# A lower bound for the second Zagreb index of trees with given Roman domination number

Document Type : Short notes

Authors

1 Universiti Malaysia Terengganu(UMT)

2 Golestan University

Abstract

For a (molecular) graph, the second Zagreb index $M_2(G)$ is equal to the sum of the products of the degrees of pairs of adjacent vertices. Roman dominating function $RDF$ of $G$ is a function $f:V(G)\rightarrow \{0,1,2\}$ satisfying the condition that every vertex with label 0 is adjacent to a vertex with label 2. The weight of an $RDF$ $f$ is $w(f)=\sum_{v\in V(G)} f(v)$. The Roman domination number of $G$, denoted by $\gamma_R (G)$, is the minimum weight among all RDF in $G$. In this paper, we present a lower bound on the second Zagreb index of trees with $n$ vertices and Roman domination number and thus settle one problem given in [On the Zagreb indices of graphs with given Roman domination number, Commun. Comb. Optim. DOI: 10.22049/CCO.2021.27439.1263 (article in press)].

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#### References

[1] A.A.S. Ahmad Jamri, R. Hasni, and S.K. Said Husin, On the Zagreb indices of graphs with given Roman domination number, Commun. Comb. Optim., in press.
[2] S. Bermudo, J.E. Nápoles, and J. Rada, Extremal trees for the Randić index with given domination number, Appl. Math. Comput. 375 (2020), ID: 125122.
[3] B. Borovićanin, K.C. Das, B. Furtula, and I. Gutman, Bounds for Zagreb indices, MATCH Commun. Math. Comput. Chem. 78 (2017), no. 1, 17–100.
[4] B. Borovićanin and Boris Furtula, On extremal Zagreb indices of trees with given domination number, Appl. Math. Comput. 279 (2016), 208–218.
[5] M. Chellali, N. Jafari Rad, S.M. Sheikholeslami, and L. Volkmann, Roman domination in graphs, Topics in Domination in Graphs (T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, eds.), Springer, Berlin/Heidelberg, 2020, pp. 365–409.
[6] E.J. Cockayne, P.A. Dreyer, S.M. Hedetniemi, and S.T. Hedetniemi, Roman domination in graphs, Discrete Maths. 278 (2004), no. 1-3, 11–22.
[7] P. Dankelmann, Average distance and domination number, Discrete Appl. Math. 80 (1997), no. 1, 21–35.
[8] K.C. Das and I. Gutman, Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem. 52 (2004), no. 1, 103–112.
[9] I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50 (2004), no. 1, 83–92.
[10] I. Gutman and N. Trinajstić, Graph theory and molecular orbits. Total π−electron energy of alternant hydrocarbons, Chem. Phsy. Lett. 17 (1972), no. 4, 535–538.
[11] T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, Topics in Domination in Graphs, Springer, Berlin/Heidelberg, 2020.
[12] D.A. Mojdeh, M. Habibi, L. Badakhshian, and Y. Rao, Zagreb indices of trees, unicyclic and bicyclic graphs with given (total) domination, IEEE Access 7 (2019), 94143–94149.
[13] S. Nikolić, G. Kovačević, A. Miličević, and N. Trinajstić, The Zagreb indices 30 years after, Croat. Chem. Acta 76 (2003), no. 2, 113–124.
[14] R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, John Wiley & Sons, 2008.