Let $p$ be prime and $alpha:x mapsto xg^x$, the Discrete Lambert Map. For $kgeq 1,$ let $ V = {0, 1, 2, . . . , p^k-1}$. The iteration digraph is a directed graph with $V$ as the vertex set and there is a unique directed edge from $u$ to $alpha(u)$ for each $uin V.$ We denote this digraph by $G(g, p^{k}),$ where $gin (mathbb{Z}/p^kmathbb{Z})^*.$  In this piece of work, we investigate the struct...

the colorful paths and rainbow paths have been considered by severalauthors.a colorful directed path in a digraph $g$ is a directed path with $chi(g)$ vertices whose colors are different. a $v$-colorful directed path is such a directed path, starting from $v$. we prove that for a given $3$-regular triangle-free digraph $g$ determining whether there is a proper $chi(g)$-coloring of $g$such that ...

We consider monotonicity problems for graph searching games. Variants of these games – defined by the type of moves allowed for the players – have been found to be closely connected to graph decompositions and associated width measures such as pathor tree-width. Of particular interest is the question whether these games are monotone, i.e. whether the cops can catch a robber without ever allowin...

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.

The harmonic index of a graph $G$, denoted by $H(G)$, is defined asthe sum of weights $2/[d(u)+d(v)]$ over all edges $uv$ of $G$, where$d(u)$ denotes the degree of a vertex $u$. Hu and Zhou [Y. Hu and X. Zhou, WSEAS Trans. Math. {bf 12} (2013) 716--726] proved that for any bicyclic graph $G$ of order $ngeq 4$, $H(G)le frac{n}{2}-frac{1}{15}$ and characterize all extremal bicyclic graphs.In this...

In this paper, the digraph valued function(digraph operator), namely the block line cut vertex digraph BLC(D) of a digraph D is defined, and the problem of reconstructing a digraph from its block line cut vertex digraph is presented. Outer planarity, maximal outer planarity, and minimally non-outer planarity properties of these digraphs are discussed.

In this paper, we present a new extension of the butterfly digraph, which is known as one of the topologies used for interconnection networks. The butterfly digraph was previously generalized from binary to d-ary. We define a new digraph by adding a signed label to each vertex of the d-ary butterfly digraph. We call this digraph the dihedral butterfly digraph and study its properties. Furthermo...

In this paper, we investigate isomorphic factorizations of the Kronecker product graphs. Using these relations, it is shown that (1) the Kronecker product of the d-out-regular digraph and the complete symmetric digraph is factorized into the line digraph, (2) the Kronecker product of the Kautz digraph and the de Bruijn digraph is factorized into the Kautz digraph, (3) the Kronecker product of b...