Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 11 3 2026 09 01 The minimum Zagreb indices for unicyclic graphs with fixed Roman domination number 959 973 14867 10.22049/cco.2024.29846.2188 EN Fateme Movahedi Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, Iran Ayu Ameliatul Shahilah Ahmad Jamri Special Interest Group on Modelling and Data Analytics (SIGMDA), Faculty of Computer Science and Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia Roslan Hasni Special Interest Group on Modelling and Data Analytics (SIGMDA), Faculty of Computer Science and Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia Mohammadhadi Akhbari Department of Mathematics, Estahban Branch, Islamic Azad University, Estahban, Iran Hailiza Kamarulhaili School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia Journal Article 2024 06 24 Let $G=(V, E)$ be a graph with the vertex set $V$ and the edge set $E$. The first Zagreb index of a graph $G$ is defined to be the sum of squares of degrees of all the vertices of the graph. The second Zagreb index of the graph $G$ is the sum of the $d(u)d(v)$ for every edge $uv \in E$, where $d(u)$ and $d(v)$ denote the degree of the vertices $u, v \in V$. In this paper, we propose new lower bounds of the Zagreb indices of unicyclic graphs in terms of the order and the Roman domination number. We prove that $4n-2\left(\gamma_{R}-\left\lceil\dfrac{2n}{3}\right\rceil\right)$ and $4n-3\left(\gamma_{R}-\left\lceil\dfrac{2n}{3}\right\rceil\right)$ are the sharp lower bounds for the first Zagreb index and the second Zagreb index, respectively. Also, we characterize the extremal trees for these lower bounds. https://comb-opt.azaruniv.ac.ir/article_14867_02da586fd53300ff836f994ef621d9f7.pdf