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On reduced second Zagreb index

Author

Listed:
  • Lkhagva Buyantogtokh

    (Mongolian National University of Education)

  • Batmend Horoldagva

    (Mongolian National University of Education
    Institute of Mathematics, National University of Mongolia)

  • Kinkar Chandra Das

    (Sungkyunkwan University)

Abstract

The reduced second Zagreb index $$RM_2$$RM2 of a graph G is defined as $$RM_2(G)=\sum _{uv\in E(G)}(d_G(u)-1)(d_G(v)-1)$$RM2(G)=∑uv∈E(G)(dG(u)-1)(dG(v)-1), where $$d_G(u)$$dG(u) is the degree of the vertex u of graph G. Furtula et al. (Discrete Appl Math 178: 83–88, 2014) studied the difference between the classical Zagreb indices of graphs and showed that it is closely related to $$RM_2$$RM2. In this paper, we obtain an upper bound in terms of order n and size m on $$RM_2$$RM2 of $$K_{r+1}$$Kr+1-free graphs. Also we prove that among all graphs of order n with chromatic number $$\chi $$χ, the Turán graph $$T_{n,\,\chi }$$Tn,χ is the unique graph having the maximum $$RM_2$$RM2. Furthermore, we completely characterize the extremal graphs with respect to $$RM_2$$RM2 among all unicyclic graphs of order n with girth g.

Suggested Citation

  • Lkhagva Buyantogtokh & Batmend Horoldagva & Kinkar Chandra Das, 2020. "On reduced second Zagreb index," Journal of Combinatorial Optimization, Springer, vol. 39(3), pages 776-791, April.
    • Handle: RePEc:spr:jcomop:v:39:y:2020:i:3:d:10.1007_s10878-019-00518-7
    DOI: 10.1007/s10878-019-00518-7
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    References listed on IDEAS

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    1. Ailin Hou & Shuchao Li & Lanzhen Song & Bing Wei, 2011. "Sharp bounds for Zagreb indices of maximal outerplanar graphs," Journal of Combinatorial Optimization, Springer, vol. 22(2), pages 252-269, August.
    2. John Estes & Bing Wei, 2014. "Sharp bounds of the Zagreb indices of k-trees," Journal of Combinatorial Optimization, Springer, vol. 27(2), pages 271-291, February.
    3. Borovićanin, Bojana & Furtula, Boris, 2016. "On extremal Zagreb indices of trees with given domination number," Applied Mathematics and Computation, Elsevier, vol. 279(C), pages 208-218.
    4. Gutman, Ivan, 2017. "On coindices of graphs and their complements," Applied Mathematics and Computation, Elsevier, vol. 305(C), pages 161-165.
    5. Shuchao Li & Licheng Zhang & Minjie Zhang, 2019. "On the extremal cacti of given parameters with respect to the difference of zagreb indices," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 421-442, August.
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    Cited by:

    1. Sakander Hayat & Farwa Asmat, 2023. "Sharp Bounds on the Generalized Multiplicative First Zagreb Index of Graphs with Application to QSPR Modeling," Mathematics, MDPI, vol. 11(10), pages 1-14, May.
    2. Lkhagva Buyantogtokh & Batmend Horoldagva & Kinkar Chandra Das, 2022. "On General Reduced Second Zagreb Index of Graphs," Mathematics, MDPI, vol. 10(19), pages 1-18, September.
    3. Kinkar Chandra Das & Yilun Shang, 2021. "Some Extremal Graphs with Respect to Sombor Index," Mathematics, MDPI, vol. 9(11), pages 1-15, May.
    4. Das, Kinkar Chandra & Gutman, Ivan, 2022. "On Sombor index of trees," Applied Mathematics and Computation, Elsevier, vol. 412(C).

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