The change of distance energy of some special complete multipartite graphs due to edge deletion

Abstract Let K p 1 , p 2 , … , p r be a complete r-partite graph with r ≥ 2 and p i ≥ 2 . Varghese, Wasin So and Vijayakumar in [Linear Algebra Appl. 553 (2018) 211-222] conjectured that the distance energy of K p 1 , p 2 , … , p r is always increased when an edge is deleted. In this paper, we prove that it is true for K p , p , … , p and the tripartite Turan graph T ( n , 3 ) . Subsequently, it is proved that, for any positive integer r, there exists a class of graphs that have r positive eigenvalues, but the distance energy of these graphs is always increased when an edge is deleted.