Proof of a conjecture on distance energy change of complete multipartite graph due to edge deletion

Abstract Let K n 1 , n 2 , … , n k be a complete k-partite graph with k ≥ 2 and n i ≥ 2 for i = 1 , 2 , … , k . The Turan graph T ( n , k ) is a complete k-partite graph of n vertices with sizes of partitions as equal as possible. The distance energy E D ( G ) of a graph G is defined as the sum of absolute values of distance eigenvalues of the graph G. Varghese et al. [Distance energy change of complete bipartite graph due to edge deletion, Linear Algebra Appl. 553 (2018) 211–222] conjectured that E D ( K n 1 , n 2 , … , n k ) E D ( K n 1 , n 2 , … , n k − e ) , where e is any edge of K n 1 , n 2 , … , n k and proved that the above relation holds for k = 2 . Very recently, Tian et al. [The change of distance energy of some special complete multipartite graphs due to edge deletion, Linear Algebra Appl. 584 (2020) 438–457] confirmed that the above conjecture holds for T ( n , k ) with n ≡ 0 ( m o d k ) and T ( n , 3 ) . They also mentioned a weaker conjecture as follows: E D ( T ( n , k ) ) E D ( T ( n , k ) − e ) , where e is any edge of T ( n , k ) and k ≥ 2 , n ≥ 2 k . In this paper, we confirm that the former conjecture is true for k ≥ 3 and then the latter conjecture follows immediately.