Resistance distance in subdivision-vertex join and subdivision-edge join of graphs

• DOI: 10.1016/j.laa.2014.06.018
• Publication Date: 2014-06-28T00:00:17Z
• Authors: Changjiang Bu; Bo Yan; Xiuqing Zhou; Jiang Zhou
• Keywords: Neighbourhood (mathematics); Join (topology)

Abstract The subdivision graph S ( G ) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let G 1 ∪ G 2 be the disjoint union of two graphs G 1 and G 2 . The subdivision-vertex join of G 1 and G 2 , denoted by G 1 ∨ ˙ G 2 , is the graph obtained from S ( G 1 ) ∪ G 2 by joining every vertex in V ( G 1 ) to every vertex in V ( G 2 ) . The subdivision-edge join of G 1 and G 2 , denoted by G 1 ⊻ G 2 , is the graph obtained from S ( G 1 ) ∪ G 2 by joining every vertex in I ( G 1 ) to every vertex in V ( G 2 ) , where I ( G 1 ) is the set of inserted vertices of S ( G 1 ) . In this paper, formulae for resistance distance in G 1 ∨ ˙ G 2 and G 1 ⊻ G 2 are obtained.