On the Ky Fan $k$-norm of the $LI$-matrix of graphs

TL;DR

This paper investigates the extremal properties of the Ky Fan $k$-norm of the $LI$-matrix of graphs, providing bounds and characterizations for various classes of graphs related to spectral graph theory.

Contribution

It introduces new bounds and characterizations for the Ky Fan $k$-norm of the $LI$-matrix, extending spectral graph theory results to this matrix parameter.

Findings

Bounds on the Ky Fan $k$-norm of $LI$-matrix for general graphs

Characterization of extremal graphs for these bounds

Bounds for trees, unicyclic, and bicyclic graphs

Abstract

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.