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Further results regarding the sum of domination number and average eccentricity

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  • Du, Zhibin

Abstract

The average eccentricity of a graph G, denoted by ecc(G), is the mean value of eccentricities of all vertices of G. Let Dn, i be the n-vertex tree obtained from a path Pn−1=v1v2⋯vn−1 by attaching a pendent vertex to vi. In [13], it was shown that the maximum value for the sum of domination number and average eccentricity among n-vertex (connected) graphs is attained by Dn, 3 when n≡0(mod3), and attained by the path Pn when n¬≡0(mod3). In this paper, we will further determine the second maximum value for the sum of domination number and average eccentricity among n-vertex (connected) graphs. It is interesting that the graphs attaining that second maximum value have three cases, which is Dn, 6 when n≡0(mod3),Dn, 3 when n≡1(mod3), and Tn when n≡2(mod3), where Tn is the n-vertex tree obtained from a path Pn−2=v1v2⋯vn−2 by attaching a pendent vertex to v3, and a pendent vertex to vn−4.

Suggested Citation

  • Du, Zhibin, 2017. "Further results regarding the sum of domination number and average eccentricity," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 299-309.
    • Handle: RePEc:eee:apmaco:v:294:y:2017:i:c:p:299-309
    DOI: 10.1016/j.amc.2016.09.014
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    References listed on IDEAS

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    1. 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.
    2. Shi, Yongtang, 2015. "Note on two generalizations of the Randić index," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 1019-1025.
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    Cited by:

    1. Brezovnik, Simon & Šumenjak, Tadeja Kraner, 2019. "Complexity of k-rainbow independent domination and some results on the lexicographic product of graphs," Applied Mathematics and Computation, Elsevier, vol. 349(C), pages 214-220.
    2. Sandi Klavžar & Kishori P. Narayankar & S. B. Lokesh, 2019. "Constructing uniform central graphs and embedding into them," Indian Journal of Pure and Applied Mathematics, Springer, vol. 50(2), pages 451-460, June.
    3. Kraner Šumenjak, Tadeja & Rall, Douglas F. & Tepeh, Aleksandra, 2018. "On k-rainbow independent domination in graphs," Applied Mathematics and Computation, Elsevier, vol. 333(C), pages 353-361.

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