Corona Of Graphs Research Articles

The Circular Number of a Graph

If I_c [S]=V(G), then a set S⊆V (G) is a circular set of G. The circular number of G, represented by cr(G), is the lowest cardinality of a circular set of G. A cr-set of G is any circular set with cardinality cr(G). In this study, we determine the circular number of certain standard graphs. It is demonstrated that there exists a connected graph G such that dn(G)=a, g(G)=b and cr(G)=c for each integer a,b, and c with a>2,b>2, and c>2. The corona of graphs and circular number of joins were also explored.

Identifying Secure Domination in the Join and Corona of Graphs

Let G be a connected simple graph. A subset S of V(G) is a dominating set of G if for every v∈V(G)\S, there exists x∈S such that xv∈E(G). An Identifying code of a graph G is a dominating set C⊆V(G) such that for every v∈V(G),N_G [v]∩C is distinct. An identifying code of a graph G is an identifying secure dominating set if for each u∈V(G)\C, there exists v∈C such that uv∈E(G) and the set (C\{v})∪{u} is a dominating set of G. The minimum cardinality of an identifying secure dominating set of G, denoted by γ_s^ID, is called the identifying secure domination number of G. In this paper, the researchers initiate the study of the concept and give some important results. In particular, the researchers show some properties of the identifying secure dominating sets in join and corona of two graphs.

Disjoint Perfect Secure Domination in the Join and Corona of Graphs

Disjoint Perfect Secure Domination in the Join and Corona of Graphs

Outer-restrained Domination in the Join and Corona of Graphs

Outer-restrained Domination in the Join and Corona of Graphs

Forcing 2-Metric Dimension in the Join and Corona of Graphs

Let G be an undirected and connected graph with vertex set V(G). An ordered set of vertices {x1,...,xk} is a 2-resolving set in G if, for each distinct vertices u,v ∈ V(G), the lists of distances (dG(u,x1),..., dG(u,xk)) and (dG(v,x1),..., dG(v,xk)) differ in at least 2 positions. The minimum size of a 2-resolving set is the 2-metric dimension dim2(G) of G. A 2-resolving set of size dim2(G) is called a 2-metric basis for G. A subset S of a 2-metric basis W of G with the property that W is the unique 2-metric basis containing S is called a forcing subset of W. The forcing number fdim2(W) of W is the minimum cardinality of forcing subsets of W. The forcing number fdim2(G) of G is the smallest forcing number among all 2-metric basis of G.
 This study deals with the forcing subsets of 2-metric basis in graphs. The 2-metric basis in graphs resulting from some binary operations such as join and corona of graphs have been characterized. These characterizations are used to determine values for the forcing number of the 2-metric dimension of each graph considered.

Bounds on Intersection Number in the Join and Corona of Graphs

In this paper, we provide an upper bound for the intersection number in the join and corona of graphs. Moreover, we give formulas for the intersection number of Kn ◦G, Pn ◦G, Cn ◦G and Crn.

Inverse fair domination in the join and corona of graphs

Let [Formula: see text] be a connected simple graph. A dominating subset [Formula: see text] of [Formula: see text] is a fair dominating set of [Formula: see text] if all the vertices not in [Formula: see text] are dominated by the same number of vertices from [Formula: see text]. Let [Formula: see text] be a minimum fair dominating set of [Formula: see text]. A fair dominating set [Formula: see text] is called an inverse fair dominating set of [Formula: see text] with respect to [Formula: see text]. The inverse fair domination number of [Formula: see text] denoted by [Formula: see text] is the minimum cardinality of an inverse fair dominating set of [Formula: see text]. In this paper, we investigate the concept and give some important results. Further, we give the characterization of an inverse fair dominating set in the join and corona of two graphs.

Global Clustering Coeffcient of the Join and Corona of Graphs

The global clustering coecient is one of the most useful indices in complex network analysis. It is another metric that somehow measures how close a graph from being a complete graph. In this paper we present some expressions for the global clustering coeffcient of the join G v H and corona G o H of arbitrary simple and undirected graphs G and H. As corollaries to these results, we will show that for the path Pm, cycle Cm, fan Fm, and wheel Wm, both Cc(Pm v Pm) and Cc(Cm v Cm) approach to 0 as m increases without bound, while Cc(Fm), Cc(Wm), Cc(Pm o Pm), and Cc(Cm o Cm) all approach to 2/3.

Forcing Subsets of Connected Co-Independent Hop Domination in the Join and Corona of Graphs

This study deals with the forcing subsets of a minimum connected co-independent hop dominating sets in graphs. Bounds or exact values of the forcing connected co-independent hop domination numbers of graphs resulting from some binary operations such asjoin and corona of graphs are determined.Some main results generated in this study include characterization of the minimum connected co-independent hop dominating sets, characterization of the forcing subsets for these types of sets, and bounds or exact values of the forcing connected co-independent hop domination numbers of the join and corona of graphs.

Restrained domination polynomial of join and corona of graphs

Let [Formula: see text] be a simple graph of order [Formula: see text]. A set [Formula: see text] is said to be a restrained set if each vertex in [Formula: see text] is adjacent to a vertex in [Formula: see text]. The restrained polynomial of a graph [Formula: see text] is [Formula: see text] where [Formula: see text] is the number of restrained sets of [Formula: see text] of cardinality [Formula: see text]. A set [Formula: see text] is said to be a restrained dominating set if each vertex in [Formula: see text] is adjacent to a vertex in [Formula: see text] and to a vertex in [Formula: see text]. The restrained domination polynomial of a graph [Formula: see text] is defined by [Formula: see text] where [Formula: see text] is the number of restrained dominating sets of [Formula: see text] of cardinality [Formula: see text]. In this paper, we derive restrained domination polynomial of join and corona of any two graphs.

(t,r) Broadcast domination number of the join and corona of graphs

(t,r) Broadcast domination number of the join and corona of graphs

On 2-Resolving Sets in the Join and Corona of Graphs

Let G be a connected graph. An ordered set of vertices {v1, ..., vl} is a 2-resolving set in G if, for any distinct vertices u, w ∈ V (G), the lists of distances (dG(u, v1), ..., dG(u, vl)) and (dG(w, v1), ..., dG(w, vl)) differ in at least 2 positions. If G has a 2-resolving set, we denote the least size of a 2-resolving set by dim2(G), the 2-metric dimension of G. A 2-resolving set of size dim2(G) is called a 2-metric basis for G. This study deals with the concept of 2-resolving set of a graph. It characterizes the 2-resolving set in the join and corona of graphs and determine theexact values of the 2-metric dimension of these graphs.

Independent resolving sets in graphs

Let be a connected graph. Let be a subset of V with an order imposed on W. The k-vector is called the resolving vector of v with respect to W. The set W is called a resolving set if for any two distinct vertices In this paper we investigate the existence of independent resolving sets in Cartesian product and corona of graphs.

Domination defect for the join and corona of graphs

Domination defect for the join and corona of graphs

Outer-Convex Dominating Set in the Corona of Graphs as Encryption Key Generator

In this paper, we present a new type of symmetric encryption by converting the classical monoalphabetic affine cipher into a polyalphabetic cipher. The proposed encryption utilizes the properties of outer-convex dominating set in the corona of graphs to generate random keys from the shared keyword to every character of the message. The new encryption eliminates the weaknesses of affine cipher, thus increasing the level of confidence for exchanging messages.

Fair Secure Dominating Set in the Corona of Graphs

In this paper, we extend the concept of fair secure dominating sets by characterizing the corona of two nontrivial connected graphs and give some important results.

Fair Restrained Dominating Set in the Corona of Graphs

In this paper, we give the characterization of a fair restrained dominating set in the corona of two nontrivial connected graphs and give some important results.

Zagreb Indices of Double Join and Double Corona of Graphs Based on the Total Graph

The Zagreb indices for a graph are defined as sum of the square of the degrees of all the vertices of the graph and sum of the product of degrees of all the pair of adjacent vertices of that graph. In this study, the Zagreb indices of two recently introduced graph operations, called double join of graphs based on the total graph and double corona of graphs based on the total graph are computed in terms of different topological indices of their factor graphs and hence some application of the derived results are also discussed.

Strong (1,\xa01,\xa02)-Kernels in the Corona of Graphs and Some Realization Problems

In this paper, we give the necessary and sufficient conditions for the existence of strong (1, 1, 2)-kernels in the corona of graphs. Moreover, we consider lower and upper strong (1, 1, 2)-kernel numbers and we prove that the difference between these parameters can be arbitrarily large.

On Strong Resolving Domination in the Join and Corona of Graphs

Let G be a connected graph. A subset S \subseteq V(G) is a strong resolving dominating set of G if S is a dominating set and for every pair of vertices u,v \in V(G), there exists a vertex w \in S such that u \in I_G[v,w] or v \in I_G[u,w]. The smallest cardinality of a strong resolving dominating set of G is called the strong resolving domination number of G. In this paper, we characterize the strong resolving dominating sets in the join and corona of graphs and determine the bounds or exact values of the strong resolving domination number of these graphs.