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...

The algebraic connectivity of a graph, which is the second-smallest eigenvalue of the Laplacian of the graph, is a measure of connectivity. We show that the problem of adding a specified number of edges to an input graph to maximize the algebraic connectivity of the augmented graph is NP-hard.

— Concerning algebraic grammars in reduced form, bisimulation in the graph ofleft dérivations is known to be décidable. The quotient of the above graph by ils greatest bisimulation, called the canonical graph, is indeed isomorphic to the graph ofleft dérivations for another algebraic grammar in reduced form, produced by an effective procedure.

In this paper, we focus on using the low-rank factorization approach to solve SDP relaxation of a graph equipartition problem, which involves an additional spectral upper bound over traditional linear SDP. We discuss equivalence between decomposed problem and original problem. also derive sufficient condition, under second order stationary point non-convex is global minimum. Moreover, constrain...