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The minimum Zagreb indices for unicyclic graphs with fixed Roman domination number | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 10 آذر 1403 اصل مقاله (413.4 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29846.2188 | ||
| نویسندگان | ||
| Fateme Movahedi1؛ Ayu Ameliatul Shahilah Ahmad Jamri2؛ Roslan Hasni* 2؛ Mohammadhadi Akhbari3؛ Hailiza Kamarulhaili4 | ||
| 1Department of Mathematics, Faculty of Sciences, Golestan University, Gorgan, Iran | ||
| 2Special Interest Group on Modelling and Data Analytics (SIGMDA), Faculty of Computer Science and Mathematics, Universiti Malaysia Terengganu, 21030 Kuala Nerus, Terengganu, Malaysia | ||
| 3Department of Mathematics, Estahban Branch, Islamic Azad University, Estahban, Iran | ||
| 4School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Penang, Malaysia | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Roman domination number؛ Unicyclic graphs؛ Zagreb indices | ||
| مراجع | ||
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