Cube Polynomial of Fibonacci and Lucas Cubes

The cube polynomial of a graph is the counting polynomial for the number of induced k-dimensional hypercubes (k≥0). We determine the cube polynomial of Fibonacci cubes and Lucas cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular they can be expressed with convolved Fibonacci numbers. Zeros of the studied cube polynomials are explicitly determined. Consequently, the coefficients sequences of cube polynomials of Fibonacci and Lucas cubes are unimodal.     

Asymptotic number of isometric generalized Fibonacci cubes

For a binary word f, let Q_d(f) be the subgraph of the d-dimensional cube Q_d induced on the set of all words that do not contain f as a factor. Let G_n be the set of words f of length n that are good in the sense that Q_d(f) is isometric in Q_d for all d. It is proved that lim_n→∞|G_n|/2ⁿ exists. Estimates show that the limit is close to 0.08, that is, about eight percent of all words are good.     

Transitive,Locally Finite Median Graphs with Finite Blocks

The subject of this paper are infinite, locally finite, vertex-transitive median graphs. It is shown that the finiteness of the Θ-classes of such graphs does not guarantee finite blocks. Blocks become finite if, in addition, no finite sequence of Θ-contractions produces new cut-vertices. It is proved that there are only finitely many vertex-transitive median graphs of given finite degree with finite blocks. An infinite family of vertex-transitive median graphs with finite intransitive blocks is also constructed and the list of vertex-transitive median graphs of degree four is presented. Sandi Klavžar: Supported by the Ministry of Science of Slovenia under the grant P1-0297. The author is also with the Faculty of Mathematics and Natural Sciences, University of Maribor, Slovenia and the Institute of Mathematics, Physics and Mechanics, Ljubljana.     

Oxy-Polybrominated Diphenyl Ethers from the Indonesian Marine Sponge,Lamellodysidea herbacea: X-ray,SAR, and Computational Studies

Polybrominated diphenyl ether (PBDE) compounds, derived from marine organisms, originate from symbiosis between marine sponges and cyanobacteria or bacteria. PBDEs have broad biological spectra; therefore, we analyzed structure and activity relationships of PBDEs to determine their potential as anticancer or antibacterial lead structures, through reactions and computational studies. Six known PBDEs (1–6) were isolated from the sponge, Lamellodysdiea herbacea; ¹³C NMR data for compound 6 are reported for the first time and their assignments are confirmed by their theoretical ¹³C NMR chemical shifts (RMSE < 4.0 ppm). Methylation and acetylation of 1 (2, 3, 4, 5-tetrabromo-6-(3′, 5′-dibromo-2′-hydroxyphenoxy) phenol) at the phenol functional group gave seven molecules (7–13), of which 10, 12, and 13 were new. New crystal structures for 8 and 9 are also reported. Debromination carried out on 1 produced nine compounds (1, 2, 14, 16–18, 20, 23, and 26) of which 18 was new. Debromination product 16 showed a significant IC₅₀ 8.65 ± 1.11; 8.11 ± 1.43 µM against human embryonic kidney (HEK293T) cells. Compounds 1 and 16 exhibited antibacterial activity against Gram-positive Staphylococcus aureus and Gram-negative Klebsiella pneumoniae with MID 0.078 µg/disk. The number of four bromine atoms and two phenol functional groups are important for antibacterial activity (S. aureus and K. pneumoniae) and cytotoxicity (HEK293T). The result was supported by analysis of frontier molecular orbitals (FMOs). We also propose possible products of acetylation and debromination using analysis of FMOs and electrostatic charges and we confirm the experimental result.     

Noninvasive localization of parathyroid adenomas: a comparison of X-ray computerized tomography, ultrasound, scintigraphy and MRI

Thirty-two (32) patients with primary hyperparathyroidism (17 with prior localization surgery, 15 without) were studied by a combination of computed tomography (CT), ultrasound (US), nuclear medicine (NM), and magnetic resonance imaging (MRI) for parathyroid adenoma localization. The sensitivity and true-positive ratio of each imaging technique and various combinations of techniques were evaluated. Of the 28 proven parathyroid adenomas (27 by surgery, 1 by digital subtraction angiography), 24 were imaged by two techniques, 19 by three techniques, and 10 by all four imaging techniques. The sensitivities were NM (65%), CT (76%), US (77%), and MRI (81%). The differences between true-positive ratios of 82%, 64%, 71%, and 77%, respectively, were not statistically significant. If multiple techniques were considered as a single test (i.e., a positive localization requires two or more tests to be positive at the same location), then sensitivity for a two-study combination was 79% and true-positive ratio 86%. Three techniques showed a sensitivity of 63% and a true-positive ratio of 92%, four modalities 40% and 100%, respectively. There was no significant difference in the various combinations of techniques employed (e.g., CT and US, US and MR, NM and MR, etc.). Thus, there appears to be an advantage in performing multiple techniques (regardless of which combination is selected) until two tests are positive at the same location.     

Vertex and Edge Orbits of Fibonacci and Lucas Cubes

The Fibonacci cube \({\Gamma_{n}}\) is obtained from the n-cube Q _n by removing all the vertices that contain two consecutive 1s. If, in addition, the vertices that start and end with 1 are removed, the Lucas cube \({\Lambda_{n}}\) is obtained. The number of vertex and edge orbits, the sets of the sizes of the orbits, and the number of orbits of each size, are determined for the Fibonacci cubes and the Lucas cubes under the action of the automorphism group. In particular, the set of vertex orbit sizes of \({\Lambda_{n}}\) is \({\{k \geq 1; k |n\} \cup \{k \geq 18; k |2n\}}\), the number of vertex orbits of \({\Lambda_{n}}\) of size k, where k is odd and divides n, is equal to \({\sum_{d | k} \mu (\frac{k}{d})F_{\lfloor{\frac{d}{2}}\rfloor+2}}\), and the number of edge orbits of \({\Lambda_{n}}\) is equal to the number of vertex orbits of \({\Gamma_{n-3}}\). Dihedral transformations of strings and primitive strings are essential tools to prove these results.     

Graphs S(n, k) and a Variant of the Tower of Hanoi Problem

For any n 1 and any k 1, a graph S(n, k) is introduced. Vertices of S(n, k) are n-tuples over {1, 2,. . . k} and two n-tuples are adjacent if they are in a certain relation. These graphs are graphs of a particular variant of the Tower of Hanoi problem. Namely, the graphs S(n, 3) are isomorphic to the graphs of the Tower of Hanoi problem. It is proved that there are at most two shortest paths between any two vertices of S(n, k). Together with a formula for the distance, this result is used to compute the distance between two vertices in O(n) time. It is also shown that for k 3, the graphs S(n, k) are Hamiltonian.     

Domination game and minimal edge cuts

In this paper a relationship is established between the domination game and minimal edge cuts. It is proved that the game domination number of a connected graph can be bounded above in terms of the size of minimal edge cuts. In particular, if $C$ a minimum edge cut of a connected graph $G$, then ${\gamma }_g\left(G\right)\le {\gamma }_g(G?C)+2{\kappa }^{\prime }\left(G\right)$. Double-Staller graphs are introduced in order to show that this upper bound can be attained for graphs with a bridge. The obtained results are used to extend the family of known traceable graphs whose game domination numbers are at most one-half their order. Along the way two technical lemmas, which seem to be generally applicable for the study of the domination game, are proved.