On the Zagreb index inequality of graphs with prescribed vertex degrees

Abstract

a b s t r a c tFor a simple graph G with n vertices and m edges, the inequality M 1 (G)/n ≤ M 2 (G)/m, where M 1 (G) and M 2 (G) are the first and the second Zagreb indices of G, is known as the Zagreb indices inequality. A set S 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 2 log s) time and O(s) space.

Degree-Based Topological Indices

“…71 Unexpectedly, research along these lines did not stop, but a long series of papers resulted. [72][73][74][75][76][77][78][79][80][81][82][83][84][85] The relation (6) is nowadays referred to as the "Zagreb index inequality". For more details see the reviews.…”

On the Zagreb indices equality

“…Soon after the announcement of this conjecture it was shown [18] that there exist graphs for which (1) does not hold. Although the work [18] appeared to completely settle Hansen's conjecture, it was just the beginning of a long series of studies [1,2,5,8,19,21,23,26,27,32,33] in which the validity or non-validity of either [18] or some generalized version of [18] was considered for various classes of graphs. These studies are summarized in two recent surveys [23,24].…”

“…Some properties of the function f have been studied in [1,2]. Easy verification shows that the Zagreb indices equality holds for regular graphs and stars.…”

Phase Changes in the Topological Indices of Scale-Free Trees

“…This index of a (chemical) graph is defined as the sum of the squares of the degrees of all vertices in the graph. The Zagreb index also attracts attention in graph theory (see, for example, Andova et al (2011), Nikiforov (2007, and Peled et al (1999)). For a survey of the application of the Zagreb index in computational chemistry, we refer the reader to Nikolić et al (2003) and the references therein.…”