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Micro-cantilever beam are now widely used in modern micro and nano devices. This device includes micro-switches and microscopes which are operated by atomic force. The dynamic analysis of cantileve...

The world is in an immediate need of treatment for coronavirus disease (COVID‐19). Chronic exposure of hydroxychloroquine in the treatment of COVID‐19 may have multiple adverse effects on human physiology, such as cardiac arrhythmias. Natural compounds need to be evaluated as treatment and preventive agents in coronavirus infection. A total of 30 compounds of Solanum tuberosum and Brassica juncea residue smoke water were selected for the virtual screening against SARS‐CoV‐1, SARS‐CoV‐2 and cellular proteins involved in the mechanism of infection. Docking analysis identified lead molecules with favorable binding energy, number of poses and hydrogen bond interactions, which indicates the effective modulation of ACE2 and TMPRSS2 receptors. Results indicated (a) curcumenol, (b) N‐desmethylselegiline, (c) phentermine and (d) sphingolipid derivatives as a selective and potent candidates in comparison to hydroxychloroquine for COVID‐19 treatment. Our in silico findings, therefore, warrant further in vitro validations of the selected compounds for the discovery of novel preventive and therapeutic drug against SARS‐CoV‐2 infection.

Present paper aims to focus on absolute mean graceful labeling in path union of various graphs. We proved path union of graphs like tree, path Pn, cycle Cn, complete bipartite graph Km, n, grid graph PM × Pn, step grid graph Stn and double step grid graph DStn are absolute mean graceful graphs.

In this paper we define graceful center of a graceful graph. We proved any graph G which admits α-labeling has at least four graceful centers. We also defined a new strong concept of universal graceful graph. Some results on ring sum of two graphs for their graceful labeling are proved.

In this paper we define arrow graph and double arrow graph. We also prove that arrow graphs $A_n^2, A_n^3, A_n^5$ are graceful and double arrow graphs $D A_n^2$ and $D A_n^3$ are graceful.

In this paper we have proved that union of three grid graphs, U3l=1(Pnl×Pml)and union of finite copies of a grid graph (Pn×Pm)are graceful. We have also given two graceful labeling functions to the grid graph (Pn×Pm).

This paper contains some results on graceful labeling. We prove that the path union of complete bipartite graph and join sum of complete bipartite graphs are graceful. We also prove that star of complete bipartite graph is graceful. Mathematics Subject Classication: 05C78

In this paper we introduce a complete star of a graph G. We prove that for a balanced cordial graph G, star of G, $P_{n}\times G$ and its complete star are balanced cordial graph. We also prove that the star of a vertex balanced graph is also a vertex balanced graph.

In this paper we prove that a star of $K_{n}$ and a cycle of $n$ copies of $K_{n}$ are cordial. We also get condition for maximum value of $e_{f}(1)-e_{f}(0)$ and highest negative value of $e_{f}(1)-e_{f}(0)$ in $K_{n}$, where $f$ is the binary vertex labeling function on the vertex set of $K_{n}$.

In this paper we perform some graph operations on smooth and semi smooth graceful graphs.We prove that star of a semi smooth graceful graph is graceful. We also prove that $K_{m,n}$ and $P(t\cdot H)$ are semi smooth graceful, where $H$ is a semi smooth graceful graph. Further we prove that the cycle graph $C(t\cdot H)$ ($t\equiv$ 0, 3 (mod $4$)) where $H$ is semi smooth graceful and step grid graph and are smooth graceful. It is also proved that every semi smooth graceful graph is odd graceful. Finally we prove that $C^{t}(m\cdot C_{n})$, $P^{t}(k\cdot T)$ where $T$ is a semi smooth graceful tree, $$, $$, $$ are graceful.

We have investigated some general classes of disconnected graceful graphs. We obtain the graceful labeling for $(P_{m} \times P_{n})~\bigcup~(P_{r} \times P_{s})$,~$C_{2f+3}~\bigcup~(P_{m} \times P_{n})~\bigcup~(P_{r} \times P_{s})$, where $ m,~n,~ r, ~s \in N - \{1\} $ and $f= 2(mn + rs) - (m + n + r + s)$. We also show that the tensor product of $P_{n}$ and $P_{3}$, where $n \in N - \{1,~2 \}$ admits graceful labeling. In addition to this we prove that a graph called star of cycle $C^{\star}_{n}$ is graceful, for $n \equiv 0$ (mod $4$).

In this paper we have investigated some grid related families of connected grace- ful graphs. We obtained graceful labeling for C(t . Pn cros Pm), (Pn cros Pm)* and path union of t copies of the grid graph(Pn cros Pm).

In this paper, we have investigate some general classes of connected graceful graphs. We prove that cycles $C_{n}, C_{m}$ joined by an arbitrary path $P_{k}$, where $m,n \equiv 0$ (mod $4$), a cycle $C_{n}$, ($n \equiv 2$ (mod $4$)) with twin chords and a path union of cycles $C_{n}$, ($n \equiv 0$ (mod $4$)) are graceful graphs.

Electrostatically actuated micro devices experience a fundamental limit on their stable travel range due to a phenomenon called as the pull-in Instability. Accurate determination of pull-in parameters (pull-in displacement and pull-in voltage) is vital in the design of electrostatic micro actuators. A systematic method of analysis of prismatic type of electrostatic beam is discussed in this paper. Using Galerkin method static analysis is carried out. Behaviour of interaction of nonlinear electrostatic force with linear restoring force of the micro cantilever beam is studied. Static analysis using COMSOL multiphysics finite element package is done to validate the results. The results are also compared with existing literature.