ISSN 1061-9208, Russian Journal of Mathematical Physics, Vol. 26, No. 2, 2019, pp. 185–205. c  Pleiades Publishing, Ltd., 2019. Electromagnetic Schr¨odinger Operators on Periodic Graphs with General Conditions at Vertices V. Rabinovich ∗ ∗ Instituto Polit´ ecnico Nacional, ESIME Zacatenco, M´ exico E-mail: vladimir.rabinovich@gmail.com Received August 29, 2018, Revised September 28, 2018, Accepted October 15, 2018; Abstract. The main aim of the paper is the study of the Fredholm property and essential spectra of electromagnetic Schr¨odinger operators H on graphs periodic with respect to a group G isomorphic to Z k . We consider the Schr¨odinger operators with nonperiodic electric and magnetic potentials and with general nonperiodic conditions on the vertices. We associate with H a family Lim(H) of limit operators H h generated by sequences h : G ∋h m −→ ∞. The main results of the paper are: (i) H is a Fredholm operator if and only if all limit operators H h of H are invertible, (ii) sp ess H =  H h ∈Lim(H) spH h (1) where sp ess H is the essential spectrum of H. Formula (1) is applied to the study of the essential spectrum of Schr¨odinger operators whose potentials are perturbations of periodic magnetic and electric potentials by slowly oscillating terms. DOI 10.1134/S1061920819020067 1. INTRODUCTION 1 0 . The main aim of the paper is the study of the Fredholm properties of electromagnetic Schr¨ odinger operators on periodic graphs with nonperiodic potentials and with general nonperiodic conditions at the vertices of the graph. To this end, we apply the limit operator method. This method and its applications to different problems of operator theory is presented in the books [20, 17]. It should be noted that the limit operator method is an effective tool of the investigation of the Fredholm properties and essential spectra of different operators of Mathematical Physics, in particular, electromagnetic Schr¨ odinger and Dirac operators on R n for wide classes of potentials [21], discrete Schr¨ odinger and Dirac operators on Z n , on periodic combinatorial graphs (see [22, 23]), and on quantum waveguides (see [24]). See also the recent papers on the essential spectra of Schr¨ odinger operators on R n with singular potentials supported on unbounded hypersurfaces in R n [26, 27]. A quantum graph is commonly understood as a metric graph equipped with differential or pseudo-differential operators on the edges and transmission conditions at the vertices. In last two decades, there is an explosive growth of interest in quantum graph theory. This is explained by the fact that quantum graphs are simplified models in mathematics, physics, chemistry, and engineering of the propagation of waves of various nature through quasi-one-dimensional (e.g. “meso” or “nano- scale”) structures. One can mention in particular photonic crystals, carbon nano-structures, thin quantum and electromagnetic waveguides, electric and neuronal networks, etc. We refer to the well-known reviews, books, and papers on this topic: [4, 7, 11, 12, 14], see also references cited in [4]. The spectral properties of periodic quantum graphs generated by Schr¨odinger operators with periodic potentials on periodic graphs have been investigated in many important works (see for instance [4, Chap.4, 6, 11, 12, 13, 15, 9], and references cited there). It was proved that the spectra of periodic quantum graphs have a band-gap structure obtained by means of the Floquet transform, just as it is done in solid state physics. In particular, a detailed spectral analysis of quantum graphs associated with carbon nano-structures was done in [6, 8, 13, 9]. In contrast to these works, we study 185