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A super edge-magic total (SEMT) labeling of a graph ℘(V, E) is a one-one map ϒ from V(℘)∪E(℘) onto {1, 2,…,|V (℘)∪E(℘) |} such that ∃ a constant “a” satisfying ϒ(υ) + ϒ(υν) + ϒ(ν) = a, for each edge υν ∈E(℘), moreover all vertices must receive the smallest labels. The super edge-magic total (SEMT) strength, sm(℘), of a graph ℘ is the minimum of all magic constants a(ϒ), where the minimum runs over all the SEMT labelings of ℘. This minimum is defined only if the graph has at least one such SEMT labeling. Furthermore, the super edge-magic total (SEMT) deficiency for a graph ℘, signified as μs(℘)$\mu_{s}(\wp)$ is the least non-negative integer n so that ℘∪nK1 has a SEMT labeling or +∞ if such n does not exist. In this paper, we will formulate the results on SEMT labeling and deficiency of fork, H -tree and disjoint union of fork with star, bistar and path. Moreover, we will evaluate the SEMT strength for trees.

An edge-magic total labeling of an (n,m)-graph G = (V,E) is a one to one map λ from V(G) ∪ E(G) onto the integers {1,2,…,n + m} with the property that there exists an integer constant c such that λ(x) + λ(y) + λ(xy) = c for any xy ∈ E(G). It is called super edge-magic total labeling if λ (V(G)) = {1,2,…,n}. Furthermore, if G has no super edge-magic total labeling, then the minimum number of vertices added to G to have a super edge-magic total labeling, called super edge-magic deficiency of a graph G, is denoted by μs(G) [4]. If such vertices do not exist, then deficiency of G will be + ∞. In this paper we study the super edge-magic total labeling and deficiency of forests comprising of combs, 2-sided generalized combs and bistar. The evidence provided by these facts supports the conjecture proposed by Figueroa-Centeno, Ichishima and Muntaner-Bartle [2].

Nanofluids are a very productive etymology of intensifying the process of heat and mass transport systems linked with the industrial and thermal engineering systems. Nanomaterials have effective thermal properties and various applications in our daily life like in heat transfer, electronic cooling systems, energy production and biomedicine and also in the food industry. Keeping the entire motivating potential ramifications of nanoparticles in mind, this work is visualized in the mathematical model developed to show the heat and mass transport behavior of swimming motile organisms in the existence of the magnetic field, heat conduction source, thermal radiation, chemical processes and viscous dissipation. The flow of mass and heat transport under consideration is governed by nonlinear partial differential equations (PDEs) transformed into ordinary differential equations (ODEs) by implementing an eminent method called similarity transform and then numerical results obtained through MATLAB inbuilt package 'bvp4c'. Numerical solution is visualized through the comparison of Casson fluid results with Newtonian fluid. The impact of numerous nondimensional parameters of temperature, heat transfer, velocity and concentration profiles involved in governing equations is debated and visualized graphically. Furthermore, the effects of parameters and local Nusselt number, motile organism's number, Biot number, Sherwood number, thermal radiation and microorganism concentration are elaborated through graphical representation. From these results, we clearly see that the velocity profile shows a decrement by raising the values of Buoyancy ratio Nr and Bioconvection Rayleigh number Nc, thermal profile depicted propagation by incrementing the values of Biot and radiation variables, concentration profile decreases by incrementing Lewis parameter Le and microorganisms profile revealed an increase and decrease by the presence of magnetic M and bioconvection Lewis variable Lb, respectively. [ABSTRACT FROM AUTHOR]