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Hypoenergetic and nonhypoenergetic digraphs

Akbari, S ; Sharif University of Technology | 2021

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  1. Type of Document: Article
  2. DOI: 10.1016/j.laa.2021.01.026
  3. Publisher: Elsevier Inc , 2021
  4. Abstract:
  5. The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. This concept was extended by Nikiforov to arbitrary complex matrices. Recall that the trace norm of a digraph D is defined as, N(D)=∑i=1nσi, where σ1≥⋯≥σn are the singular values of the adjacency matrix of D. In this paper we would like to present some lower and upper bounds for N(D). For any digraph D it is proved that N(D)≥rank(D) and the equality holds if and only if D is a disjoint union of directed cycles and directed paths. Finally, we present a lower bound on σ1 and N(D) in terms of the size of digraph D. © 2021 Elsevier Inc
  6. Keywords:
  7. Directed graphs ; Eigenvalues and eigenfunctions ; Graph structures ; Absolute values ; Adjacency matrices ; Complex matrices ; Directed paths ; Energy of a graph ; Lower and upper bounds ; Lower bounds ; Singular values ; Matrix algebra
  8. Source: Linear Algebra and Its Applications ; Volume 618 , 2021 , Pages 129-143 ; 00243795 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0024379521000458