Abstract In a graph G, a chordless path is called a monophonic path. A collection ψm of monophonic paths in G is called a monophonic graphoidal cover of G if every vertex of G is an internal vertex of at most one monophonic path in ψm and every edge of G is in exactly one monophonic path in ψm. The monophonic graphoidal covering number ηm(G) of G is the minimum cardinality of a monophonic graphoidal cover of G. In this paper, we find the monophonic graphoidal covering number of corona product of some standard graphs.

The mathematical model for wave motion in a fractured porous solid is solved for the propagation of harmonic plane wave along a general direction in 3-D space. The solution is obtained in the form of Christoffel equations, which are solved further to define the complex velocities and polarisations of five attenuated wavesin the medium. Four of these waves are longitudinal waves and the one is transverse wave. For any of these five attenuated waves, a general inhomogeneous propagation is considered with a complex specification of slowness vector involving a finite non-dimensional inhomogeneity parameter. The phase velocities and attenuation coefficients are calculated for the inhomogeneous propagation of each of the five attenuated waves in the medium. A numerical example is studies to analyse the effect of wave frequency, saturating pore-fluid, volume fraction of fractures and inhomogeneity parameter on the phase velocity and attenuation. The phenomenon of reflection is studied to calculate the partitionof wave-induced energy incident at the plane boundary of the fractured porous solid. The effect of wave frequency, volume fraction of fractures, saturating pore-fluid and inhomogeneity parameter on the energy partition are studied in the numerical example.

Syntheses of novel 3-[1-(3-nitrophenyl)-ethyl]-1-(indole-1-yl) substituted aryl/alkyl phosphinoyl/thiophosphinoyl/ selenophosphinoyl-1H-indole derivatives were accomplished in two steps. The synthetic route involves the cyclisation of equimolar quantities of 3-[1H-3-indolyl(3-nitrophenyl)methyl]-1H-indole with dichlorophenyl phosphine/ethyldichlorophosphite in the presence of triethylamine in dry acetonitrile at room temperature. These compounds were further converted to the corresponding oxides, sulphides and selenides by reacting them with hydrogen peroxide, sulphur and selenium, respectively. The structures of the novel products were established by elemental analyses, IR, ¹H, 13C and 31P NMR and mass spectroscopy. They were screened for antibacterial and antifungal activity against Staphylococcus aureus/Klebsiella pneumoniae and Pellicularia solmanicolor/Macrophomina phaseolina, respectively.