Maximal unicyclic graphs with respect to new atom-bond connectivity index.

The concept of atom-bond connectivity (ABC) index was introduced in the chemical graph theory in 1998. The atom-bond connectivity (ABC) index of a graph G defined as (see formula in text) where E(G) is the edge set and di is the degree of vertex v(i) of G. Very recently Graovac et al. define a new version of the ABC index as (see formula in text) where n(i) denotes the number of vertices of G whose distances to vertex v(i) are smaller than those to the other vertex v(j) of the edge e = v(i) v(j), and n(j) is defined analogously. In this paper we determine the maximal unicyclic graphs with respect to new atom-bond connectivity index (ABC2).

Note on the comparison of the first and second normalized zagreb eccentricity indices.

The conjecture Σuv V(G) dG(u)2 / n(G) ≤ Σuvv E(G) dG(u)dG(v) / m(G) that compares normalized Zagreb indices attracted recently a lot of attention1-9. In this paper we analyze analogous statement in which degree dG(u) of vertex u is replaced by its eccentricity δG(u) in which way we define novel first and second Zagreb eccentricity indices. We show that Σuv V(G) εG(u)2 / n(G) ≥ Σuvv E(G) εG(u)εG(v) / m(G) holds for all acyclic and unicyclic graphs and that neither this nor the opposite inequality holds for all bicyclic graphs.

A new version of atom-bond connectivity index.

The atom-bond connectivity index is a recently introduced topological index defined as [Formula: see text], where du denotes degree of vertex u. Here we define a new version of the ABC index as [Formula: see text], where nu denotes the number of vertices of G whose distances to vertex u are smaller than those to other vertex v of the edge e = uv, and nv is defined analogously. The goal of this paper is to study the ABC2 index.

Periodic cages.

Various cages are constructed by using three types of caps: f-cap (derived from spherical fullerenes by deleting zones of various size), kf-cap (obtainable by cutting off the polar ring, of size k), and t-cap ("tubercule"-cap). Building ways are presented, some of them being possible isomerization routes in the real chemistry of fullerenes. Periodic cages with ((5,7)3) covering are modeled, and their constitutive typing enumeration is given. Spectral data revealed some electronic periodicity in fullerene clusters. Semiempirical and strain energy calculations complete their characterization.

Algebraic Kekulé structures of benzenoid hydrocarbons.

An algebraic Kekulé structure of a benzenoid hydrocarbon is obtained from an ordinary Kekulé structure by inscribing into each hexagon the number of pi-electrons which (according to this Kekulé structure) belong to this hexagon. We show that in the case of catafusenes, there is a one-to-one correspondence between ordinary and algebraic Kekulé structures. On the other hand, in the case of perifusenes, one algebraic Kekulé structure may correspond to several ordinary Kekulé structures.

Nanotubes: number of Kekulé structures and aromaticity.

Carbon nanotubes (CNTs) are composed of cylindrical graphite sheets consisting of sp(2) carbons. Due to their structure CNTs are considered to be aromatic systems. In this work the number of Kekulé structures (K) in "armchair" CNTs was estimated by using the transfer matrix technique. All Kekulé structures of the cyclic variants of naphthalene and benzo[c]phenanthrene have been generated and the basic patterns have been obtained. From this information the elements of the transfer matrix were derived. The results obtained indicate that K (and the resonance energy) is greater if tubulenes are extended in the vertical than in the horizontal direction. Tubulenes are therefore more stabile than cyclic strips. An illustration, obtained by using scanning probe microscope, has been attached to affirm the existence of thin CNTs.