A   $k$-noncrossing tree is a noncrossing tree where each node receives a label in $\{1,2,\ldots,k\}$ such that the sum of labels along an ascent does not exceed $k+1,$ if we consider a path from a fixed vertex called the ...

In this article the terminal status of a vertex and terminal status connectivity indices of a connected graph have introduced. Explicit formulae for the terminal status of vertices and for terminal status connectivity ...

‎‎For an integer $k\geq 2$‎, ‎a Roman $k$-tuple dominating function‎, ‎(or just RkDF)‎, ‎in a graph $G$ is a function $f \colon V(G) \rightarrow \{0‎, ‎1‎, ‎2\}$ satisfying the condition that every vertex $u$ for which ...

For a graph $G$, an Italian dominating function is a function $f: V(G) \rightarrow \{0,1,2\}$ such that for each vertex $v \in V(G)$ either $f(v) \neq 0$, or $\sum_{u \in N(v)} f(u) \geq 2$. If a family $\mathcal{F} = ...

A dominating set $ D $ of a graph $ G=(V,E) $ is called a certified dominating set of $ G $ if $\vert N(v) \cap (V \setminus D)\vert$ is either 0 or at least 2 for all $ v \in D$. The certified domination number $\gamma_{cer}(G) ...

Topological indices are graph invariants computed usually by means of the distances or degrees of vertices of a graph. In chemical graph theory, a molecule can be modeled by a graph by replacing atoms by the vertices and ...

For a connected graph $G$ of order at least two, a  set $S$ of vertices in a graph $G$ is said to be an \textit{outer connected monophonic set} if $S$ is a monophonic set of $G$ and either $S=V$ or the subgraph induced by ...

Let $G=(E(G),V(G))$ be a (molecular) graph with vertex set $V(G)$ and edge set $E(G)$. The forgotten Zagreb index and the hyper Zagreb index of G are defined by $F(G) = \sum_{u \in V(G)} d(u)^{3}$ and $HM(G) = \sum_{uv \in ...

The eccentric graph $G_e$ of a graph $G$ is a derived graph with the vertex set same as that of $G$ and two vertices in $G_e$ are adjacent if one of them is the eccentric vertex of the other. In this paper, the concepts ...

For a simple, undirected, connected graph $G$, a function $h : V \rightarrow \lbrace 0, 1, 2 \rbrace$ is called a total Roman $\{2\}$-dominating function (TR2DF) if for every vertex $v$ in $V$ with weight $0$, either there ...

The paired domination subdivision number of a graph $G$ is the minimum number of edges that must be subdivided (where each edge in $G$ can be subdivided at most once) in order to increase the paired domination number of ...

For a graph $G$, a set $L$ of vertices is called a total liar's domination if $|N_G(u)\cap L|\geq 2$ for any $u\in V(G)$ and $|(N_G(u)\cup N_G(v))\cap L|\geq 3$ for any distinct vertices $u,v\in V(G)$. The total liar's ...

Let L(R) denote the set of all non-trivial left ideals of a ring R. The intersection graph of ideals of a ring R is an undirected simple graph denoted by G(R) whose vertices are in a one-to-one correspondence with L(R) and ...