On the Zagreb indices of graphs with given Roman domination number

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The two Zagreb indices $M_1=sum_{vin V(G)} d^2_G(v)$ and $M_2=sum_{uvin E(G)} d_G(u)d_G(v)$ are vertex degree based graph invariants that have been introduced in the 1970s and extensively studied ever since. {In this paper, we first give a lower bound on the first Zagreb index of trees with given Roman domination number and we characterize all extremal trees. Then we present upper bound for Zagreb indices of unicyclic and bicyclic graphs with given Roman domination number.