Pei Lidan, Pan Xiangfeng
School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601 China.
J Inequal Appl. 2018;2018(1):16. doi: 10.1186/s13660-017-1597-3. Epub 2018 Jan 10.
Let [Formula: see text] be a graph. A set [Formula: see text] is a distance -dominating set of if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text], where is a positive integer. The distance -domination number [Formula: see text] of is the minimum cardinality among all distance -dominating sets of . The first Zagreb index of is defined as [Formula: see text] and the second Zagreb index of is [Formula: see text]. In this paper, we obtain the upper bounds for the Zagreb indices of -vertex trees with given distance -domination number and characterize the extremal trees, which generalize the results of Borovićanin and Furtula (Appl. Math. Comput. 276:208-218, 2016). What is worth mentioning, for an -vertex tree , is that a sharp upper bound on the distance -domination number [Formula: see text] is determined.
设(G)为一个图。集合(S\subseteq V(G))是(G)的一个距离支配集，如果对于每个顶点(v\in V(G))，存在某个顶点(u\in S)使得(d(u, v)\leq k)，其中(k)是一个正整数。(G)的距离支配数(\gamma_d(G))是(G)的所有距离支配集中最小的基数。(G)的第一 Zagreb 指标定义为(M_1(G)=\sum_{uv\in E(G)} (d(u)+d(v)))，(G)的第二 Zagreb 指标为(M_2(G)=\sum_{uv\in E(G)} d(u)d(v))。在本文中，我们得到了具有给定距离支配数的(n)顶点树的 Zagreb 指标的上界，并刻画了极值树，这推广了 Borovićanin 和 Furtula（《应用数学与计算》276:208 - 218，2016）的结果。值得一提的是，对于一个(n)顶点树(T)，确定了距离支配数(\gamma_d(T))的一个精确上界。
