On the spectral closeness and residual spectral closeness of graphs

The spectral closeness of a graph G is defined as the radius matrix , whose ( u v )-entry for vertex and 2 − d u,v ) if ≠ 0 otherwise, where distance between in . residual nontrivial minimum subgraphs with one deleted. We propose local grafting operations that decrease or increase determine those graphs uniquely minimize and/or maximize some families graphs. also discuss extremal properties closeness.