete with the stable phase in mW water and that the low energy cost of particular grain boundaries might play an important role in polymorph selection

Electron paramagnetic resonance spectroscopic study of two gadolinium centres at calcium sites in synthetic fluorapatite

Gd-doped fluorapatite, synthesized from CaF2-rich melts, has been investigated as single crystals and powder samples by using X-band (9.5 GHz) and W-band (95 GHz) electron paramagnetic resonance (EPR) spectroscopy. Gd203 with natural abundances of isotopes and 157Gd-enriched Gd203 were used in the crystal synthesis. The X-band spectra obtained for the Gd-doped fluorapatite displayed a well-resolved type of Gd3+ centre (the centre 'a') caused by the Gd even isotopes (electron spin: S = 7/2; nuclear spin: I = 0), and suggested the possible presence of a second partly-resolved type of Gd3+ centre (the centre 'b') also caused by the Gd even isotopes. The latter was thoroughly disclosed in the W-band spectra. The single-crystal X- and W- band EPR spectra from three orthogonal rotation planes obtained from the Gd-doped fluorapatite allowed determination of the general spin-hamiltonian parameters for Gd3+ centres 'a' and 'b', including the spin terms of type BS (g matrix) and S2(D matrix), and the parameters associated with the high-spin terms of type S4 and S6, as well as BS3 and BS5. The validity of the spin-hamiltonian parameters was confirmed by agreement between the observed and simulated EPR spectra for both single-crystal and powder samples. The principal values of the matrices g and D for the centres 'a' and 'b' indicate that the two Gd3+-occupied sites in the synthetic fluorapatite have rhombic local symmetry. The principal directions of the D matrices suggest that the centres 'a' and 'b' correspond to substitutions of Gd3+ into Ca2 and Cal sites, respectively. These site assignments are supported by the results of pseudo-symmetry analyses using the S4 parameters. For example, the calculated pseudo-symmetry axes of the centre 'a' coincide with the local rotoinversion axis, site coordinations, as well as the faces of the coordination polyhedra of the Ca2 sites. The local structural environments of the centres 'a' and 'b' also suggest that the Gd3+ ions are incorporated into the Ca2 and Cal via Gd3+ + 02- ↔ Ca2+ +F- and 2Gd3+ + □ ↔ 3Ca2+, respectively. The vacancy (□ ) associated the centre 'b' has been shown to be located at a nearest-neighbor Ca2 site, resulting in a Gd3+--□---Gd3+ arrangement, with the cations well separated. The single-crystal X- and W-band EPR spectra of the 157Gd-doped fluorapatite revealed a well-resolved 157Gd (nuclear spin: I=3/2) hyperfine structure (HFS) of the centre 'a' and a partly-resolved 157Gd HFS of the centre 'b'. The calculated spin-hamiltonian parameters for the hyperfine, nuclear quadrupole, and nuclear Zeeman effects (i.e., matrices A,P and gn)provide further evidence for the site assignment of the centres 'a' to 157Gd nuclides at the Ca2 sites, with rhombic local symmetry. The P matrix also suggests that the electric-field gradient at the 157Gd nuclides of the centre 'a' is close to uniaxial, with the largest value along the direction of the Ca2-02 bond and almost isotropic in the horizontal plane. Moreover, single-crystal spectrum simulations have shown that the hyperfine anisotropy of the centre 'a' arises not only from A, P and gn but is also affected by terms BS (g), S2 (D), S4, S6, BS3 and BS5

Towards the Realization of Systematic, Self-Consistent Typical Medium Theory for Interacting Disordered Systems

This work is devoted to the development of a systematic method for studying electron localization. The developed method is Typical Medium Dynamical Cluster Approximation (TMDCA) using the Anderson-Hubbard model. The TMDCA incorporates non-local correlations beyond the local typical environment in a self-consistent way utilizing the momentum resolved typical-density-of-states and the non-local hybridization function to characterize the localization transition. For the (non-interacting) Anderson model, I show that the TMDCA provides a proper description of the Anderson localization transition in one, two, and three dimensions. In three-dimensions, as a function of cluster size, the TMDCA systematically recovers the re-entrance behavior of the mobility edge and obtains the correct critical disorder strength for the various disorder configurations and the associated \textit{universal order-parameter-critical-exponent} $\beta$ and in lower-dimensions, the well-knowing scaling relations are reproduced in agreement with numerical exact results. The TMDCA is also extended to treat diagonal and off-diagonal disorder by generalizing the local Blackman-Esterling-Berk and the importance of finite cluster is demonstrated. It was further generalized for multiband systems. Applying the TMDCA to weakly interaction electronic systems, I show that incorporating Coulomb interactions into disordered electron system result in two competing tendencies: the suppression of the current due to correlations and the screening of the disorder leading to the homogenizing of the system. It is shown that the critical disorder strength ($W_c^U$), required to localize all states, increases with increasing interactions ($U$); implying that the metallic phase is stabilized by interactions. Using the results, a soft pseudogap at the intermediate $W$ close to $W_c^U$ is predicted independent of filling and dimension, and I demonstrate in three-dimensions that the mobility edge is preserved as long as the chemical potential, $\mu$, is at or beyond the mobility edge energy ($\omega_\epsilon$). A two-particle formalism of electron localization is also developed within the TMDCA and used to calculate the direct-current conductivity, enabling direct comparison with experiments. Note significantly, the TMDCA benchmarks well with numerical exact results with a dramatic reduction in computational cost, enabling the incorporation of material\u27s specific details as such provide an avenue for the possibility of studying electron localization in real materials