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

Abstract

The Laplacian energy of a digraph $G$ is defined as $\sum_{i=1}^n λ_i^2$, where $λ_i$ 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 Turán problems to spectral Turán 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.

Figures (8)

• Figure 1: Digraph $\overrightarrow{F_{n,k}}$ ($q$ copies of $\overleftrightarrow{K_k}$ and one copy of $\overleftrightarrow{K_r}$)
• Figure 2: The two $\overrightarrow{C_3}$ in $G_1$ of Case 2.1
• Figure 3: The two $\overrightarrow{C_3}$ in $G_1$ of Case 2.1.1
• Figure 4: The $\overrightarrow{C_3}$ in $G_1$ of Case 2.1.2
• Figure 5: The example of $\overrightarrow{C_3}$ in $G_2$ of Case 2'

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Lemma 2.1
• proof
• Claim 1
• Lemma 3.1