The Bounds of Energies of Rough Complemented Graph

Abstract

The main objective of this paper is to study the various energies and their bounds of the Rough complemented graph corresponding to the given Rough semiring. In this paper, for a given approximation space I=(U,R) where U is the nonempty finite set of objects and R is an equivalence relation on U, the Rough semiring (T,∆,∇) is taken for study. The Rough complemented graph of T  denoted by GRC(T) is a graph whose vertices are V(GRC(T))={RS(Y)|Y∈〖℘(E)〗^1 }   be the set of equivalence classes induced by I and two distinct vertices RS(X) and RS(Y) are adjacent iff RS(X)∇RS(Y)=RS(∅). Note that there will be 2^n-2  vertices in GRC(T). Also Randic, seidel, minimum dominating, maximal independent and dominating energies of GRC(T) are obtained, the lower and upper bounds of these energies are also established. These energies are obtained through Python programming, and a bar diagram is used to conduct a comparative study for various values of n. All the illustrated concepts are explained with suitable examples.