‎let $g$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎we define an undirected simple graph $delta(g)$ whose‎ ‎vertices are the proper subgroups of $g$, which are not contained in the‎ ‎frattini subgroup of $g$ and two vertices $h$ and $k$ are joined by an edge‎ ‎if and only if $g=langle h‎ , ‎krangle$‎. ‎in this paper we classify finite groups with planar graph‎. ‎...

In the title compound, C(14)H(10)BrF(3)O(4), the coumarin ring system is almost plannar (r.m.s. deviation = 0.025 Å) and a short C-H⋯F contact occurs. The propano-ate fragment is orientated almost perpendicular to the ring [dihedral angle = 71.80 (12)°]. In the crystal, mol-ecules are linked by C-H⋯O hydrogen bonds, generating [100] chains.

‎Let $G$ be a finite group which is not a cyclic $p$-group‎, ‎$p$ a prime number‎. ‎We define an undirected simple graph $Delta(G)$ whose‎ ‎vertices are the proper subgroups of $G$, which are not contained in the‎ ‎Frattini subgroup of $G$ and two vertices $H$ and $K$ are joined by an edge‎ ‎if and only if $G=langle H‎ , ‎Krangle$‎. ‎In this paper we classify finite groups with planar graph‎. ‎...

Abstract—An ultra-wideband (UWB) planar monopole antenna integrated with a narrow-band (NB) cylindrical dielectric resonator antenna (DRA) is presented. The proposed antenna consists of a UWB monopole excited by a coplanar waveguide (CPW) transmission line, acting as a ground for a DRA excited by a slot. The mode HEM11δ is excited in the NB DRA. To validate the concept of integration, an antenn...

let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...