The edge Szeged index is a new molecular structure descriptor equal to the sum of products mu(e)mv(e) over all edges e = uv of the molecular graph G, where mu(e) is the number of edges which its distance to vertex u is smaller than the distance to vertex v, and nv(e) is defined analogously. In this paper, the edge Szeged index of one-pentagonal carbon nanocone CNC5[n] is computed for the first ...

Zeolites represent inorganic solid-state materials with porous structures of fascinating complexity. Recently, significant progress was made by reticular synthesis of related organic solid-state materials, such as metal-organic or covalent organic frameworks. Herein we go a step further and report the first example of a fluid honeycomb mimicking a zeolitic framework. In this unique self-assembl...

A recurrent formula is presented, for the enumeration of the compositions of positive integers as sums over multisets of positive integers, that closely resembles Euler’s recurrence based on the pentagonal numbers, but where the coefficients result from a discrete integration of Euler’s coefficients. Both a bijective proof and one based on generating functions show the equivalence of the subjec...

Recently a new mechanism of proton tunneling in a prism-like water hexamer was revealed [Richardson et al., Science, 2016, 351, 1310]. The tunneling motion involves the concerted breaking of two hydrogen bonds and rotations of two nearest water molecules. Eventually, this structural transformation means flipping one of the hydrogen bonds without the creation of defects in the hydrogen bond netw...

Various diverse extracellular proteins possess Ca(2+)-binding epidermal growth factor (EGF)-like domains, the function of which remains uncertain. We have determined, at high resolution (1.5 A), the crystal structure of such a domain, from human clotting factor IX, as a complex with Ca2+. The Ca2+ ligands form a classic pentagonal bipyramid with six ligands contributed by one polypeptide chain ...

the eigenvalues of a graph is the root of its characteristic polynomial. a fullerene f is a 3-connected graphs with entirely 12 pentagonal faces and n/2 -10 hexagonal faces, where n isthe number of vertices of f. in this paper we investigate the eigenvalues of a class of fullerenegraphs.