On squared distance matrix of complete multipartite graphs

Let G = K n 1 , n 2 , ⋯ , n t be a complete t -partite graph on n = ∑ i = 1 t n i vertices. The distance between vertices i and j in G , denoted by d ij is defined to be the length of the shortest path between i and j . The squared distance matrix Δ ( G ) of G is the n × n matrix with ( i , j ) th entry equal to 0 if i = j and equal to d ij 2 if i ≠ j . We define the squared distance energy E Δ ( G ) of G to be the sum of the absolute values of its eigenvalues. We determine the inertia of Δ ( G ) and compute the squared distance energy E Δ ( G ) . More precisely, we prove that if n i ≥ 2 for 1 ≤ i ≤ t , then E Δ ( G ) = 8 ( n - t ) and if h = | { i : n i = 1 } | ≥ 1 , then 8 ( n - t ) + 2 ( h - 1 ) ≤ E Δ ( G ) < 8 ( n - t ) + 2 h . Furthermore, we show that for a fixed value of n and t , both the spectral radius of the squared distance matrix and the squared distance energy of complete t -partite graphs on n vertices are maximal for complete split graph S n , t and minimal for Turán graph T n , t .