The commuting graph of bounded linear operators on a Hilbert space

On the lower bound for diameter of commuting graph of prime-square sized matrices

“…On the other hand, every finite graph is an induced subgraph of a commuting graph of a finite group, see [18]. Unaware of this result, the authors proved in [3] a similar statement, valid also for countably infinite graphs by essentially the same technique, for induced subgraphs of commuting graph of B( 2 ), the algebra of bounded operators on a complex separable Hilbert space. It was also shown in [3] that not all finite graphs are induced subgraphs of a commuting graph of complex matrices if the size is kept fixed.…”

Realizability Problem for Commuting Graphs

“…The base step is given by (4). To prove the induction step, assume we have already shown that vertices a, c with d(a, c) = 2 and with labels +1 satisfy…”

“…Finally suppose that G ∼ = N ⋊ H with H ∼ = A 5 ∼ = SL (2,4) and with N = t 1 Z 4 2 the direct sum of t copies of the natural module Z 4 2 ∼ = GF(4) 2 for SL (2,4). Then v is a Sylow 3-subgroup of G and so there are precisely |G|/|N G ( v )| = (|V (Γ)| + 1)/6 isolated edges (use that S 3 ⊆ H, so v and v 2 are conjugate).…”

“…However, the vertices of this subgraph, in general, do not form a semigroup as they may not be closed under multiplication. Unaware of [15], the second author with collaborators [2] proved that every finite simple graph is isomorphic to the induced subgraph of the commuting graph of complex matrices with sufficiently large size. The vertices of this subgraph can be taken to be projections.…”

Graphs Generated by Measures

“…Lots of graphs are constructed from algebraic objects such as semigroups ( [1]), groups ( [2,3]), and rings ( [4,5]). Recently some graphs have been constructed from Hilbert and topological spaces ( [6,7]) which have nice graph theoretical properties. This motivates us to construct a bigger category of graphs which has more properties.…”