## 1 Introduction  
Kekule-patterned graphene represents an unusual and technologically very promising modification of conventional graphene energy band structure due to coupling of its orbital with spin degrees of freedom. In fact, a periodic Y-shape modification of the bond strengths of graphene, as illustrated in Figure 1, leads to chiral electronic states as well as two nonequivalent coupled Dirac cones with different Fermi velocities [1; 2; 3]. Practically, such a unique feature can be realized by depositing graphene on a special copper-based substrate [4]. Here, we focus on Kek-Y graphene, which is further distinguished from Kek-O graphene by a finite bandgap [4; 5; 6]. The most crucial feature of Kek-Y graphene is its valley-momentum locking, i.e., two eigenstate modes of electrons are chiral (helical) and have zero bandgap, while both sublattice pseudospin and isospin are locked to the same group velocity [1]. Recently, there have been a number of crucial research publications aimed at addressing these unusual electronic, optical and transport properties of Kekule-distorted graphene [3; 7; 8; 9; 10]. Even though the mismatch between slopes of two Dirac cones appears fixed in Kekule-patterned graphene, i.e., \(v_{F,1}/v_{F,2}\sim 0.9\), we still consider a general case for Kek-Y graphene with two nonequivalent Dirac cones and having two arbitrary Fermi velocities \(v_{F,1}\) and \(v_{F,2}\). Kekule-patterned graphene is the newest member in the family of Dirac-cone materials, including graphene [11; 12], silicene [13; 14; 15] and transition-metal dichalcogenides [16; 17], as well as the most unusual group of \(\alpha\)-\(7_{3}\) materials [18; 19; 20]. The most uncommon feature of the \(\alpha\)-\(\mathcal{T}_{3}\) model is the presence of a flat band in its energy dispersions, giving rise to the most unusual electronic and optical properties [21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34]. On the other hand, the magnetic properties and magnetic-quantization behavior of \(\alpha\)-\(\mathcal{T}_{3}\) materials have also attracted a lot of attention [35; 36; 37; 38; 39], including a phase transition from a diamagnetic to a paramagnetic susceptibility by continuously varying the parameter \(\alpha\)[40]. In fact, the low-energy dispersion of the \(\alpha\)-\(\mathcal{T}_{3}\) model looks somewhat similar to that of the generalized Kek-Y graphene model if the Fermi velocity of the lower Dirac cone is greatly reduced compared to the larger one (i.e., \(v_{F,2}\ll v_{F,1}\)). Apart from Kek-Y graphing, there is another famous modification of Kekule-patterned graphene, which is referred to as Kek-O. In the latter case, Kekule dimerization takes place in benzene rings of hexagons in graphene. The energy spectrum of Kek-O graphene demonstrates a large bandgap. Therefore, the case of Kek-O graphene could be considered similar to regular graphene in the presence of circular polarized irradiation discussed above. A substantial bandgap in the energy dispersions of Kek-O graphene evidently results in a tremendous loss of electron transmission over square barrier, as well as significant drop of its optical and Boltzmann conductivity. The Wentzel-Kramers-Brillouin (WKB), or the semi-classical, theory provides an intuitive but realistic description for dynamical behaviors of electrons in a variety of complex quantum systems under the condition of a high-kinetic energy. In this case, the properties of such electrons are not much different from their classical counterparts and could be effectively described by classical dynamics. However, in contrast to a perturbation theory, the interaction between a particle and an external potential can be very strong, implying that in a number of cases, the WKB description can still be valid even as other types of approximation fail [41; 42; 43; 44; 45; 46]. Therefore, from this point of view, it is essential and crucial to develop a semi-classical WKB theory for each newly discovered Dirac lattice, such as \(\alpha\)-\(\mathcal{T}_{3}\) or Kekule-patterned graphene.