On the Zagreb index inequality of graphs with prescribed vertex degrees

For a simple graph G with n vertices and m edges, the inequality M1(G)/n ≤ M2(G)/m, where M1(G) and M2(G) are the first and the second Zagreb indices of G, is known as Zagreb indices inequality. According to this inequality, a set S of integers is good if for every graph whose degrees of vertices are in S, the inequality holds. We characterize that an interval [a, a+n] is good if and only if a ≥ n(n−1) 2 or [a, a+n] = [1, 4]. We also present an algorithm that decides if an arbitrary set S of cardinality s is good, which requires O(s log s) time and O(s) space. 1 Pr ep ri n t se ri es , I M FM , I S S N 2 23 220 94 , n o. 1 12 0, M ay 2 5, 2 01 0