Computing Correlation among the Graphs under Lexicographic Product via Zagreb Indices

A topological index (TI) is a numerical descriptor of a molecule structure or graph that predicts its different physical, biological, and chemical properties in a theoretical way avoiding the difficult and costly procedures of chemical labs. In this paper, for two connected (molecular) graphs G 1 and G 2 , we define the generalized total-sum graph consisting of various (molecular) polygonal chains by the lexicographic product of the graphs T k G 1 and G 2 , where T k G 1 is obtained by applying the generalized total operation T k on G 1 with k ≥ 1 as some integral value. Moreover, we compute the different degree-based TIs such as first Zagreb, second Zagreb, forgotten Zagreb, and hyper-Zagreb. In the end, a comparison among all the aforesaid TIs is also conducted with the help of certain statistical tools for some particular families of generalized total-sum graphs under lexicographic product.

Radio Labelings of Lexicographic Product of Some Graphs

Labeling of graphs has defined many variations in the literature, e.g., graceful, harmonious, and radio labeling. Secrecy of data in data sciences and in information technology is very necessary as well as the accuracy of data transmission and different channel assignments is maintained. It enhances the graph terminologies for the computer programs. In this paper, we will discuss multidistance radio labeling used for channel assignment problems over wireless communication. A radio labeling is a one-to-one mapping ℘ : V G ⟶ ℤ + satisfying the condition | ℘ μ − ℘ μ ′ | ≥ diam G + 1 − d μ , μ ′ : μ , μ ′ ∈ V G for any pair of vertices μ , μ ′ in G . The span of labeling ℘ is the largest number that ℘ assigns to a vertex of a graph. Radio number of G , denoted by r n G , is the minimum span taken over all radio labelings of G . In this article, we will find relations for radio number and radio mean number of a lexicographic product for certain families of graphs.

Movable resolving domination in graphs

Let [Formula: see text] be a connected graph. Brigham et al., Resolving domination in graphs, Math. Bohem. 1 (2003) 25–36 defined a resolving dominating set as a set [Formula: see text] of vertices of a connected graph [Formula: see text] that is both resolving and dominating. A resolving dominating is a [Formula: see text]-movable resolving dominating set of [Formula: see text] if for every [Formula: see text], either [Formula: see text] is a resolving dominating set or there exists a vertex [Formula: see text] such that [Formula: see text] is a resolving dominating set of [Formula: see text]. The minimum cardinality of a [Formula: see text]-movable resolving dominating set of [Formula: see text], denoted by [Formula: see text] is the [Formula: see text]-movable[Formula: see text]-domination number of [Formula: see text]. A [Formula: see text]-movable resolving dominating set with cardinality [Formula: see text] is called a [Formula: see text]-set of [Formula: see text]. In this paper, we characterize the [Formula: see text]-movable resolving dominating sets in the join and lexicographic product of two graphs and determine the bounds or exact values of the [Formula: see text]-movable resolving domination number of these graphs.

Exact Values of Zagreb Indices for Generalized T-Sum Networks with Lexicographic Product

The use of numerical numbers to represent molecular networks plays a crucial role in the study of physicochemical and structural properties of the chemical compounds. For some integer k and a network G , the networks S k G and R k G are its derived networks called as generalized subdivided and generalized semitotal point networks, where S k and R k are generalized subdivision and generalized semitotal point operations, respectively. Moreover, for two connected networks, G 1 and G 2 , G 1 G 2 S k and G 1 G 2 R k are T -sum networks which are obtained by the lexicographic product of T G 1 and G 2 , respectively, where T ε S k , R k . In this paper, for the integral value k ≥ 1 , we find exact values of the first and second Zagreb indices for generalized T -sum networks. Furthermore, the obtained findings are general extensions of some known results for only k = 1 . At the end, a comparison among the different generalized T -sum networks with respect to first and second Zagreb indices is also included.

On Resolving Hop Domination in Graphs

A set S of vertices in a connected graph G is a resolving hop dominating set of G if S is a resolving set in G and for every vertex v ∈ V (G) \ S there exists u ∈ S such that dG(u, v) = 2. The smallest cardinality of such a set S is called the resolving hop domination number of G. This paper presents the characterizations of the resolving hop dominating sets in the join, corona and lexicographic product of two graphs and determines the exact values of their corresponding resolving hop domination number.

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.