Topological and Spectral Properties of Wavy Zigzag Nanoribbons

Low-dimensional graphene-based nanomaterials are interesting due to their cutting-edge electronic and magnetic properties. Their large surface area, strong mechanical resistance, and electronic properties have enabled potential pharmaceutical and opto-electronic applications. Graphene nanoribbons (GNRs) are graphene strips of nanometer size possessing zigzag and armchair edge geometries with tunable widths. Despite the recent developments in the characterization, design and synthesis of GNRs, the study of electronic, magnetic and topological properties, GNRs continue to pose a challenge owing to their multidimensionality. In this study, we obtain the topological and electronic properties of a series of wave-like nanoribbons comprising nanographene units with zigzag-shaped edges. The edge partition techniques based on the convex components are employed to compute the mathematical formulae of molecular descriptors for the wave-like zigzag GNRs. We have also obtained the spectral and energetic properties including HOMO-LUMO gaps, bond delocalization energies, resonance energies, 13C NMR and ESR patterns for the GNRs. All of these computations reveal zero to very low HOMO-LUMO gaps that make these nanoribbons potential candidates for topological spintronics

Tight Bounds on 1-Harmonious Coloring of Certain Graphs

Graph coloring is one of the most studied problems in graph theory due to its important applications in task scheduling and pattern recognition. The main aim of the problem is to assign colors to the elements of a graph such as vertices and/or edges subject to certain constraints. The 1-harmonious coloring is a kind of vertex coloring such that the color pairs of end vertices of every edge are different only for adjacent edges and the optimal constraint that the least number of colors is to be used. In this paper, we investigate the graphs in which we attain the sharp bound on 1-harmonious coloring. Our investigation consists of a collection of basic graphs like a complete graph, wheel, star, tree, fan, and interconnection networks such as a mesh-derived network, generalized honeycomb network, complete multipartite graph, butterfly, and Benes networks. We also give a systematic and elegant way of coloring for these structures

Szegedski topološki indeksi in učinkovitost prerezne metode: primer melemskih struktur

The Szeged index is a bond-additive topological descriptor that quantifies each bond\u27s terminal atoms based on their closeness sets which is measured by multiplying the number of atoms in the closeness sets. Based on the high correlation between the Szeged index and the physico-chemical properties of chemical compounds, Szeged-like indices have been proposed by considering closeness sets with bond counts and other mathematical operations like addition and subtraction. As there are many ways to compute the Szeged-like indices, the cut method is predominantly used due to its complexity compared to other approaches based on algorithms and interpolations. Yet, we here analyze the usefulness of the cut method in the case of melem structures and find that it is less effective when the size and shape of the cavities change in the structures.Szegedski indeks je topološki deskriptor, ki kvantificira končne atome vsake vezi na podlagi njihovih množic bližine, ki se merijo s številom atomov v množicah bližine. Na podlagi visoke korelacije med szegedskim indeksom in fizikalno-kemijskimi lastnostmi kemičnih spojin so bili predlagani podobni indeksi in sicer z upoštevanjem množic bližine, štetjem vezi ter drugimi matematičnimi operacijami, kot sta seštevanje in odštevanje. Ker obstaja veliko načinov za izračun szegedskih indeksov, se prerezna metoda pretežno uporablja zaradi svoje kompleksnosti v primerjavi z drugimi pristopi, ki temeljijo na algoritmih in interpolacijah. Tukaj analiziramo uporabnost prerezne metode v primeru melemskih struktur in ugotavljamo, da je manj učinkovita, če se v strukturah spremenita velikost in oblika votlin

Wirelength of Enhanced Hypercube into Windmill and Necklace Graphs

An embedding of an interconnection network into another is one of the main issues in parallel processing and computing systems. Congestion, dilation, expansion and wirelength are some of the parameters used to analyze the efficiency of an embedding in which resolving the wirelength problem reduces time and cost in the embedded design. Due to the potential topological properties of enhanced hypercube, it has become constructive in recent years, and a lot of research work has been carried out on it. In this paper, we use the edge isoperimetric problem to produce the exact wirelengths of embedding enhanced hypercube into windmill and necklace graphs

Resistance Distance in <i>H</i>-Join of Graphs <i>G</i>₁,<i>G</i>₂,<i>…</i>,<i>G</i>_k

In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H-join of graphs G 1 , G 2 , &#8230; , G k . Recall that H is an arbitrary graph with V ( H ) = { 1 , 2 , &#8230; , k } , and G 1 , G 2 , &#8230; , G k are disjoint graphs. Then, the H-join of graphs G 1 , G 2 , &#8230; , G k , denoted by ⋁ H { G 1 , G 2 , &#8230; , G k } , is a graph formed by taking G 1 , G 2 , &#8230; , G k and joining every vertex of G i to every vertex of G j whenever i is adjacent to j in H. Here, we first give the Laplacian matrix of ⋁ H { G 1 , G 2 , &#8230; , G k } , and then give a { 1 } -inverse L ( ⋁ H { G 1 , G 2 , &#8230; , G k } ) { 1 } or group inverse L ( ⋁ H { G 1 , G 2 , &#8230; , G k } ) # of L ( ⋁ H { G 1 , G 2 , &#8230; , G k } ) . It is well know that, there exists a relationship between resistance distance and entries of { 1 } -inverse or group inverse. Therefore, we can easily obtain resistance distance in ⋁ H { G 1 , G 2 , &#8230; , G k } . In addition, some applications are presented in this paper

Guanidinium and hydrogen carbonate rosette layers: Distance and degree topological indices, Szeged-type indices, entropies, and NMR spectral patterns

Supramolecular chemistry explores non-covalent interactions between molecules, and it has facilitated the design of functional materials and understanding of molecular self-assembly processes. We investigate a captivating class of supramolecular structures, the guanidinium and hydrogen carbonate rosette layers. These rosette layers are composed of guanidinium cations and carbonate anions, exhibiting intricate hydrogen-bonding networks that lead to their unique structural properties. Topological and entropy indices unveil the connectivity and complexity of the structures, providing valuable insights for diverse applications. We have developed the cut method technique to deconstruct the guanidinium and hydrogen carbonate rosette layers into smaller components and obtain the distance, Szeged-type and entropy measures. Subsequently, we conducted a comparative analysis between topological indices and entropies which contributes to a deeper understanding of the structural complexity of these intriguing supramolecular systems. We have derived the degree based topological indices and entropies of the underlying rosette layers. Furthermore, our computations reveal several isentropic structures associated with degree and entropy indices. We have employed distance vector sequence-based graph theoretical techniques in conjunction with symmetry-based combinatorial methods to enumerate and construct the various NMR spectral patterns which are demonstrated to contrast the isomers and networks of the rosettes

Entropy structural characterization of zeolites BCT and DFT with bond-wise scaled comparison

Abstract Entropy of a connected network is a quantitative measure from information theory that has triggered a plethora of research domains in molecular chemistry, biological sciences and computer programming due to its inherent capacity to explore the structural characteristics of complex molecular frameworks that have low structural symmetry as well as high diversity. The analysis of the structural order is greatly simplified through the topological indices based graph entropy metrics, which are then utilized to predict the structural features of molecular frameworks. This predictability has not only revolutionized the study of zeolitic frameworks but has also given rise to new generations of frameworks. We make a comparative study of two versatile framework topologies namely zeolites BCT and DFT, which have been widely utilized to create a new generation of frameworks known as metal organic frameworks. We discuss bond-additive topological indices and compute entropy measure descriptors for zeolites BCT and DFT using degree and degree-sum parameters. In addition, we perform bond-wise scaled comparative analysis between BCT and DFT which shows that zeolite BCT has greater entropy values compared to zeolite DFT

Wiener Polarity Index Calculation of Square-Free Graphs and Its Implementation to Certain Complex Materials

Molecular topology is a portion of mathematical chemistry managing the logarithmic portrayal of chemical materials, permitting a tremendous yet straightforward characterization of the compounds. Concerning the traditional physical-chemical descriptors, it is conceivable to set up direct quantitative structure-activity relationship methods to associate with such descriptors termed topological indices. In this study, we have developed the mathematical technique to study the Wiener polarity index of chemical materials without squares. We have taken the cancer treatment drugs such as lenvatinib and cabozantinib to illustrate our approach. In addition, we explored the inherent property of silicate, Sierpiński, and octahedral-related complex materials that the edge set can be decomposed in such a way that any edge in the same part of the decomposition has an equal number of neighboring vertices and applied the technique to derive the formulae for these materials