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

Let K_n1,n2,…,nk be a complete k-partite graph with k≥2 and n_i≥2 for i=1,2,…,k. The Turán 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. (2018) [11] conjectured that E_D(K_n1,n2,…,nk)<E_D(K_n1,n2,…,nk−e), where e is any edge of K_n1,n2,…,nk and proved that the above relation holds for k=2. Very recently, Tian et al. (2020) [10] confirmed that the above conjecture holds for T(n,k) with n≡0 (modk) 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≥2k. In this paper, we confirm that the former conjecture is true for k≥3 and then the latter conjecture follows immediately.