The Primary Decomposition of the Factor Group

The main purpose of this work is to find the primary decomposition of factor group K (The factor group of all z-valued class functions module the group of Z-valued generalized characters for elementary abelian group G),where G is a finite abelian group of type , p=3. This work depends on finding the rational valued character matrix from the character table of and finding the invariant factors of this matrix. We have used the MATLAB program to calculate some results of this work

Doubly connected pitchfork domination and it’s inverse in graphs

Let G be a finite, simple, undirected graph and without isolated vertices. A subset D of V is a pitchfork dominating set if every vertex v ∈ D dominates at least j and at most k vertices of V − D, for any j and k integers. A subset Dˉ¹ of V − D is an inverse pitchfork dominating set if Dˉ¹ is a dominating set. The domination number of G, denoted by γpf (G) is a minimum cardinality over all pitchfork dominating sets in G. The inverse domination number of G, denoted by γˉ¹ pf (G) is a minimum cardinality over all inverse pitchfork dominating sets in G. In this paper, a special modified pitchfork dominations called doubly connected pitchfork domination and it’s inverse are introduced when j = 1 and k = 2. Some properties and bounds are studied with respect to the order and the size of the graph. These modified dominations are applied and evaluated for several well known and complement graphs.Publisher's Versio

Changing and unchanging on Tadpole domination number in G − e, G + e graphs

In this paper, the effect of “Tadpole domination” change is examined, which is one of the domination types, when a graph is modified, by deleting or adding an edge in G. This edge deletion may represent a cut in a network. The occurrence of any interruption in connections of this network may lead to the pause of data transmission in the network and thus affect the work of the entire network, as well as the need to create new necessary connections within the network or excluding others for the possibility of reducing the cost. Based on these criteria “Tadpole domination number” change has been examined. The increase, decrease, and non-increase or decrease was determined for this number, in case of deletion or addition, and we have proved some basic cases for this domination change.Publisher's Versio