On the spectrum and energy of digraphs with self-loops

2025/08/09 by Fung, Kevin, Lim, Johnny
#05C50 #05C90 #05C92 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.07004

Abstract

A digraph with self-loops D_S with vertex set V is a simple digraph with a self-loop attached at every vertex in S ⊂ V. In this paper, we study the energy E(D_S) of D_S and its properties, which extend several classical results on simple directed graphs. If D₁,...,D_k are the strong components of D_S, we establish a necessary and sufficient conditions for E(D_S) ≤ ∑^k_i=1 E(D_i), for which the strict inequality exists for S≠ ∅. We also provide several bounds and characterizations for the energy and spectral radius of D_S, including the McClelland type bound. Lastly, we propose a notion of the complement of D_S and establish some formulae describing the relationship between the energy and spectrum of regular digraphs with their complement.

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