HF-Index and Y-Index of Some New Graph of Operations

In this paper we derive HF index of some graph operations containing join, Cartesian Product, Corona Product of graphs and compute the Y index of new operations of graphs related to the join of graphs.

Computing some role assignments of Cartesian Product of Graphs

In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. As a simple graph models a social network role assignment rises to the decision problem called r -Role Assignment whether it exists such an assignment of r distinct roles to the vertices of the graph. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is one of the most studied operation on graphs and has numerous applications in diverse areas, such as Mathematics, Computer Science, Chemistry and Biology. In this paper, we determine the computational complexity of r -Role Assignment restricted to Cartesian product of graphs, for r = 2,3. In fact, we show that the Cartesian product of graphs is always 2-role assignable, however the problem of 3-Role Assignment is still NP-complete for this class.

Forcing Total dr-Power Domination Number of Graphs Under Some Binary Operations

In this paper, the total dr-power domination number of graphs such as complete bipartite graph, generalized fan and generalized wheel are obtained. The forcing total dr-power domination number of graphs resulting from some binary operations such as join, corona and lexicographic product of graphs were determined.

Resolving Restrained Domination in Graphs

Let G be a connected graph. Brigham et al. [3] defined a resolving dominating setas a set S of vertices of a connected graph G that is both resolving and dominating. A set S ⊆ V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or hV (G) \ Si has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these graphs.

Total Perfect Hop Domination in Graphs Under Some Binary Operations

Let G = (V (G), E(G)) be a simple graph. A set S ⊆ V (G) is a perfect hop dominating set of G if for every v ∈ V (G) \ S, there is exactly one vertex u ∈ S such that dG(u, v) = 2. The smallest cardinality of a perfect hop dominating set of G is called the perfect hop domination number of G, denoted by γph(G). A perfect hop dominating set S ⊆ V (G) is called a total perfect hop dominating set of G if for every v ∈ V (G), there is exactly one vertex u ∈ S such that dG(u, v) = 2. The total perfect hop domination number of G, denoted by γtph(G), is the smallest cardinality of a total perfect hop dominating set of G. Any total perfect hop dominating set of G of cardinality γtph(G) is referred to as a γtph-set of G. In this paper, we characterize the total perfect hop dominating sets in the join, corona and lexicographic product of graphs and determine their corresponding total perfect hop domination number.