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

Communications in Combinatorics and Optimization

article

Ayu Ameliatul Shahilah Ahmad Jamri¹  Fateme Movahedi²  Roslan Hasni¹  Mohammad Hadi Akhbari³ 

[1] Universiti Malaysia Terengganu;Golestan University;Islamic Azad University

关键词: Second Zagreb index;    Roman domination number;    tree;   

DOI  :  10.22049/cco.2022.27553.1288

学科分类：社会科学、人文和艺术（综合）

来源: Azarbaijan Shahide Madani Universit

【 摘 要 】

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)].

【 授权许可】

CC BY-SA