‎in this paper we introduce mixed unitary cayley graph $m_{n}$‎ ‎$(n>1)$ and compute its eigenvalues‎. ‎we also compute the energy of‎ ‎$m_{n}$ for some $n$‎.

As Heftmann, Bennett, and Bonner have indicated (l), considerable information has accumulated concerning the biosynthesis of sterols in animals (2) and fungi (3), yet practically nothing is known about the biogenesis of typically higher plant sterols or the numerous pentacyclic triterpenes that are widely distributed throughout the plant kingdom. As yet, it cannot be assumed that the over-all b...

Divergent total syntheses of (-)-kopsifoline D and (-)-deoxoapodine are detailed from a common pentacyclic intermediate 15, enlisting the late-stage formation of two different key strategic bonds (C21-C3 and C21-O-C6) unique to their hexacyclic ring systems that are complementary to its prior use in the total syntheses of kopsinine (C21-C2 bond formation) and (+)-fendleridine (C21-O-C19 bond fo...

We consider an edge-isoperimetric problem (EIP) on the cartesian powers of graphs. One of our objectives is to extend the list of graphs for whose cartesian powers the lexicographic order provides nested solutions for the EIP. We present several new classes of such graphs that include as special cases all presently known graphs with this property. Our new results are applied to derive best poss...

We introduce nite relational structures called sketches, that represent edge crossings in drawings of nite graphs. We consider the problem of characterizing sketches in Monadic Second-Order logic. We answer positively the question for framed sketches, i.e., for those representing drawings of graphs consisting of a planar connected spanning subgraph (the frame) augmented with additional edges th...

Let c : E(G) → [k] be an edge-coloring of a graph G, not necessarily proper. For each vertex v, let c̄(v) = (a1, . . . , ak), where ai is the number of edges incident to v with color i. Reorder c̄(v) for every v in G in nonincreasing order to obtain c∗(v), the color-blind partition of v. When c∗ induces a proper vertex coloring, that is, c∗(u) 6= c∗(v) for every edge uv in G, we say that c is col...

Euphorbia wallichii of the family Euphorbiaceae yielded a new rearranged pentacyclic triterpene of the multiflorane class which we have named spirowallichiione. The structure of this natural spirocompound was elucidated with the aid of modern spectroscopic techniques, including 2D-NMR.

A concise, asymmetric synthesis of the pentacyclic framework of the cortistatins has been accomplished in 12 steps from commercially available starting materials, employing a highly diastereoselective intramolecular (4+3) cycloaddition of epoxy enolsilanes as the key step.