Vol. 24 No. 1 (2025): Mapana Journal of Sciences
Research Articles

Degree-Eccentricity Matrix of Graphs and Some Properties

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  • Akhil B.,
  • Manju V. N.,
  • Roy John
Akhil B.
Department of Mathematics, University of Kerala, Kariavattom, Thiruvananthapuram, 695581, Kerala, India
Manju V. N.
Department of Mathematics, University of Kerala, Kariavattom, Thiruvananthapuram, 695581, Kerala, India
Roy John
Department of Mathematics, St. Stephen’s College, Pathanapuram, Kollam, 689695, Kerala, India
DOI: https://doi.org/10.12723/mjs.72.7

Published 2025-04-05

Keywords

  • Degree-Eccentricity matrix,
  • Degree-Eccentricity energy,
  • Irreducibility,
  • Primitivity

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Abstract

This paper presents a new matrix for a given graph called the Degree-Eccentricity (DE) matrix, which consists of the degree and eccentricity of a vertex. Properties such as irreducibility and primitivity of this matrix are discussed. Further we obtain the spectrum and energy of DE matrices associated with various classes of graphs and some graphs obtained through graph operations. Also, we try to develop an algorithm to construct a new class of graph with DE energy equal to one. Further, we made an attempt to discover few graphs with DE energy equal to one. Also, an upper bound for the eigenvalues of DE matrix is obtained.

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References

  1. A. Abdollahi, S. Janbaz, M. R. Oboudi, Graphs cospectral with a friendship graph or its complement, Transactions on Combinatorics, Vol. 2 No. 4 (2013), pp 37-52.
  2. A. Elumalai, Some graph energies of research, Malaya Journal of Mathematik, Vol. S, No. 2, 4013-4016, 2020.
  3. Bo Zhou, Ivan Gutman, Jose Antonio de la Pena, Juna Rada and Leonel Mendoza, On Spectral Moments and Energy of Graphs, MATCH Commun, Math. Comput. Chem. 57 (2007) 183-191.
  4. Dario Fasino, Orthogonal Cauchy-like matrices, Numerical Algorithms (2023) 92: 619-637
  5. Fumio Hiai, D´enes Petz, Introduction to Matrix Analysis and Applications, Hindustan Book Agency, India (2014)
  6. G. Suresh Singh, Graph Theory, PHI, New Delhi (2022).
  7. Indulal G, Studies on the Spectrum and the Energy of Graphs, Ph.D thesis, Department of Mathematics, Cochin University of Science and Technology, March 2007.
  8. Ivan Gutman, Boris Furtula, Graph Energies and their Applications Bulletin T. CLII de l’Acad′emie serbe des sciences et des arts - 2019 Classe des Sciences math′ematiques et naturelles Sciences math´ematiques, No 44, 29-45.
  9. N. J. Pullman, Matrix Theory and its Applications, Selected Topics, Pure and Applied Mathematics, A series of Monographs and Textbook, Marcel Dekker, Inc, New York, 1976.
  10. R. Balakrishnan, K. Ranganathan, A Textbook of Graph Theory, Second Edition, Springer, 2012.
  11. S Pirzada, An Introduction to Graph Theory, Universities Press (India) Private
  12. Limited, 2009.
  13. S.Meenakshi and S. Lavanya, A Survey on Energy of Graphs, Annals of Pure and Applied Mathematics Vol. 8, No. 2, 2014, 183-191
  14. Xingzhi Zhan, Matrix Theory, Graduate Studies in Mathematics, Vol 147, American Mathematical Society Providence, Rhode Island.