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On the A α -Spectral Radii of Cactus Graphs

Author

Listed:
  • Chunxiang Wang

    (School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
    These authors contributed equally to this work.)

  • Shaohui Wang

    (Department of Mathematics, Louisiana College, Pineville, LA 71359, USA
    These authors contributed equally to this work.)

  • Jia-Bao Liu

    (School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
    These authors contributed equally to this work.)

  • Bing Wei

    (Department of Mathematics, University of Mississippi, University, MS 38677, USA
    These authors contributed equally to this work.)

Abstract

Let A ( G ) be the adjacent matrix and D ( G ) the diagonal matrix of the degrees of a graph G , respectively. For 0 ≤ α ≤ 1 , the A α -matrix is the general adjacency and signless Laplacian spectral matrix having the form of A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) . Clearly, A 0 ( G ) is the adjacent matrix and 2 A 1 2 is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The A α -spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results.

Suggested Citation

  • Chunxiang Wang & Shaohui Wang & Jia-Bao Liu & Bing Wei, 2020. "On the A α -Spectral Radii of Cactus Graphs," Mathematics, MDPI, vol. 8(6), pages 1-9, May.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:6:p:869-:d:364346
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