a lower bound for the second zagreb index of trees with given roman domination number

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 $w(f)=sum_{vin v(g)} f(v)$. 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)].