Relation Between the Fractional Domination Number of Some Graphs and their Line Graphs

Abstract

Let G is the simple connected graph with order n and its line graph noted as L(G). Let the fractional dominating number denoted by γ_(f ) (G)  of graph G and the upper fractional dominating number Г_(f ) (G). In this study we have obtained union and join on〖  γ〗_(f ) (G)  and γ_(f ) (L(G))  exploring the graphs for upper fractional domination number including Cycle, complete, Star, Bi-Star graph, wheel graph, Cubic graph, Graphs of Cartesian product like (K_2  ×〖 P〗_(n )), (K_3  ×〖 P〗_(n )) and (Cm×Cn) with consideration of the computational complexity. We have taken parameters related to       fractional domination in line graphs towards generalization. The goal of this paper is to provide a generalized results of sum 〖 γ〗_(f ) (G)+γ_(f ) (L(G)),  Г_(f ) (G)+Г_(f ) L(G) and product 〖 γ〗_(f ) (G)*γ_(f ) (L(G)),  Г_(f ) (G)*Г_(f ) L(G) for some specific graph classes.