Commuting involution graphs of linear groups

Abstract

The commuting involution graph of a special linear group has been determined when the dimension is low and the base field is finite. But the general structure of the graph of the group based on arbitrary field and dimension has not been classified. As for general linear groups, which is larger than special linear groups, even the restricted classification has not been done. We determine the diameter of the commuting involution graphs of general linear groups over an arbitrary field. By using the properties of an involution matrix, we represent the matrix as the product of block-lower and block-upper triangular matrices, which enables us to prove the diameter. As a corollory of this, we verify the diameter of the graph of certain projective special linear groups over finite field. Further, we prove the diameter of the graph of the set of all involutions in the general linear group over a field of characteristic 2, completing the diameter of the graph of the set of all involutions over an arbitrary field. Finally, as an application, we classify the structure of the commuting involution graphs of four-dimensional general linear groups over finite field.