Twin Minus Total Domination Numbers In Directed Graphs

Let $D = (V,A)$ be a finite simple directed graph (shortly, digraph). A function $f : V \rightarrow {−1, 0, 1}$ is called a twin minus total dominating function (TMTDF) if $f(N^−(v)) \ge 1$ and $f(N^+(v)) \ge 1$ for each vertex $v \in V$. The twin minus total domination number of $D$ is $\gamma_{mt}^\ast (D) = \text{min} \{ w(f) | f$ is a TMTDF of $D \}$. In this paper, we initiate the study of twin minus total domination numbers in digraphs and we present some lower bounds for $\gamma_{mt}^\ast (D)$ in terms of the order, size and maximum and minimum in-degrees and out-degrees. In addition, we determine the twin minus total domination numbers of some classes of digraphs

Reformulated F-index of graph operations

The first general Zagreb index is defined as $M_1^\lambda(G)=\sum_{v\in V(G)}d_{G}(v)^\lambda$ where $\lambda\in \mathbb{R}-\{0,1\}$‎. ‎The case $\lambda=3$‎, ‎is called F-index‎. ‎Similarly‎, ‎reformulated first general Zagreb index is defined in terms of edge-drees as $EM_1^\lambda(G)=\sum_{e\in E(G)}d_{G}(e)^\lambda$ and the reformulated F-index is $RF(G)=\sum_{e\in E(G)}d_{G}(e)^3$‎. ‎In this paper‎, ‎we compute the reformulated F-index for some graph operations‎

An introduction to the twin signed total k -domination numbers in directed graphs

Let D = (V,A) be a finite simple directed graph (shortly digraph), N−(v) and N+(v) denote the set of in-neighbors and out-neighbors of a vertex v ∈ V, respectively. A function f:V − → { − 1,1 } is called a twin signed total k-dominating function (TSTkDF) if ∑ u ∈ (N−(v))f(u) ≥ k and ∑ u ∈ (N+(v))f(u) ≥ k for each vertex v ∈ V. The twin signed total k-domination number of D is γ∗stk(D) = min{ω(f) | f is a TSTkDF of D }, where ω(f) = ∑ v ∈ Vf(v) is the weight of f. In this paper, we initiate the study of twin signed total k-domination in digraphs and present different bounds on γ∗stk(D). In addition, we determine the twin signed total k-domination number of some classes of digraphs. Our results are mostly extensions of well-known bounds of the twin signed total domination numbers of directed graphs

Lower Bounds on the Entire Zagreb Indices of Trees

For a (molecular) graph G, the first and the second entire Zagreb indices are defined by the formulas M1εG=∑x∈VG∪EGdx2 and M2εG=∑x is either adjacent or incident to ydxdy in which dx represents the degree of a vertex or an edge x. In the current manuscript, we establish some lower bounds on the first and the second entire Zagreb indices and determine the extremal trees which achieve these bounds

The Third Leap Zagreb Index for Trees

The third leap Zagreb index is the sum of the products of vertex degrees and second degrees. In this paper, a lower bound on the third leap Zagreb index is established, and the extremal trees achieving this bound are characterized

THE ROMAN BONDAGE NUMBER OF A DIGRAPH

[[abstract]]Let D=(V,A)D=(V,A) be a finite and simple digraph. A Roman dominating function on DD is a labeling f:V(D)→{0,1,2}f:V(D)→{0,1,2} such that every vertex with label 0 has an in-neighbor with label 2. The weight of an RDF ff is the value ω(f)=∑v∈Vf(v)ω(f)=∑v∈Vf(v). The minimum weight of a Roman dominating function on a digraph DD is called the Roman domination number, denoted by γR(D)γR(D). The Roman bondage number bR(D)bR(D) of a digraph DD with maximum out-degree at least two is the minimum cardinality of all sets A′⊆AA′⊆A for which γR(D−A′)>γR(D)γR(D−A′)>γR(D). In this paper, we initiate the study of the Roman bondage number of a digraph. We determine the Roman bondage number in several classes of digraphs and give some sharp bounds