## 1 Introduction  
Investigations of interactions between nanoparticles and material surfaces of different nature are of profound importance for physics and its applications in nanotechnology, including bioelectronics (see, e.g., the following articles and reviews [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17]). The microparticle-surface interaction includes several contributions, among which are mechanical contact forces, Born repulsion, and attractive dispersion forces [18; 19]. At separations between a nanoparticle and a surface exceeding several nanometers, the dispersion forces, which are also called the van der Waals or Casimir-Polder forces, become dominant. These are determined by the quantum and thermal fluctuations in the electromagnetic field. An entirely new material that finds increasing use in nanotechnology, is graphene, i.e., the plane sheet of carbon atoms arranged in a hexagonal lattice [20; 21]. The dispersion (Casimir-Polder) interaction of graphene with different atomic systems [23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36] and nanoparticles [37; 38; 39; 40; 41; 42] has been the subject of much investigation. The obtained results are finding ever-widening applications in bioelectronics [43; 44; 45; 46]. The implementation of interactions between nanoparticles and graphene to new generation of nanodevices calls for the development of theoretical methods, which will make it possible to calculate the dispersion force as a function of all relevant parameters. These methods were developed in the framework of the Lifshitz theory [47; 48; 49] by expressing the dispersion force between an atom or a nanoparticle and a graphene sheet via the atomic (nanoparticle) electric polarizability and the polarization tensor of graphene [23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36]. In so doing, the polarization tensor of graphene was found [50; 51; 52; 53] on the basis of the first principles of thermal quantum field theory in the framework of the Dirac model [20; 21; 22]. The Lifshitz theory of dispersion forces is formulated for the case when the interacting bodies are in the state of thermal equilibrium with the environment. This condition, however, is violated when both of the interacting bodies (or at least one of them) are kept at temperatures different from that of the environment. The formalism generalizing the Lifshitz theory for systems out of thermal equilibrium was developed in [54; 55; 56; 57; 58; 59; 54; 59]. In recent years, different aspects of the nonequilibrium dispersion forces acting between two material plates, a small sphere or an atom and a material plate, and between two spheres were investigated using this formalism [60; 61; 62; 63; 64; 65; 66]. Specifically, the temperature-dependent response functions of the interacting bodies was considered in [63; 64]. The nonequilibrium dispersion force acting on spherical nanoparticles on the source side of an ideal (pristine) freestanding-in-vacuum graphene sheet was investigated quite recently [67]. The pristine character of graphene assumed in [67] means that its crystal lattice does not include any foreign atoms and the quasiparticles are massless, as was supposed in the original Dirac model [20; 21; 22]. Reference [67] suggested that the temperature of nanoparticles is the same as that of the environment, whereas the graphene sheet can be either cooler or hotter than the environment. It was shown that the impact of the nonequilibrium effects of the dispersion force decreases with increasing graphene-nanoparticle separation distance. Furthermore, according to the obtained results, at relatively short separations the effects of nonequilibrium may change the sign of the dispersion force by making it repulsive [67]. In this article, we applied the theory of nonequilibrium dispersion interaction to investigate the force acting on nanoparticles kept at the environmental temperature on the source side of the gapped graphene described by the Dirac model with light but massive quasiparticles.