## 1 Introduction  
Booming all-optical communication has revolutionized information transmission technology drastically, and in the all-optical communicating network, the all-optical switch is one of the key types of components. A typical all-optical switch could be realized by optical bistability, tristability or multistability effects, in which any given value of input intensity can correspond to two, three or multiple stable output states, respectively [1]. Optical bistability, tristability and multistability can also be utilized for optical memory [2], sensors [3] or logic gates [4]. Although the hysteresis phenomena originate from magnetic effects, investigations demonstrate that bistable effects are widespread in optics [5], thermotics [6], and mechanics [7,8]. The Fabry-Perot cavity is the simplest structure to realize optical bistability [9]. Recently, Fano resonance [10] and surface plasmon polaritons (SPPs) [11] have been proposed to achieve optical bistability too. The common characteristic of those systems is that they can localize the light field and enhance the third-order nonlinear effect of materials. Localization of light field can be also observed in many other constructions or effects, such as defective photonic crystals [12], nanocavity [13], topological bound modes [14, 15, 16, 17] and solitons [18, 19]. Intense laser light can be utilized, not for optical bistability, but for probing the internal structure of molecules [20, 21, 22].  
Graphene is an excellent two-dimensional material. It possesses predominant mechanical properties [23, 24] and superconductivity [25]. Furthermore, it has a considerable Kerr coefficient [26]. Past research has shown that low-threshold optical bistability and multistability could be achieved through embedding graphene into defective photonic crystals or combining graphene with dielectrics. Moreover, the bistable thresholds and the intervals between the lower threshold and upper threshold could be regulated conveniently by the external voltage on graphene.  
Since single sheets of graphene are arrayed periodically to form one-dimensional photonic crystals, there are photonic bandgaps in wave-vector space, similar to the semiconductor bands structure. Explorations prove that arrays of graphene sheets could significantly localize the light field and optical solitons may arise at the edge of the bandgaps [27; 28]. The nonlinear optical effects, such as optical bistability, tristability and multistability, all run based on localization enhancement of light field. Therefore, it inspires our interest to explore optical bistability, tristability and multistability in arrays of graphene sheets.  
The tunable optical bistability, tristability and multistability are explored here in arrays of graphene. Single sheets of graphene array periodically along the horizontal direction. The linear and nonlinear transmittances are firstly investigated. Then, we give the bistable relationship between the output and input intensity. To illustrate the regulation on bistability by the external voltage, we simulate the surface conductivity of graphene in the parametric space composed of the external voltage and the incident wavelength. By further increasing the incident intensity of light, optical tristability and multistability can also be induced. This research can be applied to all-optical switches and optical storages.