On metric dimension of cube of trees

Let $G=(V,E)$ be a connected graph and $d_{G}(u,v)$ be the shortest distance between the vertices $u$ and $v$ in $G$. A set $S=\{s_{1},s_{2},\cdots,s_{n}\}\subset V(G)$ is said to be a {\em resolving set} if for all distinct vertices $u,v$ of $G$, there exist an element $s\in S$ such that $d(s,u)\neq d(s,v)$. The minimum cardinality of a resolving set for a graph $G$ is called the {\em metric dimension} of $G$ and it is denoted by $\beta{(G)}$. A resolving set having $\beta{(G)}$ number of vertices is named as {\em metric basis} of $G$. The metric dimension problem is to find a metric basis in a graph $G$, and it has several real-life applications in network theory, telecommunication, image processing, pattern recognition, and many other fields. In this article, we consider {\em cube of trees} $T^{3}=(V, E)$, where any two vertices $u,v$ are adjacent if and only if the distance between them is less than equal to three in $T$. We establish the necessary and sufficient conditions of a vertex subset of $V$ to become a resolving set for $T^{3}$. This helps determine the tight bounds (upper and lower) for the metric dimension of $T^{3}$. Then, for certain well-known cubes of trees, such as caterpillars, lobsters, spiders, and $d$-regular trees, we establish the boundaries of the metric dimension. Further, we characterize some restricted families of cube of trees satisfying $\beta{(T^{3})}=\beta{(T)}$. We provide a construction showing the existence of a cube of tree attaining every positive integer value as their metric dimension

Study of Thermo-Sensitive In-Situ Gels for Ocular Delivery

The aim of the present study was the development of thermo-sensitive in-situ gels for in-vitro evaluation of ophthalmic delivery systems of ketorolac tromethamine (KT), based on methylcellulose (MC) in combination with hydroxypropylmethyl cellulose (HPMC). The gel temperature of 1% MC solution was observed at 60°C. It was found that 6% oral rehydration salt without dextrose (ORS) was capable to reduce the gel temperature below physiological temperature. HPMC was added to increase viscosity and drug release time. The results indicated a large increase in viscosity at 37°C with addition of HPMC whch provided sustained release of the drug over a 4h period. From in-vitro release studies, it could be concluded that the developed systems were thus a better alternative to conventional eye drops

DESIGN OF HIGH PASS FILTER USING OTA

Volume 1 Issue 7 (September 2013

Burning a binary tree and its generalization

Graph burning is a graph process that models the spread of social contagion. Initially, all the vertices of a graph $G$ are unburnt. At each step, an unburnt vertex is put on fire and the fire from burnt vertices of the previous step spreads to their adjacent unburnt vertices. This process continues till all the vertices are burnt. The burning number $b(G)$ of the graph $G$ is the minimum number of steps required to burn all the vertices in the graph. The burning number conjecture by Bonato et al. states that for a connected graph $G$ of order $n$, its burning number $b(G) \leq \lceil \sqrt{n} \rceil$. It is easy to observe that in order to burn a graph it is enough to burn its spanning tree. Hence it suffices to prove that for any tree $T$ of order $n$, its burning number $b(T) \leq \lceil \sqrt{n} \rceil$ where $T$ is the spanning tree of $G$. It was proved in 2018 that $b(T) \leq \lceil \sqrt{n + n_2 + 1/4} +1/2 \rceil$ for a tree $T$ where $n_2$ is the number of degree $2$ vertices in $T$. In this paper, we provide an algorithm to burn a tree and we improve the existing bound using this algorithm. We prove that $b(T)\leq \lceil \sqrt{n + n_2 + 8}\rceil -1$ which is an improved bound for $n\geq 50$. We also provide an algorithm to burn some subclasses of the binary tree and prove the burning number conjecture for the same

The Hidden Microplastic A New Insight into Degradation of Plastic in Marine Environment

Plastic is usually used in essential areas like packaging, industries electronic, construction, building, healthcare, transport, etc. gradually pollution is increasing in the world. Plastic makes a high level of pollution that is affecting both the life on earth and the marine organisms. Around the world, many scientists and environmentalists have been developing various technologies to deal with the constant increase of this threat to the environment. Various bio-based solutions are to be kept in the account to mitigate the foreseen problem of micro-plastic pollution. The indigenous microbes (exposed to plastic) form the dense bio-film around the plastic and degrade it with the help of active catalytic enzymes. Therefore, in this review, the authors have discussed the source, the harmful impact of micro-plastic, biodegradation of plastic, and future eco-friendly approaches which might help in the removal of plastic from the marine environment