In this note, we show how the determinant of the q-distance matrix Dq(T ) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85–88]. Further, by means of the result, w...

Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. We call such a graph, which is derived from a chemical compound, a molecular graph. Evidence shows that the vertex-weighted Wiener number, which is defined over this molecular graph, is ...

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...

A dumbbell graph, denoted by Da,b,c, is a bicyclic graph consisting of two vertexdisjoint cycles Ca, Cb and a path Pc+3 (c > −1) joining them having only its end-vertices in common with the two cycles. In this paper, we study the spectral characterization w.r.t. the adjacency spectrum of Da,b,0 (without cycles C4) with gcd(a, b) > 3, and we complete the research started in [J.F. Wang et al., A ...

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.