Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 8 1 2023 03 01 On the Zagreb indices of graphs with given Roman domination number 141 152 14324 10.22049/cco.2021.27439.1263 EN Ayu Ameliatul Shahilah Ahmad Jamri MENGGABANG TELIPOT KUALA NERUS Roslan Hasni Universiti Malaysia Terengganu(UMT), Malaysia Sharifah Kartini Said Husain Universiti Putra Malaysia(UPM) Journal Article 2021 09 02 Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The two Zagreb indices $M_1=\sum_{v\in V(G)} d^2_G(v)$ and $M_2=\sum_{uv\in E(G)} d_G(u)d_G(v)$ 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. https://comb-opt.azaruniv.ac.ir/article_14324_9c5f1f9cdee97c3e03b866d32fa67a9e.pdf