Books by Dr. M Kamal Kumar
A Subset S of the vertex set of a graph G is called a dominating set of G, if each vertex of G is... more A Subset S of the vertex set of a graph G is called a dominating set of G, if each vertex of G is either in S or adjacent to at least one vertex in S. A partition D = D 1 D 2 D k of the vertex set of G is said to be a domatic partition or simply a d-partition of G, if each class of D i of D is a dominating set in G. The maximum cardinality taken over all d-partitions of G is called the domatic number of G denoted by d G . A graph G is said to be domatically critical or d-critical if for every edge x in G, d G − x < d G otherwise G is said to be domatically non d-critical. The embedding index of a non d-critical graph G is defined to be the smallest order of d-critical graph H containing G as an induced sub graph denoted by G . In this paper, we find the upper bound of G for few well known classes of graphs.
Motivated by the article in Scientific American [7], Michael A Henning and Stephen T Hedetniemi e... more Motivated by the article in Scientific American [7], Michael A Henning and Stephen T Hedetniemi explored the strategy of defending the Roman Empire. Cockayne defined Roman dominating function (RDF) on a Graph G = (V, E) to be a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0 is adjacent to at least one vertex v for which f (v) = 2. For a real valued function f : V → R the weight of f is w(f ) = P v∈V f (v). The Roman domination number (RDN) denoted by γ R (G) is the minimum weight among all RDF in G. If V − D contains a roman dominating function f 1 : V → {0, 1, 2}. "D" is the set of vertices v for which f (v) > 0. Then f 1 is called Inverse Roman Dominating function (IRDF) on a graph G w.r.t. f . The inverse roman domination number (IRDN) denoted by γ 1 R (G) is the minimum weight among all IRDF in G. In this paper we find few results of IRDN.
A subset S of the vertex set of a graph G is called a dominating set of G if each vertex of G is ... more A subset S of the vertex set of a graph G is called a dominating set of G if each vertex of G is either in S or adjacent to at least one vertex in S. A partition D = {D1, D2, …, Dk} of the vertex set of G is said to be a domatic partition or simply a d-partition of G if each class Di of D is a dominating set in G. The maximum cardinality taken over all d-partitions of G is called the domatic number of G denoted by d(G). A graph G is said to be domatically critical or d-critical if for every edge x in G, d(G-x) < d(G), otherwise G is said to be domatically non d-critical. The embedding index of a non d-critical graph G is defined to be the smallest order of a d-critical graph H containing G as an induced subgraph denoted by θ(G). In this paper, we find the upper bound of θ(G) for grid graphs.
Consider a simple connected graph G(V, E), a set S is a dominating set if for every vertex u OE V... more Consider a simple connected graph G(V, E), a set S is a dominating set if for every vertex u OE V -S, there exists a vertex v OE S such that u is adjacent to v. i.e. for every vertex u OE V -S, d(u, S) = 1. A dominating set D in G is a minimal dominating set if no proper subset of D is a dominating set. The minimum cardinality among all the minimal dominating sets is called domination number of the graph G denoted by g (G).
Graph theory concepts applicable in more real life problems. The 3 rd dimensional product of vert... more Graph theory concepts applicable in more real life problems. The 3 rd dimensional product of vertex measurable graph is applicable in supply chain network. Assigning secondary warehouses and maintaining the selling stores plays important role in supply chain network. In this paper, the mathematical model of 3 rd dimensional product of vertex measurable graph is developed and the numerical example is taken to illustrate the model. Subject Classification: 05C69, 90B06, 90B50, 90C05, 62P30
Representation of a set of vertices in a graph by means of a matrix was introduced by Sampath Kum... more Representation of a set of vertices in a graph by means of a matrix was introduced by Sampath Kumar. Let G(V, E) be a graph and S ⊆ V be a set of vertices, we can represent the set S by means of a matrix as follows, in the adjacency matrix A(G) of G replace the aii element by 1 if and only if vi ∈ S. In this paper we define set energy and find its properties and also study the special case of set S being a dominating set and corresponding domination energy of some special class of graphs.
The minimum cardinality among all the minimal dominating sets is called domination number of the ... more The minimum cardinality among all the minimal dominating sets is called domination number of the graph G denoted by γ(G).
Motivated by the article in Scientific American [8], Michael A Henning and Stephen T. Hedetniemi ... more Motivated by the article in Scientific American [8], Michael A Henning and Stephen T. Hedetniemi explored the strategy of defending the Roman Empire. Cockayne defined Roman dominating function (RDF) on a Graph G = (V, E) to be a function f : V → {0, 1, 2} satisfying the condition that every vertex u for which f (u) = 0. is adjacent to at least one vertex v for which f (v) = 2. For a real valued function f : V → R the weight of f is ( ) ( ). v V w f f v ∈ = ∑ The Roman Domination Number (RDN) denoted by γR (G) is the minimum weight among all RDF in G. If V -D contains a Roman dominating function f 1 : V → {0, 1, 2}, where D is the set of vertices v for which f (v) > 0. Then f 1 is called inverse Roman dominating function (IRDF) on a graph G w.r.t. f. The inverse Roman domination number (IRDN) denoted by γ 1 R(G) is the minimum weight among all IRDF in G. In this paper we find few results of RDN and IRDN.
Representing a subset of vertices in a graph by means of a matrix was introduced by E. Sampath Ku... more Representing a subset of vertices in a graph by means of a matrix was introduced by E. Sampath Kumar. Let G(V, E) be a graph and S ⊆ V be a set of vertices. We can represent the set S by means of a matrix as follows, in the adjacency matrix A(G) of G replace the aii element by 1 if and only if, vi ∈ S. In this paper we study the set S being dominating set and corresponding domination energy of some class of graphs.
A Subset S of the vertex set of a graph G is called a dominating set of G if each vertex of G is ... more A Subset S of the vertex set of a graph G is called a dominating set of G if each vertex of G is either in S or adjacent to at least one vertex in S. A partition D = {D 1 , D 2 , …, D k } of the vertex set of G is said to be a domatic partition or simply a d-partition of G if each class D i of D is a dominating set in G. The maximum cardinality taken over all d-partitions of G is called the domatic number of G denoted by d (G). A graph G is said to be domatically critical or d-critical if for every edge x in G, d (G -x) < d (G), otherwise G is said to be domatically non d-critical. The embedding index of a non d-critical graph G is defined to be the smallest order of a d-critical graph H containing G as an induced subgraph denoted by ) (G θ . In this paper, we find the ) (G θ for the Barbell graph, the Lollipop graph and the Tadpole graph.
Domination and other related concepts in undirected graphs are well studied. Although domination ... more Domination and other related concepts in undirected graphs are well studied. Although domination and related topics are extensively studied, the respective analogies on digraphs have not received much attention. Such studies in the directed graphs have applications in game theory and other areas.
It well known that radius of curvature of a plano convex lens can be determined using Newton's Ri... more It well known that radius of curvature of a plano convex lens can be determined using Newton's Rings set up making use of Interference by division of amplitude principle. The general method widely used involves measurement of diameter of several circular dark fringes. We propose two alternative methods involving different formulae. It is based on the fact that, the formation of bright and dark interference fringes is a measure of the thickness of the air film at that point. The methods use simpler geometry and the formulae are easier to derive. We present the experimental data. The results from the experiment are in agreement with results obtained through the general method.
The Mathematical Combinatorics (International Book Series) is a fully refereed international book... more The Mathematical Combinatorics (International Book Series) is a fully refereed international book series and published in USA quarterly comprising 100-150 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.
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Books by Dr. M Kamal Kumar