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On Spectral Radius and Energy of a Graph with Self-Loops

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  • Deekshitha Vivek Anchan
  • Gowtham H. J.
  • Sabitha D’Souza
  • Xinan Hao

Abstract

The spectral radius of a square matrix is the maximum among absolute values of its eigenvalues. Suppose a square matrix is nonnegative; then, by Perron–Frobenius theory, it will be one among its eigenvalues. In this paper, Perron–Frobenius theory for adjacency matrix of graph with self-loops AGS will be explored. Specifically, it discusses the nontrivial existence of Perron–Frobenius eigenvalue and eigenvector pair in the matrix AGS−σnI, where σ denotes the number of self-loops. Also, Koolen–Moulton type bound for the energy of graph GS is explored. In addition, the existence of a graph with self-loops for every odd energy is proved.

Suggested Citation

  • Deekshitha Vivek Anchan & Gowtham H. J. & Sabitha D’Souza & Xinan Hao, 2024. "On Spectral Radius and Energy of a Graph with Self-Loops," Mathematical Problems in Engineering, Hindawi, vol. 2024, pages 1-7, May.
    • Handle: RePEc:hin:jnlmpe:7056478
    DOI: 10.1155/2024/7056478
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