The concept of Sombor index SO(G) was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by SO(G). This paper introduces a new matrix for a graph G, called the Sombor matrix, and defines a new variant of graph energy called Sombor energy ES(G) of a graph G. The striking feature of this new matrix is that it is related to well-known degree-based topological indices called forgotten indices. When ES(G) values of some molecules containing hetero atoms are correlated with their total πelectron energy, we got a good correlation with the correlation coefficient r = 0.976. Further, we found some bounds and characterizations on the largest eigenvalue of S(G) and Sombor energy of graphs.

In this paper, we prove the unique solvability of an analogue problem Darboux for a loaded integro-differential equation with Caputo operator by method of integral equations. The problem is equivalently reduced to a system of integral equations, which is unconditionally and uniquely solvable.

Asymptotics of resonances for the Dirac operator with different potentials on edges of a quantum graph with the Kirchhoff coupling conditions at vertices is studied. The results are obtained for a quantum graph that consists of a compact interior and a finite number of exterior edges of infinite length connected to the interior.

In this paper, we investigate the interaction of ultrashort pulses with anisotropic optical media with carbon nanotubes in the presence of a magnetic field. Based on Maxwell’s equations, taking into account the second polarization of the light wave, an effective equation for the vector potential of the electromagnetic field is obtained. The dependence of the pulse shape on the magnetic field is revealed, and the Fourier spectra of the pulse are analyzed.

We demonstrated a Q-switched thulium-doped fiber laser (TDFL) based on Mxene Ti₃C₂T_x saturable absorber (SA) as Q-switcher. Ti₃C₂T_x was obtained using selective etching and embedded into polyvinyl alcohol (PVA) film. As the film was added into a TDFL cavity, a stable Q-switched pulse train operating at 1996 nm was produced within a single mode 1552 nm pump power range from 161.8 to 237.1 mW. When the pump power was varied within this range, the repetition rate increased from 19.6 to 33.3 kHz while the pulse width decreases from 6.71 to 3.55 µs. To the best of our knowledge, this is the first report of a Ti₃C₂T_x SA for passively generating Q-switched pulses in the 2 µm wavelength region.

Quantum Dot Cellular Automata is an emerging technology in the field of nanotechnology and has the potential to replace the existing CMOS technology. CMOS technology has its limitations in terms of high leakage current. However, QCA technology has higher speed of operation and very low power consumption. In this paper, designs for 2-4 and 3-8 decoder circuits have been made using a novel inverter circuit design which helps in decreasing the energy dissipation of the circuits. Finally, the proposed designs are compared to previously made designs. All the circuits in this paper have been simulated using QCA Designer software.