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On the Generalized Adjacency Spread of a Graph

Author

Listed:
  • Maryam Baghipur

    (Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-163, Iran)

  • Modjtaba Ghorbani

    (Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-163, Iran)

  • Shariefuddin Pirzada

    (Department of Mathematics, University of Kashmir, Srinagar 192101, India)

  • Najaf Amraei

    (Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University, Tehran 16785-163, Iran)

Abstract

For a simple finite graph G , the generalized adjacency matrix is defined as A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) , α ∈ [ 0 , 1 ] , where A ( G ) and D ( G ) are respectively the adjacency matrix and diagonal matrix of the vertex degrees. The A α -spread of a graph G is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the A α ( G ) . In this paper, we answer the question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on the A α -spread of a graph. Electron. J. Linear Algebra 2020 , 36 , 214–227). Furthermore, we show that the path graph, P n , has the smallest S ( A α ) among all trees of order n . We establish a relationship between S ( A α ) and S ( A ) . We obtain several bounds for S ( A α ) .

Suggested Citation

  • Maryam Baghipur & Modjtaba Ghorbani & Shariefuddin Pirzada & Najaf Amraei, 2023. "On the Generalized Adjacency Spread of a Graph," Mathematics, MDPI, vol. 11(6), pages 1-9, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1416-:d:1097812
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    References listed on IDEAS

    as
    1. Guo, Haiyan & Zhou, Bo, 2020. "On adjacency-distance spectral radius and spread of graphs," Applied Mathematics and Computation, Elsevier, vol. 369(C).
    2. Das, Kinkar Ch. & Gutman, Ivan & Furtula, Boris, 2017. "On spectral radius and energy of extended adjacency matrix of graphs," Applied Mathematics and Computation, Elsevier, vol. 296(C), pages 116-123.
    3. Maryam Baghipur & Modjtaba Ghorbani & Hilal A. Ganie & Yilun Shang, 2021. "On the Second-Largest Reciprocal Distance Signless Laplacian Eigenvalue," Mathematics, MDPI, vol. 9(5), pages 1-12, March.
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