  Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix
of a graph $G$, respectively. Then $L(G)=D(G)-A(G)$ is called Laplacian matrix
of the graph $G$. Let $G$ be a graph with $n$ vertices and $m$ edges. Then the
$LI$-matrix of $G$ are defined as $LI(G)=L(G)-\frac{2m}{n}I_n$, 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 of graphs, which is closely related to
the well known problems and results in spectral graph theory, such as the
Laplacian spectral radius, the Laplacian spread, the sum of the $k$ largest
Laplacian eigenvalues, the Laplacian energy, and other parameters. Some bounds
on the Ky Fan $k$-norm of the $LI$-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 of trees, unicyclic graphs and bicyclic graphs are
determined, and the corresponding extremal graphs are characterized.
