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Graph Theory pp 177-281 | Cite as

An Annotated Glossary of Graph Theory Parameters, with Conjectures

  • Ralucca GeraEmail author
  • Teresa W. Haynes
  • Stephen T. Hedetniemi
  • Michael A. Henning
Chapter
Part of the Problem Books in Mathematics book series (PBM)

Abstract

This glossary contains an annotated listing of some 300 parameters of graphs, together with their definitions, and, for most of these, a reference to the authors who introduced them. Let G = (V, E) be an undirected graph having order n = |V | vertices and size m = |E| edges. Two graphs G and H are isomorphic, denoted G ≃ H, if there exists a bijection ϕ : V (G) → V (H) such that two vertices u and v are adjacent in G if and only if the two vertices ϕ(u) and ϕ(v) are adjacent in H. For the purposes of this paper, we shall say that a parameter of a graph G is any integer-valued function \(f: \mathcal {G} \rightarrow \mathcal {Z}\) from the class of all finite graphs \(\mathcal {G}\) to the integers \(\mathcal {Z}\), such that for any two graphs G and H, if G is isomorphic to H then f(G) = f(H). This glossary also contains a listing of some 70 conjectures related to these parameters, more than 26 new parameters and open problem areas for study, and some 600 references to papers in which these parameters were introduced and then studied.

Notes

Acknowledgements

The authors would like to thank Robert Beeler, Gary Chartrand, Mustapha Chellali, Jean Dunbar, Renu Laskar, C.M. Mynhardt, and Ping Zhang for their constructive criticism of the manuscript, which greatly improved this chapter.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Ralucca Gera
    • 1
    Email author
  • Teresa W. Haynes
    • 2
  • Stephen T. Hedetniemi
    • 3
  • Michael A. Henning
    • 4
  1. 1.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA
  2. 2.Department of MathematicsEast Tennessee State UniversityJohnson CityUSA
  3. 3.School of ComputingClemson UniversityClemsonUSA
  4. 4.Department of Pure and Applied MathematicsUniversity of JohannesburgAuckland ParkSouth Africa

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