Silicon is a semiconductor material with several advantages over the other similar materials, such as its low cost, nontoxicity, and almost limitless availability, as well as many years of expertise in its purification, manufacture, and device development. It is used in practical for all of the most recent electrical products. Graph theory can be used to depict a chemical structure, with vertices representing atoms and edges representing chemical bonds. Molecular descriptors are important in mathematical chemistry, particularly in QSPR/QSAR research. In this research, by using two graph operations, namely; double and strong double graph, we computed the closed formulas for some degree-based topological indices of silicon carbide. Furthermore, we also compare topological indices numerically and graphically.

The buried hetero-structure (BH) InGaAsP-InP waveguide is used for asystem of double microring resonators (DMR). The light transmission and location ofresonant peaks are determined for six different sets of ring radii with different ordermode numbers. The effect of changing middle coupling coefficient on the box likeresponse is studied. It is found that the surge of coupling coefficient to the lower valuesmakes the through port resonance peaks sharper and for a larger amount of middlecoupling values, the transmission decreases according to the order mode numbers. TheDMR design with a small middle coupling and close values for rings perimeters cangenerate practical pass bandwidth of the resonant transmission peak. Moreover, anymodification in resonant mode numbers and middle coupling coefficient can change thewidth and height of the box like response. A DMR simulated results with the freespectral range (FSR) of 10. 2 nm is validated by comparing with the experimental data. Achieved results are practical in the filtering process of optical communication.

A tree containing exactly two non-pendant vertices is called a doublestar. Let k1 and k2 be two positive integers. The double-star with degree sequence (k1 + 1; k2 + 1; 1; : : :; 1) is denoted by Sk1; k2. It is known that a cubic graph has an S1; 1-decomposition if and only if it contains a perfect matching. In this paper, we study the S1; 2-decomposition of cubic graphs. We present some necessary and some sufficient conditions for the existence of an S1; 2-decomposition in cubic graphs.

The independence graph Ind(G) of a graph G is the graph with vertices as maximum independent sets of G and two vertices are adjacent, if and only if the corresponding maximum independent sets are disjoint. In this work, we find the independence graph of Cartesian product of d copies of complete graphs Kq, which is known as the Hamming graph H(d, q). Greenwell and Lovasz [7] found that the independence number of direct product of d copies of Kq as qd−1. We prove that the independence number of Hamming graph H(d, q), which is cartesian product of d copies of Kq, is also qd−1. As an application of our findings, we find answers for rook problem in higher dimensional square chess board.

‎Graph coloring is the assignment of one color to each vertex of a graph so that two adjacent vertices are not of the same color‎. ‎The graph coloring problem (GCP) is a matter of combinatorial optimization‎, ‎and the goal of GCP is determining the chromatic number $\chi(G)$‎. ‎Since GCP is an NP-hard problem‎, ‎then in this paper‎, ‎we propose a new approximated algorithm for finding the coloring number (it is an approximation of chromatic number) by using a graph adjacency matrix to colorize or separate a graph‎. ‎To prove the correctness of the proposed algorithm‎, ‎we implement it in MATLAB software‎, ‎and for analysis in terms of solution and execution time‎, ‎we compare our algorithm with some of the best existing algorithms that are already implemented in MATLAB software‎, ‎and we present the results in tables of various graphs‎. ‎Several available algorithms used the largest degree selection strategy‎, ‎while our proposed algorithm uses the graph adjacency matrix to select the vertex that has the smallest degree for coloring‎. ‎We provide some examples to compare the performance of our algorithm to other available methods‎. ‎We make use of the Dolan-Mor\'e performance profiles to assess the performance of the numerical algorithms‎, ‎and demonstrate the efficiency of our proposed approach in comparison with some existing methods‎.