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

3 Islamic Azad 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

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Communications in Combinatorics and Optimization
Volume 8, Issue 2
June 2023
Pages 391-396

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