Bulletin of the International Mathematical Virtual Institute

1-7

Authors: I. H. Nagaraja Rao, G. Venkata Rao and S. Rajesh

The aim of this paper is to show that the comment of Rashwan and Maustafa [5] on a result of Sharma and Deshpande [8] is not true by pointing out the inconsistency in their claimed example. Further the results of the former authors are generalised and are supported by examples.

17-27

Authors: Aysun Ayatac and Tufan Turaci

In communication networks, "vulnerability" indicates the resistance of a network to disruptions in communication after a breakdown of some pro- cessors or communication links. We may use graphs to model networks, as graph theoretical parameters can be used to describe the stability and reliability of communication networks If we think of a graph as model- ing a network, the average lower independence number of a graph is one measure of graph vulnerability. For a vertex v of a graph G = (V;E), the lower independence number iv(G) of G relative to v is the minimum cardinality of a maximal independent set of G that contains v. The av- erage lower independence number of G, denoted by iav(G), is the value iav(G) = 1 jV (G)j P v²V (G) iv(G). In this paper, we de¯ned and examined this parameter and considered the average lower independence number of special graphs and theirs total graphs.

43-46

Authors: Ivan D. Aranđelović and D. Krtinić

In 1920 H. Steinhaus [Sur les distances des points de mesure posi- tive, Fundamenta Mathematicae 1 (1920) 93-104.] proved the following result: "Let A be a Lebesgue measurable set of positive measure. Then there exist at least two points in A such that the distance between them is a rational number". In this paper we shall prove that there exists a sequence (xn)n>1 of di®erent points in A such that the distance between any two of them is a rational number. Further, we shall extend our result to the case when A is a set with the Baire property (non-necessarily Lebesgue measurable).

59-67

Authors: I. H. Nagaraja Rao, S. Rajesh and G. Venkata Rao

In this paper, we claim that the comment of Sastry et.al[2] on a result of Servet Kutukcu and Sushil Sharma[5] is not true by pointing out their non observation of a condition in the hypothesis of the result. Further, the results of [5] are generalized and supported by examples.

77-86

Authors: R. A. Rashwan and S. M. Altwqi

In this paper, we introduce a new one-step iterative scheme to approximate a common ¯xed point of three multivalued nonexpansive map- pings in a uniformly convex real Banach space and establish strong and weak convergence theorems for the proposed process. Our results extend important results.

93-100

Authors: R. Sundareswaran and V. Swaminathan

C.A. Barefoot, et. al. introduced the concept of the integrity of a graph. It is an useful measure of vulnerability and it is de¯ned as follows I(G) = minfjSj +m(G ¡ S) : S ½ V (G)g, where m(G ¡ S) denotes the order of the largest component in G ¡ S. The integrity of the set S is de¯ned as jSj + m(G ¡ S) and is denoted by IS, where m(G ¡ S) denotes the order of maximum component in G¡S. A partition of V (G) into subsets V1; V2; ¢ ¢ ¢ ; Vt such that IVi ; 1 6 i 6 t is a constant is called equi-integrity partition of G. The maximum cardinality of such a partition is called equi-integrity partition number of G and is denoted by EI(G). Since V (G) itself is an equi-integrity partition of G, the existence of EI-partition is guaranteed. In this paper, a study of this new parameter is initiated.

109-116

Authors: Kuppusamy Markandar Dharmalingam

Let G = (V;E) be a simple graph. Let H be the graph constructed from G as follows: V (H) = V (G), two points u and v are adjacent in H if and only if u and v are adjacent and degree equitable in G. H is called the adjacency inherent equitable graph of G or equitable associate of G and is denoted by e(G). This Paper aims at the study of a new concept called equitable associate graph of a graph. In this paper we show that there is relation between e(G) and e¡(G). Further results on the new parameter e(G),e¡(G) and complexity of equitable dominating set are discussed.

1-11

Authors: Ersin Aslan and Alpay Kirlangic

In a communication network, the vulnerability measures the resis- tance of the network to disruption of operation after the failure of certain stations or communication links. To measure the vulnerability we have some parameters which are connectivity, toughness, scattering number, integrity, tenacity and their edge analogues. This paper includes several results on the toughness and the scattering number of a gear graph as a communication network. Firstly, we compute the scattering number and toughness of a gear graph. In addition, the scattering number and tough- ness of the complement of a gear graph, the cartesian product of two gear graphs and the sequential join of gear graphs are computed. Finally, we compare the results for the scattering number and toughness.

21-26

Authors: Daniel A. Romano

We associate with every anti-ordered set ((X;=; 6=); ®) with ®\®¡1 = ; a partial groupoid ((X;=; 6=); ¢) in such a way that (x; y) 2 ® () x¢y = y and (x; y) ./ ® () x ¢ y = x for two elements x; y 2 X such that x 6= y.

39-43

Authors: Y.B.Venkatakrishnan amd V.Swaminathan

Let G = (X; Y;E) be a bipartite graph. A subset S of X is a comple- ment free dominating set if S is an X¡dominating set and X ¡ S is not an Y ¡dominating set. A subset S of X is called a minimal complement free dominating set if any proper subset of S is not a complement free dominating set. The minimum cardinality of a minimal complement free dominating set is called the complement free domination number of G and is denoted by °cf (G). In this paper some results on complement free domination number are obtained.

53-57

Authors: Zygfryd Kominek

We consider the following pexiderized version of Jensen-Hossz¶u equa- tion of the form 2f( x + y 2 ) = g(x + y ¡ xy) + h(xy); where f; g; h are unknown real-valued functions of a real variable. We prove that f; g; h are a±ne functions and, moreover, we prove that these equation is stable in the Hyers-Ulam sense.