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On the minimum energy of regular graphs

Aashtab, A ; Sharif University of Technology | 2019

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  1. Type of Document: Article
  2. DOI: 10.1016/j.laa.2019.07.001
  3. Publisher: Elsevier Inc , 2019
  4. Abstract:
  5. The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. Gutman et al. proved that for every cubic graph of order n, E(G)≥n. Here, we improve this result by showing that if G is a connected subcubic graph of order n≥8, then E(G)≥1.01n. Also, we prove that if G is a traceable subcubic graph of order n≥8, then E(G)>1.1n. Let G be a connected cubic graph of order n≥8, it is shown that E(G)>n+2. It was proved that if G is a connected cubic graph of order n, then E(G)≤1.65n. Also, in this paper we would like to present the best lower bound for the energy of a connected cubic graph. We introduce an infinite family of connected cubic graphs whose for each element of order n, say G, E(G)≥1.24n, and conjecture that if 6|n, then minimum energy occurs just for each element of this family. We conjecture that there exists N such that for every connected cubic graph G of order n≥N, E(G)≥1.24n. © 2019
  6. Keywords:
  7. Subcubic graph ; Eigenvalues and eigenfunctions ; Absolute values ; Adjacency matrices ; Energy ; Energy of a graph ; Minimum energy ; Non-hypoenergetic ; Regular graphs ; Subcubic graphs ; Graph theory
  8. Source: Linear Algebra and Its Applications ; Volume 581 , 2019 , Pages 51-71 ; 00243795 (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/abs/pii/S0024379519302824