On Schatten p-norm of the distance matrices of graphs

For a connected simple graph G, the generalized distance matrix is defined by $$ D_{\alpha }:= \alpha Tr(G)+(1-\alpha ) D(G), ~0\le \alpha \le 1 $$ D α : = α T r ( G ) + ( 1 - α ) D ( G ) , 0 ≤ α ≤ 1 , where Tr(G) is the diagonal matrix of vertex transmissions and D(G) is the distance matrix. For particular values of $$ \alpha $$ α , we obtain the distance matrix, the distance Laplacian matrix and the distance signless Laplacian matrix and other uncountable distance based matrices. Let $$ \partial _{1}\ge \partial _{2}\ge \dots \ge \partial _{n} $$ ∂ 1 ≥ ∂ 2 ≥ ⋯ ≥ ∂ n be the $$ D_{\alpha } $$ D α eigenvalues of G and $$ p\ge 2 $$ p ≥ 2 a real number, the Schatten p-norm is the p-th root of the sum of p-th powers of eigenvalues of $$ D_{\alpha }, ~\alpha \in [\frac{1}{2},1] $$ D α , α ∈ [ 1 2 , 1 ] , that is, $$ \Vert D_{\alpha }\Vert _{p}^{p} =\partial _{1}^{p}+\partial _{2}^{p}+\dots +\partial _{n}^{p}$$ ‖ D α ‖ p p = ∂ 1 p + ∂ 2 p + ⋯ + ∂ n p . In this paper, we obtain various bounds for $$ \Vert D_{\alpha }\Vert _{p}^{p} $$ ‖ D α ‖ p p in terms of different graph parameters and characterize the corresponding extremal graphs.