A lower bound for the second Zagreb index of trees with given Roman domination number

Ahmad Jamri, Ayu Ameliatul Shahilah; Movahedi, Fateme; Hasni, Roslan; Akhbari, Mohammad Hadi

doi:10.22049/cco.2022.27553.1288

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	A lower bound for the second Zagreb index of trees with given Roman domination number
Communications in Combinatorics and Optimization
مقاله 8، دوره 8، شماره 2، شهریور 2023، صفحه 391-396 اصل مقاله (361.24 K)
نوع مقاله: Short notes
شناسه دیجیتال (DOI): 10.22049/cco.2022.27553.1288
نویسندگان
Ayu Ameliatul Shahilah Ahmad Jamri¹؛ Fateme Movahedi²؛ Roslan Hasni^* ¹؛ Mohammad Hadi Akhbari³
¹Universiti Malaysia Terengganu(UMT)
²Golestan University
³Islamic Azad University
چکیده
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 $w(f)=\sum_{v\in V(G)} f(v)$. 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)].
کلیدواژه‌ها
Second Zagreb index؛ Roman domination number؛ tree

مراجع
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A lower bound for the second Zagreb index of trees with given Roman domination number