If G is a simple graph (a non-oriented graph without loops or multiple edges), its (0, 1)-adjacency matrix A is symmetric and roots of the characteristic polynomial PG (λ) = det (λI −A) (the eigenvalues of G, making up its spectrum) are all real numbers, for which we assume their non-increasing order: λ1 ≥ λ2 ≥ · · · ≥ λn. In a connected graph for the largest eigenvalue λ1 (the index of the gra...

Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees adjacent vertices graph Ω provides multiplicative first Zagreb index graph. In context theory, generalized is defined as product sum αth powers vertex Ω, where α real number such that α≠0 and α≠1. The focus this work extremal for several classes including trees, unicyclic,...

A domino reaction has been developed for the synthesis of oxygen bridged bicyclic ethers through the coupling of 4-(2-hydroxyethyl)cyclohex-3-enols with aldehydes in the presence of 10 mol% of molecular iodine in dichloromethane at 25 °C. This method is highly diastereoselective affording the corresponding bicyclic ethers, i.e. octahydro-4a,7-epoxyisochromenes in good yields with high selectivi...

The fragmentation reaction of differently functionalized [2.2.2]- and [2.2.1]bicyclic systems that leads to substituted five membered heterocycles and five/six membered carbocycles is broadly studied. This reaction is carried out through a retro-Dieckmann-type condensation on strained [2.2.1]bicyclic β-ketosulfones and their counterparts β-ketoesters under very mild catalytic acid or basic cond...

The enantioselective synthesis of 7-epi-incarvilline for formal syntheses of (-)-incarvilline, (+)-incarvine C, and (-)-incarvillateine is described. The key features of our synthesis involve (1) stereoselective construction of the optically active bicyclic lactone utilizing Pd(0)-catalyzed allylic alkylation, (2) efficient transformation of the bridged bicyclic lactone to the key bicyclic lact...

Abstract This thesis is concerned with the synthesis of -sultams and the development of new routes for the synthesis of bicyclic versions of these molecules as potential anti-bacterials. The synthesis of 1-azetines, 1,2-thiazetin-1,1-dioxides and isothiazol-1,1-dioxides as precursors of bicyclic heterocycles is described. 1-Azetines were synthesised from azetidin-2-ones prepared via the [2+2] c...

The study of the maximum and minimal characteristics graphs is focus significant field mathematics known as extreme graph theory. Finding biggest or smallest that meet specified criteria main goal this discipline. There are several applications extremal theory in various fields, including computer science, physics, chemistry. Some important include: Computer networking, social chemistry physics...

We developed a 1,3-dipolar cycloaddition reaction of azomethine imines with terminal alkynes catalyzed by group 11 metal amides to provide N,N-bicyclic pyrazolidinone derivatives. This reaction afforded the cycloadducts in a unique 5,7-disubstituted manner. Furthermore, we succeeded in applying this catalysis to asymmetric reactions, and the desired heterocycles were produced in high yields wit...

A b s t r a c t. Let K3 and K ′ 3 be two complete graphs of order 3 with disjoint vertex sets. Let B∗ n(0) be the 5-vertex graph, obtained by identifying a vertex of K3 with a vertex of K ′ 3 . Let B∗∗ n (0) be the 4-vertex graph, obtained by identifying two vertices of K3 each with a vertex of K ′ 3 . Let B∗ n(k) be graph of order n , obtained by attaching k paths of almost equal length to the...

The oral delivery of protein and peptide drugs is limited by their proteolytic degradation and the poor absorption across the intestinal epithelia. In this work, we exposed a phage library of small bicyclic peptides (<1.5 kDa) to a pancreatic extract of proteases prior to affinity selection to enrich binders with higher stability in the intestinal environment. Panning with the therapeutic targe...