Ranks of Regular Graphs Under Certain Unary Operations

Abstract

We consider the rank of the adjacency matrix of some classes of regular graphs that are transformed under certain unary operations. In particular, we study the ranks of the subdivision graph, the connected cycle graph, the connected subdivision graph, and the total graph of the following families of graphs: cycles, complete graphs, complete bipartite and multipartite graphs, circulant graphs of degrees three and four, and some Cartesian graph products.