Resistance distance and Kirchhoff index of the Q-vertex (or edge) join graphs

Abstract Graph operations have been extensively applied to the analysis of complex networks with properties abstracted from the real world. The resistance distance is a distance function which is related to the hitting time and random walks on graphs. In this paper, we focus on the resistance distance of a Q-vertex (or edge) join operation G 1 〈 v 〉 G 2 (or G 1 〈 e 〉 G 2 ) of graphs G 1 and G 2 . When G 1 is a regular graph, we give the resistance distance of the Q-vertex (or edge) join graph in terms of the number of walks in G 1 〈 v 〉 G 2 (or G 1 〈 e 〉 G 2 ).