  Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix
of a graph $G$, respectively. Then $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ are
called Laplacian matrix and signless Laplacian matrix of the graph $G$,
respectively. Let $G$ be a graph with $n$ vertices and $m$ edges. Then the
$LI$-matrix and $QI$-matrix of $G$ are defined as $$LI(G)=L(G)-\frac{2m}{n}I_n
\quad \text{and} \quad QI(G)=Q(G)-\frac{2m}{n}I_n,$$ where $I_n$ is the
identity matrix. In this paper, we are interested in extremal properties of the
Ky Fan $k$-norm of the $LI$-matrix and $QI$-matrix of graphs, which are closely
related to the well-known problems and results in spectral graph theory, such
as the (signless) Laplacian spectral radius, the (signless) Laplacian spread,
the sum of the $k$ largest (signless) Laplacian eigenvalues, the (signless)
Laplacian energy, and other parameters. Upper and lower bounds on the Ky Fan
$k$-norm of the $LI$-matrix and $QI$-matrix of graphs are given, and the
extremal graphs are partly characterized. In addition, upper and lower bounds
on the Ky Fan $k$-norm of $LI$-matrix and $QI$-matrix of trees, unicyclic
graphs and bicyclic graphs are determined, and the corresponding extremal
graphs are characterized.
