The domination number and the least QQ-eigenvalue

A vertex set $D$ of a graph $G$ is said to be a dominating set if every vertex of $V(G)\setminus D$ is adjacent to at least a vertex in $D$, and the domination number $\gamma(G)$ ($\gamma$, for short) is the minimum cardinality of all dominating sets of $G$. For a graph, the least $Q$-eigenvalue is the least eigenvalue of its signless Laplacian matrix. In this paper, for a nonbipartite graph with both order $n$ and domination number $\gamma$, we show that $n\geq 3\gamma-1$, and show that it contains a unicyclic spanning subgraph with the same domination number $\gamma$. By investigating the relation between the domination number and the least $Q$-eigenvalue of a graph, we minimize the least $Q$-eigenvalue among all the nonbipartite graphs with given domination number.Comment: 13 pages, 3 figure

Breaking a novel colour image encryption algorithm based on chaos

Recently, a colour image encryption algorithm based on chaos was proposed by cascading two position permutation operations and one substitution operation, which are all determined by some pseudo-random number sequences generated by iterating the Logistic map. This paper evaluates the security level of the encryption algorithm and finds that the position permutation-only part and the substitution part can be separately broken with only $\lceil (\log_2(3MN))/8 \rceil$ and 2 chosen plain-images, respectively, where $MN$ is the size of the plain-image. Concise theoretical analyses are provided to support the chosen-plaintext attack, which are verified by experimental results also.Comment: 5 pages, 1 figur

4-(4-Fluoro­phen­yl)-2-oxo-1,2,5,6-tetra­hydro­benzo[h]quinoline-3-carbonitrile

In the mol­ecule of the title compound, C20H13FN2O, the fluoro­phenyl ring is oriented at a dihedral angle of 72.76 (3)° with respect to the fused benzene ring. In the crystal structure, inter­molecular N—H⋯O, C—H⋯O and C—H⋯F inter­actions link the mol­ecules into chains. π–π contacts between the quinoline and benzene rings [centroid–centroid distance = 3.918 (3) Å] may further stabilize the structure. A weak C—H⋯π inter­action is also present. The O atom and two of the CH2 groups of the quinoline ring system are disordered over two positions. The O atom was refined with occupancies of 0.489 (17) and 0.511 (17), while C and H atoms were refined with occupancies of 0.435 (13) and 0.565 (13)

Spins of the supermassive black hole in M87: new constraints from TeV observations

The rapid TeV $\gamma-$ray variability detected in the well-known nearby radio galaxy M87 implies an extremely compact emission region (5-10 Schwarzschild radii) near the horizon of the supermassive black hole in the galactic center. TeV photons are affected by dilution due to interaction with the radiation field of the advection-dominated accretion flow (ADAF) around the black hole, and can thus be used to probe the innermost regions around the black hole. We calculate the optical depth of the ADAF radiation field to the TeV photons and find it strongly depends on the spin of the black hole. We find that transparent radii of 10 TeV photons are of $5R_{\rm S}$ and $13R_{\rm S}$ for the maximally rotating and non-rotating black holes, respectively. With the observations, the calculated transparent radii strongly suggest the black hole is spinning fast in the galaxy. TeV photons could be used as a powerful diagnostic for estimating black hole spins in galaxies in the future.Comment: 4 pages, 4 figures. to appear in ApJ

The CJT calculation in studying nuclear matter beyond mean field approximation

We have introduced a CJT calculation in studying nuclear matter beyond mean field approximation. Based on the CJT formalism and using Walecka model, we have derived a set of coupled Dyson equations of nucleons and mesons. Neglecting the medium effects of the mesons, the usual MFT results could be reproduced. The beyond MFT calculations have been performed by thermodynamic consistently determining the meson effective masses and solving the coupled gap equations for nucleons and mesons. The numerical results for the nucleon and meson effective masses at finite temperature and chemical potential in nuclear matter are discussed.Comment: 8 pages, 8 figure