Extremal Laplacian energy of $\overrightarrow{C_{k+1}}$-free digraphs

TL;DR

This paper investigates the maximum Laplacian energy of digraphs that do not contain a directed cycle of length k+1, extending spectral Turán problems to directed graphs and characterizing extremal structures.

Contribution

It determines the maximum Laplacian energy and characterizes extremal $ ightarrow$C_{k+1}$-free digraphs, advancing spectral Turán theory for directed graphs.

Findings

Identified the maximum Laplacian energy for $ ightarrow$C_{k+1}$-free digraphs.

Characterized the structure of extremal digraphs achieving this maximum.

Extended Turán-type spectral problems to directed graph settings.

Abstract

The Laplacian energy of a digraph $G$ is defined as $\sum_{i=1}^n \lambda_i^2$, where $\lambda_i$ are the eigenvalues of the Laplacian matrix of $G$. A (di)graph $G$ is said to be $H$-free if it does not contain a copy of the fixed (di)graph $H$ as a sub(di)graph. In this paper, we extend the Tur\'{a}n problems to spectral Tur\'{a}n problems in digraphs: what is the maximal Laplacian energy of an $H$-free digraph of given order? In particular, we determine the maximum Laplacian energy and characterize the extremal digraphs of $\overrightarrow{C_{k+1}}$-free digraphs.