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Wiener–Hosoya Matrix of Connected Graphs

Author

Listed:
  • Hassan Ibrahim

    (Department of Mathematics Statistics & Computer Science, Federal University of Agriculture, Makurdi P.M.B 2373, Nigeria)

  • Reza Sharafdini

    (Department of Mathematics, Faculty of Intelligent Systems Engineering and Data Science, Persian Gulf University, Bushehr 75169, Iran)

  • Tamás Réti

    (Bánki Donát Faculty of Mechanical and Safety Engineering, Óbuda University Bécsiút 96/B, H-1034 Budapest, Hungary)

  • Abolape Akwu

    (Department of Mathematics Statistics & Computer Science, Federal University of Agriculture, Makurdi P.M.B 2373, Nigeria)

Abstract

Let G be a connected (molecular) graph with the vertex set V ( G ) = { v 1 , ⋯ , v n } , and let d i and σ i denote, respectively, the vertex degree and the transmission of v i , for 1 ≤ i ≤ n . In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G , which is defined as the n × n matrix whose ( i , j ) -entry is equal to σ i 2 d i + σ j 2 d j if v i and v j are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

Suggested Citation

  • Hassan Ibrahim & Reza Sharafdini & Tamás Réti & Abolape Akwu, 2021. "Wiener–Hosoya Matrix of Connected Graphs," Mathematics, MDPI, vol. 9(4), pages 1-12, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:359-:d:497490
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