Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $Gamma(R)$, is a graph with vertex set $RZ(R)$ and two vertices $a$ and $b$ are adjacent iff $ab=ba$. In this paper, we consider the commuting graph of non-commutative rings of order pq and $p^2q$ with Z(R) = 0 and non-commutative rings with unity of order $p^3q$. It is proved that $C_R(a)$ is a commutative ring...

Quantum heat cycles and quantum refrigerators are analyzed using various systems as their working mediums. For example, to evaluate the efficiency work done of Carnot cycle in regime, one can consider harmonic oscillator a medium. all these well-defined substances (which commutative space structure), engine is not up mark efficiency. So, inevitable question arise, observe catalytic effect on en...