M. Nalliah

Bio: M. Nalliah is an academic researcher from VIT University. The author has contributed to research in topics: Mathematics & Chromatic scale. The author has an hindex of 1, co-authored 5 publications receiving 7 citations.

Antimagic labelings of graphs and digraphs

Abstract: By a graph G = (V,E), we mean a finite, undirected graph with neither loops nor multiple edges and without isolated vertices. The order |V | and the size |E| of G are denoted by p and q respectively. For graph theoretic terminology we refer to Chartrand and Lesniak [19]. Graph labeling is one major research area in graph theory. New results are being discovered and published at a rapidly increasing rate. Further we have an enormous number of open problems and conjectures on graph labelings. For an excellent and up to date dynamic survey on graph labeling we refer to Gallian [23]. Most of the graph labeling methods trace their origin to the concept of β-valuation introduced by Rosa [32]. The same concept was introduced by Golomb who called it a graceful labeling [24]. Various types of graph labelings such as graceful labeling, harmonious labeling, equitable labeling, cordial labeling, arithmetic labeling, Skolem graceful labeling, set labeling, magic labeling, antimagic labeling, set-magic labeling, Σ-labeling, α-labeling, multiplicative and strongly multiplicative labeling, prime labeling, mean labeling and orthogonal labeling have been investigated by several authors. The concept of graph labeling has a wide range of applications to other branches of science such as X-ray crystallography, coding theory, cryptography, astronomy, circuit design and communication networks design. Informally, by a graph labeling we mean an assignment of numbers to graph elements such as vertices or edges or both subject to some conditions. These conditions are normally expressed on the basis of some values (weights) of an evaluating function. One situation is all the vertex weights or all the edge weights are same. In such case we call the labeled graph as vertex magic or edge magic respectively. Another situation is all the vertex weights or edge weights are different. In such case we call the graph as vertex antimagic or edge antimagic respectively. For an exhaustive study on magic labelings we refer to the book by Wallis [41]. A variety of antimagic labelings with lot of open problems are given in Baca and Miller [10]. Hefetz et al. [27] studied antimagic labeling in digraphs. In this thesis we concentrate mainly on two types of antimagic labelings of graphs and antimagic labelings of digraphs. The notion of antimagic labeling was introduced by Hartsfield and Ringel [26] in 1990. A graph G is antimagic if the edges of G can be labeled by the numbers 1, 2, 3, . . . , q such that the sums of the labels of the edges incident to each vertex (called weight of a vertex) are distinct. Also they conjectured that every connected graph different from K2 is antimagic. This conjecture is still open. Even if we restrict ourselves to trees, it is not known

Local vertex antimagic chromatic number of some wheel related graphs

Abstract: Let G = (V,E) be a graph of order p and size q having no isolated vertices. A bijection ƒ : E → {1, 2, 3, ..., q} is called a local antimagic labeling if for all uv ∈ E we have w(u) ≠ w(v), the weight w(u) = ∑e∈E(u) f(e) where E(u) is the set of edges incident to u. A graph G is local antimagic if G has a local antimagic labeling. The local antimagic chromatic number χla(G) is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we determine the local antimagic chromatic number for some wheel related graphs.

Local antimagic vertex coloring for generalized friendship graphs

Abstract: Let G = (V, E) be a graph of order p and size q without isolated vertices. A bijection f: E → {1, 2, … , q} is called a local antimagic labeling if w(u) ≠ w(v) for all uv ∈ E, where the vertex weight w(u) = Ʃe∈E(u) f(e) and E(u) is the set of edges incident to the vertex u ∈ V. The local antimagic chromatic number χla(G) is defined to be the minimum number of colors(vertex weights) taken over all colorings of G induced by local antimagic labelings of G. In this paper, we study the local chromatic number for generalized friendship graph of complete graphs and cycles.

Complete characterization of s-bridge graphs with local antimagic chromatic number 2

Abstract: An edge labeling of a connected graph G = ( V, E ) is said to be local antimagic if it is a bijection f : E → { 1 , . . ., | E |} such that for any pair of adjacent vertices x and y , f + ( x ) 6 = f + ( y ), where the induced vertex label f + ( x ) = P f ( e ), with e ranging over all the edges incident to x . The local antimagic chromatic number of G , denoted by χ la ( G ), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G . In this paper, we characterize s -bridge graphs with local antimagic chromatic number 2.

(a, d)-Distance Antimagic Labeling of Some Types of Graphs

Abstract: We analyze the necessary existence conditions for (a, d)-distance antimagic labeling of a graph G = (V, E) of order n. We obtain theorems that expand the family of not (a, d) -distance antimagic graphs. In particular, we prove that the crown Pn ? P1 does not admit an (a, 1)-distance antimagic labeling for n ? 2 if a ? 2. We determine the values of a at which path Pn can be an (a, 1)-distance antimagic graph. Among regular graphs, we investigate the case of a circulant graph.

On ( a , d )-Distance Anti-Magic and 1-Vertex Bimagic Vertex Labelings of Certain Types of Graphs

Abstract: The results for the corona P n ∘ P1 are generalized, which make it possible to state that P n ∘ P1 is not an ( a, d)-distance antimagic graph for arbitrary values of a and d. A condition for the existence of an ( a, d)-distance antimagic labeling of a hypercube Q n is obtained. Functional dependencies are found that generate this type of labeling for Q n . It is proved by the method of mathematical induction that Q n is a (2 n + n − 1, n − 2) -distance antimagic graph. Three types of graphs are defined that do not allow a 1-vertex bimagic vertex labeling. A relation between a distance magic labeling of a regular graph G and a 1-vertex bimagic vertex labeling of G ∪ G is established.

A Super (A,D)-Bm-Antimagic Total Covering of Ageneralized Amalgamation of Fan Graphs

Abstract: All graph in this paper are finite, simple and undirected. Let G, H be two graphs. A graph G is said to be an ( a,d )- H -antimagic total graph if there exist a bijective function such that for all subgraphs H’ isomorphic to H , the total H -weights form an arithmetic progression where a, d > 0 are integers and m is the number of all subgraphs H’ isomorphic to H . An ( a, d )- H -antimagic total labeling f is called super if the smallest labels appear in the vertices. In this paper, we will study a super ( a, d )- B m -antimagicness of a connected and disconnected generalized amalgamation of fan graphs on which a path is a terminal.

Some results on (a,d)-distance antimagic labeling

Abstract: Let G = (V,E) be a graph of order N and f : V → {1, 2,...,N} be a bijection. For every vertex v of graph G, we define its weight w(v) as the sum ∑u∈N(v) f(u), where N(v) denotes the open neighborhood of v. If the set of all vertex weights forms an arithmetic progression {a, a + d, a + 2d, . . . , a + (N − 1)d}, then f is called an (a, d)-distance antimagic labeling and the graph G is called (a, d)-distance antimagic graph. In this paper we prove the existence or non-existence of (a, d)- distance antimagic labeling of some well-known graphs.