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M-polynomial revisited: Bethe cacti and an extension of Gutman’s approach

  • Emeric Deutsch
  • Sandi KlavžarEmail author
Original Research

Abstract

The M-polynomial of a graph G is defined as \(\sum _{i\le j} m_{i,j}(G)x^iy^j\), where \(m_{i,j}(G)\), \(i,j\ge 1\), is the number of edges uv of G such that \(\{d_v(G), d_u(G)\} = \{i,j\}\). Knowing the M-polynomial, formulas for bond incident degree indices (an important subclass of degree-based topological indices) can be obtained by means of specific operators defined on differentiable functions in two variables. This is illustrated on three infinite families of Bethe cacti. Gutman’s approach for the computation of the coefficients of the M-polynomial is also recalled and an extension of it is given. This extension is used to determine the M-polynomial of a two-parameter infinite family of lattice graphs.

Keywords

M-polynomial Bethe cacti Degree-based topological index Bond incident degree index Graph polynomial 

Mathematics Subject Classification

05C07 05C31 92E10 

Notes

Acknowledgements

Sandi Klavžar acknowledges the financial support from the Slovenian Research Agency (Research Core Funding No. P1-0297).

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Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Polytechnic Institute of New York UniversityBrooklynUSA
  2. 2.Faculty of Mathematics and PhysicsUniversity of LjubljanaLjubljanaSlovenia
  3. 3.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  4. 4.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

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