Mathematics Interdisciplinary Research
Volume:3 Issue: 1, Spring 2018

We propose that high efficient terahertz (THz) multilayer structures are composed of ‎DSTMS/SiO2 and OH1/SiO2 at 3 and 6THz frequencies. We show that the efficiencies of these ‎structures are higher than DAST/SiO2 structure in both of 3 and 6THz frequencies. OH1/SiO2 ‎structure at 6THz has an efficiency as large as 10-1; at 3THz frequency, DSTMS/SiO2 structure ‎has an efficiency as large as 10-2. Meanwhile bulk OH1 has an efficiency as large as 10-3 at ‎‎3THz due to perfect phase matching whose efficiency is lower than DSTMS/SiO2 structure. ‎We also show that other structures, namely DSTMS/ZnTe at 3THz and DAST/GaP at 8THz, ‎have low efficiency, so they are not suitable as THz sources.‎

In this paper, we introduce a new class of contractive mappings in a fuzzy metric space and we present fixed point results for this class of maps.

Let G*H be the product * of G and H. In this paper we determine the rth power of the graph G*H in terms of G^r, H^r and G^r*H^r, when * is the join, Cartesian, symmetric difference, disjunctive, composition, skew and corona product. Then we solve the equation (G*H)^r=G^r*H^r. We also compute the Wiener index and Wiener polarity index of the skew product.

The concept of average degree-eccentricity matrix ADE(G) of a connected graph $G$ is introduced. Some coefficients of the characteristic polynomial of ADE(G) are obtained, as well as a bound for the eigenvalues of ADE(G). We also introduce the average degree-eccentricity graph energy and establish bounds for it.

Let G and H be graphs. The strong product GH of graphs G and H is the graph with vertex set V(G)V(H) and u=(u1, v1) is adjacent with v= (u2, v2) whenever (v1 = v2 and u1 is adjacent with u2) or (u1 = u2 and v1 is adjacent with v2) or (u1 is adjacent with u2 and v1 is adjacent with v2). In this paper, we first collect the earlier results about strong product and then we present applications of these results in working with some important graphs such as Fence graphs.

‎A tree containing exactly two non-pendant vertices is called a double-star‎. ‎Let k_1 and k_2 be two positive integers‎. ‎The double-star with degree sequence (k_1+1‎, ‎k_2+1‎, ‎1‎, ‎ldots‎, ‎1) is denoted by S_{k_1‎, ‎k_2}‎. ‎It is known that a cubic graph has an S_{1,1}-decomposition if and only if it contains a perfect matching‎. ‎In this paper‎, ‎we study the S_{1,2}-decomposition of cubic graphs‎. ‎We present some necessary and some sufficient conditions for the existence of an S_{1‎, ‎2}-decomposition in cubic graphs‎.