S. Arumugam

FRACTIONAL GLOBAL DOMINATION IN GRAPHS

Let G = (V; E) be a graph. A function g : V ! [0; 1] is called a global dominating function (GDF ) of G, if for every v 2 V; g(N[v]) = P u2N[v] g(u) 1 and g(N(v)) = P u= 2N(v) g(u) 1. A GDF g of a

On the uniqueness of d-vertex magic constant

Abstract Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r ≥ 4.

ON FRACTIONAL METRIC DIMENSION OF GRAPHS

A vertex x in a connected graph G = (V, E) is said to resolve a pair {u, v} of vertices of G if the distance from u to x is not equal to the distance from v to x. The resolving neighborhood for the pair {u, v} is defined as R{u, v} = {x ∈ V : d(u, x) ≠ d(v, x)}. A real valued function f : V → [0, 1] is a resolving function (RF) of G if f(R{u, v}) ≥ 1 for any two distinct vertices u, v ∈ V. The weight of f is defined by |f| = f(V) = ∑u∈Vf(v). The fractional metric dimension dimf(G) is defined by dimf(G) = min{|f| : f is a resolving function of G}. In this paper, we characterize graphs G for which . We also present several results on fractional metric dimension of Cartesian product of two connected graphs.

Bounds on Graphoidal Length of a Graph

Abstract A graphoidal cover of a graph G is a set Ψ of non-trivial paths (which are not necessarily open) in G such that every vertex of G is an internal vertex of at most one path in Ψ and every edge of G is in exactly one path in Ψ. We denote the set of all graphoidal covers of graph G by G G . In this paper we introduce a parameter gl(G), called graphoidal length of the graph G and is defined as g l ( G ) = max Ψ ∈ G G ⁡ { min P ∈ Ψ ⁡ l ( P ) } . We give bounds for the parameter gl(G) in terms of the well known and well studied parameter η ( G ) , graphoidal covering number of the graph and show that the bounds are sharp.

A New Measure for Gracefulness of Graphs

Abstract Let G = ( V , E ) be a ( p , q ) -graph without isolated vertices. The gracefulness grac ( G ) of G is the smallest positive integer k for which there exists an injective function f : V → { 0 , 1 , 2 , … , k } such that the edge induced function g f : E → { 1 , 2 , … , k } defined by g f ( u v ) = | f ( u ) − f ( v ) | is also injective. Let c ( f ) = max ⁡ { i : 1 , 2 , … , i are edge labels } and let m ( G ) = max f ⁡ { c ( f ) } where the maximum is taken over all injective functions f : V → N ∪ { 0 } such that g f is also injective. This new measure m ( G ) determines how close G is to being graceful. We determine m ( G ) for a fe families of nongraceful graphs.

The distance magic index od a graph

Abstract Let G be a graph of order n and let S be a set of positive integers with |S| = n. Then G is said to be S-magic if there exists a bijection ϕ : V (G) → S satisfying ∑x∈N(u) ϕ(x) = k (a constant) for every u ∈ V (G). Let α(S) = max{s : s ∈ S}. Let i(G) = min α(S), where the minimum is taken over all sets S for which the graph G admits an S-magic labeling. Then i(G) − n is called the distance magic index of the graph G. In this paper we determine the distance magic index of trees and complete bipartite graphs.

Theoretical Computer Science and Discrete Mathematics

A vertex-deleted subgraph of a graph G is called a card of G. A card of G with which the degree of the deleted vertex is also given is called a degree associated card (or dacard) of G. The degree associated reconstruction number of a graph G (or drn(G)) is the size of the smallest collection of dacards of G that uniquely determines G. It is shown that drn(G) = 1 or 2 for all biregular bipartite graphs with degrees d and d+ k, k ≥ 2 except the bistar B2,2 on 6 vertices and that drn(B2,2) = 3.

Global Secure Domination in Graphs

Let \(G=(V,E)\) be a graph. A subset S of V is called a dominating set of G if every vertex in \(V\backslash S\) is adjacent to a vertex in S. A dominating set S is called a secure dominating set if for every vertex \(v\in V-S,\) there exists \(u\in S\) such that \(uv\in E\) and \((S-\{u\})\cup \{v\}\) is a dominating set of G. If S is a secure dominating set of both G and its complement \(\overline{G},\) then S is called a global secure dominating set (gsd-set) of G. The minimum cardinality of a gsd-set of G is called the global secure domination number of G and is denoted by \(\gamma _{gs}(G).\) In this paper we present several basic results on \(\gamma _{gs}(G)\) and interesting problems for further investigation.

On Nearly Distance Magic Graphs

Let \(G=(V,E)\) be a graph on n vertices. A bijection \(f: V \rightarrow \{1,2,\ldots , n\}\) is called a nearly distance magic labeling of G if there exists a positive integer k such that \(\sum _{x \in N(v)} f(x)=k \ or \ k+1\) for every \(v \in V\). The constant k is called a magic constant of the graph and any graph which admits such a labeling is called a nearly distance magic graph. In this paper we present several basic results on nearly distance magic graphs and compute the magic constant k in terms of the fractional total domination number of the graph.

Distance Antimagic Labelings of Graphs

Let \(G=(V,E)\) be a graph of order n. Let \(f: V(G)\rightarrow \{1,2,\dots ,n\}\) be a bijection. For any vertex \(v \in V,\) the neighbor sum \(\sum \limits _{u\in N(v)}f(u)\) is called the weight of the vertex v and is denoted by w(v). If \(w(x) \ne w(y)\) for any two distinct vertices x and y, then f is called a distance antimagic labeling. A graph which admits a distance antimagic labeling is called a distance antimagic graph. If the weights form an arithmetic progression with first term a and common difference d, then the graph is called an (a, d)-distance antimagic graph.