Let $G=(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. An {outer-independent total $2$-rainbow dominating function of a graph $G$ is a function $f$ from $V(G)$ to the set of all subsets of $\{1,2\}$ such ...

A coalition in a graph $G = (V, E)$ consists of two disjoint sets $V_1$ and $V_2$ of vertices, such that neither $V_1$ nor $V_2$ is a dominating set, but the union $V_1 \cup V_2$ is a dominating set of $G$. A coalition ...

The $2S3$ transformation, which was first described for positive integers, has been defined for dyadic rational numbers in the open interval $(0,1)$  in this study.  The set of dyadic rational numbers  is a Prüfer 2-group. ...

The concept of topology defined on a set can be extended to the field of graph theory by defining the notion of graph topologies on graphs where we consider a collection of subgraphs of a graph $G$ in such a way that this ...

For a (molecular) graph, the second Zagreb index $M_2(G)$ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Roman dominating function $RDF$ of $G$ is a function $f:V(G)\rightarrow \{0,1,2\}$ ...

Let $\mathscr{D}$ be a simple connected digraph with $n$ vertices and $m$ arcs and let $W(\mathscr{D})=\mathscr{D},w)$ be the weighted digraph corresponding to $\mathscr{D}$, where the weights are taken from the set of ...

Cop Robber game is a two player game played on an undirected graph. In this game, the cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee ...

Let $S = (G,\sigma)$ be a signed graph. A function $f: V \rightarrow \{0,1,2\}$ is a Roman dominating function on $S$ if $(i)$ for each $v \in V,$ $f(N[v]) = f(v) + \sum_{u \in N(v)} \sigma(uv ) f(u) \geq 1$ and $(ii)$ for ...

In this paper, we have punctured unit $\mathbb{Z}_q$-Simplex code  and constructed a new code called unit $\mathbb{Z}_q$-Simplex code of type $\alpha$. In particular, we find the parameters of  these codes and have proved ...

In this paper we characterize the commutative rings with unity for which line signed graph of signed unit graph is balanced and consistent. To do this, first we derive some sufficient conditions for balance and consistency ...

The Sombor index of the graph $G$ is a degree based topological index, defined as $SO = \sum_{uv \in \mathbf E(G)}\sqrt{d_u^2+d_v^2}$, where $d_u$ is the degree of the vertex $u$, and $\mathbf E(G)$ is the edge set of $G$. ...

A tractable method of solving 3-person games in which players' pure strategies are staircase functions is suggested. The solution is meant to be Pareto-efficient. The method considers any 3-person staircase-function game ...