Some lower bounds for the energy of graphs

Akbari, S ; Sharif University of Technology | 2020

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  1. Type of Document: Article
  2. DOI: 10.1016/j.laa.2020.01.001
  3. Publisher: Elsevier Inc , 2020
  4. Abstract:
  5. The singular values of a matrix A are defined as the square roots of the eigenvalues of A⁎A, and the energy of A denoted by E(A) is the sum of its singular values. The energy of a graph G, E(G), is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A=(BDD⁎C), then E(A)≥2E(D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E(H) is an edge cut of G, then E(H)≤E(G), i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E(G)≥4m+n(n−2)|det⁡(A)|[Formula presented]. Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E(G) given in Oboudi (2019) [14]. © 2020 Elsevier Inc
  6. Keywords:
  7. Energy of graph ; Hermitian matrix ; Singular values ; Eigenvalues and eigenfunctions ; Graph theory ; Absolute values ; Adjacency matrices ; Bipartite graphs ; Connected graph ; Energy of a graph ; Energy of graphs ; Hermitian matrices ; Singular values ; Matrix algebra
  8. Source: Linear Algebra and Its Applications ; Volume 591 , 2020 , Pages 205-214
  9. URL: https://www.sciencedirect.com/science/article/abs/pii/S002437952030001X