Molecules that trace the high-density regions of the interstellar medium have been observed in (Ultra-)Luminous Infrared Galaxies, in order to initiate multiple-molecule multiple-transition studies to evaluate the physical and chemical environment of the nuclear medium and its response to the ongoing nuclear activity. The HCN(1-0), HNC(1-0), HCO+(1-0), CN(1-0) and CN(2-1), CO(2-1), and CS(3-2) transitions were observed in sources covering three decades of infrared luminosity including sources with known OH megamaser activity. The data for the molecules that trace the high-density regions have been augmented with data available in the literature. The integrated emissions of high-density tracer molecules show a strong relation to the infrared luminosity. Ratios of integrated line luminosities have been used for a first order diagnosis of the integrated molecular environment of the evolving nuclear starbursts. Diagnostic diagrams display significant differentiation among the sources that relate to initial conditions and the radiative excitation environment. Initial differentiation has been introduced between the FUV radiation field in photon-dominated-regions and the X-ray field in X-ray-dominated-regions. The galaxies displaying OH megamaser activity have line ratios typical of PDRs.Read less <