Development of A Soil Resistivity Calculator

Soil resistivity is an important parameter for designing electrical earthing system. The measurement of soil resistivity is greatly influenced by moisture content, temperature, porosity, degree of saturation, number of soil layer(s), and frequency of lightning current. Researchers have proposed various methodologies to provide an approximation of soil resistivity using the listed parameters. In order to ease the process of estimating soil resistivity at a particular area, there is a pressing need to devise a simple tool that enables the calculation of soil resistivity in the most accurate manner. As such, this research proposes a reliable tool for quick evaluation of soil resistivity based on various methodologies using Microsoft Excel’s built-in-functions and Visual Basic Application (VBA) Next, the developed tool was validated using two methods, in which the output value of the calculator was compared with data retrieved from IEEE Std 142-2007 and data reported in past studies. The validation results revealed that the developed calculator may serve as a significant application in future due to its time-saving and cost-effective attributes.

Saturation Number of $tK_{l,l,l}$ in the Complete Tripartite Graph

For fixed graphs $F$ and $H$, a graph $G\subseteq F$ is $H$-saturated if there is no copy of $H$ in $G$, but for any edge $e\in E(F)\setminus E(G)$, there is a copy of $H$ in $G+e$. The saturation number of $H$ in $F$, denoted $sat(F,H)$, is the minimum number of edges in an $H$-saturated subgraph of $F$. In this paper, we study saturation numbers of $tK_{l,l,l}$ in complete tripartite graph $K_{n_1,n_2,n_3}$. For $t\ge 1$, $l\ge 1$ and $n_1,n_2$ and $n_3$ sufficiently large, we determine $sat(K_{n_1,n_2,n_3},tK_{l,l,l})$ exactly.

Saturation Number of Berge Stars in Random Hypergraphs

Let $G$ be a graph. We say an $r$-uniform hypergraph $H$ is a Berge-$G$ if there exists a bijection $\phi: E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each $e\in E(G)$. Given a family of $r$-uniform hypergraphs $\mathcal{F}$ and an $r$-uniform hypergraph $H$, a spanning sub-hypergraph $H'$ of $H$ is $\mathcal{F}$-saturated in $H$ if $H'$ is $\mathcal{F}$-free, but adding any edge in $E(H)\backslash E(H')$ to $H'$ creates a copy of some $F\in\mathcal{F}$. The saturation number of $\mathcal{F}$ is the minimum number of edges in an $\mathcal{F}$-saturated spanning sub-hypergraph of $H$. In this paper, we asymptotically determine the saturation number of Berge stars in random $r$-uniform hypergraphs.

Nearly-Regular Hypergraphs and Saturation of Berge Stars

Given a graph $G$, we say a $k$-uniform hypergraph $H$ on the same vertex set contains a Berge-$G$ if there exists an injection $\phi:E(G)\to E(H)$ such that $e\subseteq\phi(e)$ for each edge $e\in E(G)$. A hypergraph $H$ is Berge-$G$-saturated if $H$ does not contain a Berge-$G$, but adding any edge to $H$ creates a Berge-$G$. The saturation number for Berge-$G$, denoted $\mathrm{sat}_k(n,\text{Berge-}G)$ is the least number of edges in a $k$-uniform hypergraph that is Berge-$G$-saturated. We determine exactly the value of the saturation numbers for Berge stars. As a tool for our main result, we also prove the existence of nearly-regular $k$-uniform hypergraphs, or $k$-uniform hypergraphs in which every vertex has degree $r$ or $r-1$ for some $r\in \mathbb{Z}$, and less than $k$ vertices have degree $r-1$.

Hypergraph Saturation Irregularities

Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $G$ is called $\mathcal{F}$-saturated if it does not contain any members of $\mathcal{F}$ but adding any edge creates a copy of some $r$-graph in $\mathcal{F}$. The saturation number $\operatorname{sat}(\mathcal{F},n)$ is the minimum number of edges in an $\mathcal{F}$-saturated graph on $n$ vertices. We prove that there exists a finite family $\mathcal{F}$ such that $\operatorname{sat}(\mathcal{F},n) / n^{r-1}$ does not tend to a limit. This settles a question of Pikhurko.