Symmetry and Combinatorial Concepts for Cyclopolyarenes, Nanotubes and 2D-Sheets: Enumerations, Isomers, Structures Spectra & Properties

This review article highlights recent developments in symmetry, combinatorics, topology, entropy, chirality, spectroscopy and thermochemistry pertinent to 2D and 1D nanomaterials such as circumscribed-cyclopolyarenes and their heterocyclic analogs, carbon and heteronanotubes and heteronano wires, as well as tessellations of cyclopolyarenes, for example, kekulenes, septulenes and octulenes. We establish that the generalization of Sheehan’s modification of Pólya’s theorem to all irreducible representations of point groups yields robust generating functions for the enumeration of chiral, achiral, position isomers, NMR, multiple quantum NMR and ESR hyperfine patterns. We also show distance, degree and graph entropy based topological measures combined with techniques for distance degree vector sequences, edge and vertex partitions of nanomaterials yield robust and powerful techniques for thermochemistry, bond energies and spectroscopic computations of these species. We have demonstrated the existence of isentropic tessellations of kekulenes which were further studied using combinatorial, topological and spectral techniques. The combinatorial generating functions obtained not only enumerate the chiral and achiral isomers but also aid in the machine construction of various spectroscopic and ESR hyperfine patterns of the nanomaterials that were considered in this review. Combinatorial and topological tools can become an integral part of robust machine learning techniques for rapid computation of the combinatorial library of isomers and their properties of nanomaterials. Future applications to metal organic frameworks and fullerene polymers are pointed out.

𝒫-energy of graphs

Given a graph G = (V, E), with respect to a vertex partition 𝒫 we associate a matrix called 𝒫-matrix and define the 𝒫-energy, E𝒫 (G) as the sum of 𝒫-eigenvalues of 𝒫-matrix of G. Apart from studying some properties of 𝒫-matrix, its eigenvalues and obtaining bounds of 𝒫-energy, we explore the robust(shear) 𝒫-energy which is the maximum(minimum) value of 𝒫-energy for some families of graphs. Further, we derive explicit formulas for E𝒫 (G) of few classes of graphs with different vertex partitions.

Uniquely monopolar-partitionable block graphs

Special issue PRIMA 2013 International audience As a common generalization of bipartite and split graphs, monopolar graphs are defined in terms of the existence of certain vertex partitions. It has been shown that to determine whether a graph has such a partition is NP-complete for general graphs and polynomial for several classes of graphs. In this paper, we investigate graphs that admit a unique such partition and call them uniquely monopolar-partitionable graphs. By employing a tree trimming technique, we obtain a characterization of uniquely monopolar-partitionable block graphs. Our characterization implies a polynomial time algorithm for recognizing them.

An Exact Algorithm for Finding K-Biclique Vertex Partitions of Bipartites

Biclustering has been extensively studied in many fields such as data mining, e-commerce, computational biology, information security, etc. Problems of finding bicliques in bipartite, which are variants of biclustering, have received much attention in recent years due to its importance for biclustering. The k-biclique vertex partition problem proposed by Bein et al. is one of finding bicliques problems in bipartite. Its aim is to find k bicliques (kk) such that each vertex of the bipartite occurs in exactly one member of these bicliques. First, we give a sufficient condition of the k-biclique vertex partition problem. Moreover, we present an exact algorithm for finding k-biclique vertex partitions of a bipartite. Finally, we propose a method to generate simulated datasets used to test the algorithm. Experimental results on simulated datasets show that the algorithm can find k-biclique vertex partitions of a bipartite with relatively fast speed.