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On some properties of graph irregularity indices with a particular regard to the σ-index

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  • Réti, Tamás

Abstract

Some well-known graph irregularity indices of a connected graph G are investigated. Our study is focused mainly on the comparison of the Bell’s degree-variance (Var(G)) and the Collatz–Sinogowitz irregularity index (CS(G)) with the recently introduced σ(G) irregularity index. It is a degree-based topological invariant calculated as σ(G)=F(G)−2M2(G) where M2(G) is the second Zagreb index, F(G)=∑d3(v), and d(v) is the degree of the vertex v in G. By introducing the notion of the complete split-like graphs representing a broad subclass of bidegreed connected graphs, it is shown that for these graphs the equality σ(G)=n2Var(G) holds.

Suggested Citation

  • Réti, Tamás, 2019. "On some properties of graph irregularity indices with a particular regard to the σ-index," Applied Mathematics and Computation, Elsevier, vol. 344, pages 107-115.
    • Handle: RePEc:eee:apmaco:v:344-345:y:2019:i::p:107-115
    DOI: 10.1016/j.amc.2018.10.010
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    1. Aouchiche, M. & Bell, F.K. & Cvetkovic, D. & Hansen, P. & Rowlinson, P. & Simic, S.K. & Stevanovic, D., 2008. "Variable neighborhood search for extremal graphs. 16. Some conjectures related to the largest eigenvalue of a graph," European Journal of Operational Research, Elsevier, vol. 191(3), pages 661-676, December.
    2. Hansen, Pierre & Mladenovic, Nenad & Moreno Pérez, Jos´e A., 2008. "Variable neighborhood search," European Journal of Operational Research, Elsevier, vol. 191(3), pages 593-595, December.
    3. Gutman, Ivan, 2018. "Stepwise irregular graphs," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 234-238.
    4. van Dam, E.R., 1997. "Nonregular Graphs with Three Eigenvalues," Research Memorandum 746, Tilburg University, School of Economics and Management.
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