Effective low-energy Hamiltonians for interacting nanostructures

We present a functional renormalization group (fRG) treatment of trigonal graphene nanodiscs and composites thereof, modeled by finite-size Hubbard-like Hamiltonians with honeycomb lattice structure. At half filling, the noninteracting spectrum of these structures contains a certain number of half-filled states at the Fermi level. For the case of trigonal nanodiscs, including interactions between these degenerate states was argued to lead to a large ground state spin with potential spintronics applications. Here we perform a systematic fRG flow where the excited single-particle states are integrated out with a decreasing energy cutoff, yielding a renormalized low-energy Hamiltonian for the zero-energy states that includes effects of the excited levels. The numerical implementation corroborates the results obtained with a simpler Hartree-Fock treatment of the interaction effects within the zero-energy states only. In particular, for trigonal nanodiscs the degeneracy of the one-particle-states with zero-energy turns out to be very robust against influences of the higher levels. As an explanation, we give a general argument that within this fRG scheme the zero-energy degeneracy remains unsplit under quite general conditions and for any size of the trigonal nanodisc. We furthermore discuss the differences in the effective Hamiltonian and their ground states of single nanodiscs and composite bow-tie-shaped systems.Comment: 13 page

The fractional chromatic number of triangle-free subcubic graphs

Heckman and Thomas conjectured that the fractional chromatic number of any triangle-free subcubic graph is at most 14/5. Improving on estimates of Hatami and Zhu and of Lu and Peng, we prove that the fractional chromatic number of any triangle-free subcubic graph is at most 32/11 (which is roughly 2.909)

The Randic index and the diameter of graphs

The {\it Randi\'c index} $R(G)$ of a graph $G$ is defined as the sum of 1/\sqrt{d_ud_v} over all edges $uv$ of $G$, where $d_u$ and $d_v$ are the degrees of vertices $u$ and $v,$ respectively. Let $D(G)$ be the diameter of $G$ when $G$ is connected. Aouchiche-Hansen-Zheng conjectured that among all connected graphs $G$ on $n$ vertices the path $P_n$ achieves the minimum values for both $R(G)/D(G)$ and $R(G)- D(G)$. We prove this conjecture completely. In fact, we prove a stronger theorem: If $G$ is a connected graph, then $R(G)-(1/2)D(G)\geq \sqrt{2}-1$, with equality if and only if $G$ is a path with at least three vertices.Comment: 17 pages, accepted by Discrete Mathematic

On a conjecture of the Randić index

AbstractThe Randić index of a graph G is defined as R(G)=∑u∼v(d(u)d(v))−12, where d(u) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u, v. A conjecture on R(G) for connected graph G is as follows: R(G)≥r(G)−1, where r(G) denotes the radius of G. We proved that the conjecture is true for biregular graphs, connected graphs with order n≤10 and tricyclic graphs

Topological Frustration in Graphene Nanoflakes: Magnetic Order and Spin Logic Devices

Magnetic order in graphene-related structures can arise from size effects or from topological frustration. We introduce a rigorous classification scheme for the types of finite graphene structures (nano-flakes) which lead to large net spin or to antiferromagnetic coupling between groups of electron spins. Based on this scheme, we propose specific examples of structures that can serve as the fundamental (NOR and NAND) logic gates for the design of high-density ultra-fast spintronic devices. We demonstrate, using ab initio electronic structure calculations, that these gates can in principle operate at room temperature with very low and correctable error rates.Comment: Typo in title fixe

On Maximum Matchings and Eigenvalues of Benzenoid Graphs

In August 2003 the computer program GRAFFITI made conjecture 1001 stating that for any benzenoid graph, the size of a maximum matching equals the number of positive eigenvalues. Later, the authors learned that this conjecture was already known in 1982 to I. Gutman (Kragujevac). Here we present a proof of this conjecture and of a related theorem. The results are of some relevance in the theory of (unsaturated) polycyclic hydrocarbons

Mathematical applications of inductive logic programming

Accepted versio

Few simple rules governing hydrogenation of graphene dots

We investigated binding of hydrogen atoms to small Polycyclic Aromatic Hydrocarbons (PAHs) - i.e. graphene dots with hydrogen-terminated edges - using density functional theory and correlated wavefunction techniques. We considered a number of PAHs with 3 to 7 hexagonal rings and computed binding energies for most of the symmetry unique sites, along with the minimum energy paths for significant cases. The chosen PAHs are small enough to not present radical character at their edges, yet show a clear preference for adsorption at the edge sites which can be attributed to electronic effects. We show how the results, as obtained at different level of theory, can be rationalized in detail with the help of few simple concepts derivable from a tight-binding model of the $\pi$ electrons

O maksimalnom sparivanju i svojstvenim vrijednostima benzenoidnih grafova

In August 2003 the computer program GRAFFITI made conjecture 1001 stating that for any benzenoid graph, the size of a maximum matching equals the number of positive eigenvalues. Later, the authors learned that this conjecture was already known in 1982 to I. Gutman (Kragujevac). Here we present a proof of this conjecture and of a related theorem. The results are of some relevance in the theory of (unsaturated) polycyclic hydrocarbons.U kolovozu 2003. uporabom kompjutorskoga programa GRAFFITI naslućeno je da je za bilo koji benzenoidni graf maksimalno sparivanje jednako broju pozitivnih svojstvenih vrijednosti. Kasnije su autori saznali da je taj rezultat bio poznat već 1982. Ivanu Gutmanu (Kragujevac). U članku je dan rigorozan dokaz toga rezultata i odgovarajući teorem. Taj je rezultat od određene važnosti u teoriji policikličkih ugljikovodika

Bipartizing fullerenes

A fullerene graph is a cubic bridgeless planar graph with twelve 5-faces such that all other faces are 6-faces. We show that any fullerene graph on n vertices can be bipartized by removing O(sqrt{n}) edges. This bound is asymptotically optimal.Comment: 14 pages, 4 figure