The matching energy of a graph was introduced by Gutman and Wagner in 2012 and defined as the sum of the absolute values of zeros of its matching polynomial. Let θ(r, s, t) be the graph obtained by fusing two triples of pendant vertices of three paths Pr+2, Ps+2 and Pt+2 to two vertices. The graph obtained by identifying the center of the star Sn−g with the degree 3 vertex u of θ(1, g−3, 1) is ...

Let $g$ be a bounded symmetric measurable nonnegative function on $[0,1]^2$, and $\left\lVert g \right\rVert = \int_{[0,1]^2} g(x,y) dx dy$. For graph $G$ with vertices $\{v_1,v_2,\ldots,v_n\}$ edge set $E(G)$, we define
 \[ t(G,g) \; \int_{[0,1]^n} \prod_{\{v_i,v_j\} \in E(G)} g(x_i,x_j) \: dx_1 dx_2 \cdots dx_n .\]
 We conjecture that $t(G,g) \geq \left\lVert \right\rVert^{|E(G)|}$ ...

A simple and mild process was developed for the highly stereoselective synthesis of halogenated bicyclic [4.3.0] and [3.3.0] γ-lactams, possessing four stereocenters, from easily available cyclic 2-enynamides. The reaction required only an inexpensive iron(II) halide under dry air and was tolerant of aryl, heteroaryl, and alkyl groups at the alkyne terminus.

An asymmetric Heck reaction allows desymmetrization of substituted cyclic olefins in high dr and ee. A bisphosphine oxide is uniquely stereoselective for this purpose. Desymmetrization of bicyclic olefins via hydroarylation can also be realized in high ee.

A few facile synthetic pathways for bicyclic aminopyridinol antioxidants are presented. Attachment of a long alkyl chain to the bicyclic pyridinol scaffold was established using ester linkage. Non-substituted pyrrolopyridinols and 1,3-oxazine-fused pyridinols were also synthesized as novel antioxidant scaffolds. Antioxidant activities were measured by a radical clock method and new compounds pr...

The molybdenum-mediated cleavage reactions of isoxazoline rings fused in bicyclic frameworks were investigated. A tandem N-O bond cleavage-retro aldol reaction of an isoxazoline ring fused in a bicyclic framework led to the cleavage of the bicyclic framework. These reactions provide a novel stereoselective synthesis of substituted cyclopentene rings, cyclopentane rings, and attached-ring system...

e=uv∈E(nu(e)+n0(e)/2)(nv(e)+n0(e)/2), where nu(e) and nv(e) are, respectively, the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u, and n0(e) is the number of vertices equidistant to u and v. Hansen used the AutoGraphiX and made the following conjecture about the revised Szeged index for a connected bicy...

Topological indices play an important role in Mathematical Chemistry especially in the quantitative structure-property relationship (QSPR) and quantitative structureactivity relationship (QSAR). Recent research indicates that the augmented Zagreb index (AZI) possess the best correlating ability among several topological indices. The main purpose of the current study is to establish some mathema...

Let Sz(G), Sz(G) and W (G) be the Szeged index, revised Szeged index and Wiener index of a graph G. In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order n > 10 are characterized; and the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results...

Bicyclic peptides are promising scaffolds for the development of inhibitors of biological targets that proved intractable by typical small molecules. So far, access to bioactive bicyclic peptide architectures is limited due to a lack of appropriate orthogonal ring-closing reactions. Here, we report chemically orthogonal ring-closing olefin (RCM) and alkyne metathesis (RCAM), which enable an eff...