the wiener index is the sum of distances between all pairs of vertices in a connected graph. in this paper, explicit expressions for the expected value of the wiener index of three types of random pentagonal chains (cf. figure 1) are obtained.

The Wiener index is the sum of distances between all pairs of vertices in a connected graph. In this paper, explicit expressions for the expected value of the Wiener index of three types of random pentagonal chains (cf. Figure 1) are obtained.

Two one-dimensional (1-D) chain complexes with pentagonal bipyramidal Dy(III) centers have been synthesized and magnetically characterized. Field-induced single molecule magnet behavior has been revealed in both compounds, which is still rarely reported in a lanthanide compound with a pentagonal bipyramidal coordination geometry. Their crystal field parameters and orientations of the magnetic e...

Spin canting, antiferromagnetic ordering, metamagnetism and single-chain magnetism were verified in a cyano-bridged Fe(II) compound synthesized from the pentagonal bipyramidal Fe(II) starting material in the presence of excessive BF4(-) anions.

Pentagonal conjugates of tryptophane zipper-forming peptide (CKTWTWTE) with a pentaazacyclopentadecane core (Pentagonal-Gly-Trpzip and Pentagonal-Ala-Trpzip) were synthesized and their self-assembling behaviors were investigated in water. Pentagonal-Gly-Trpzip self-assembled into nanofibers with the width of about 5 nm in neutral water (pH 7) via formation of tryptophane zipper, which irreversi...

The petroleum industry has a complex, inflexible and challenging supply chain (SC) that impacts both the national economy as well people’s daily lives with range of services, including transportation, heating, electricity, lubricants, chemicals petrochemicals. In industry, management presents several challenges, especially in logistics sector, are not found other industries. addition, logistica...

The structure of uranopilite, [(UO2)6(SO4)O2(OH)6(H2O)6](H2O)8, space group P1̄, a 8.896(2), b 14.029(3), c 14.339(3) Å, 96.610(4), 98.472(4), 99.802(4)°, V 1726.1(4) Å3, Z = 2, has been solved and refined on the basis of F2 for all unique data collected with monochromatic MoK X-radiation and a CCD-based detector to an agreement factor (R1) of 7.0%, calculated using 3907 unique observed reflecti...

Boron-based heteroatomic rings can have exotic chemical bonding, in which the p lone-pairs of heteroatoms manage to participate in delocalized p bonding, compensating for boron’s electrondeficiency. We explore herein the bonding properties of ternary B–N–H systems with a pentagonal ring, using the B3N2H5 0/ /2 clusters as examples. Computational structural searches lead to perfectly planar C2v ...

Let Pn be a pentagonal chain with 2n pentagons in which two edges common can regarded as adding one vertex and to hexagon. Thus, the linear derivation chains QPn represent graph obtained by attaching four-membered rings every of Pn. In this article, Laplacian spectrum consisting eigenvalues symmetric matrices is determined. Next, formulas for invariants that represented spectrum, namely, Kirchh...