The first and second Zagreb indices were first introduced by Gutman and Trinajsti？ (1972). It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. Recently, the first and second Zagreb coindices, a new pair of invariants, were introduced in Do？li？ (2008). In this paper we introduce the and ( )-analogs of the above Zagreb indices and coindices and investigate the relationship between the enhanced versions to get a unified theory. 1. Introduction Like in many other branches of mathematics, one tries to find in graph theory certain invariants of graphs which depend only on the graph itself (or in other cases, in addition to, an embedding into the plane or some other manifold), see, for example, [1] and the references given therein. A graph invariant is any function on a graph that does not depend on a labeling of its vertices. A big number of different invariants have been employed to date in graphs structural studies as well as through a broad range of applications including molecular biology, organic chemistry, nuclear physics, neurology, psychology, linguistics, logistics, and economics, see [2]. Here we are interested in the theory of Zagreb indices and Zagreb coindices. The first and second kinds of Zagreb indices were first introduced in [3] (see also [4]). It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in other practical aspects. Recently, the first and second Zagreb coindices, a new pair of invariants, were introduced in [5]. In this paper we introduce the and -analogs of the above Zagreb indices and coindices and investigate the relationship between the enhanced versions to get a unified theory. Insightfully, this theory will have its place and influence in the territory of mathematical chemistry. Throughout this work we consider only simple and finite graphs, that is, finite graphs without multiedges or loops. For terms and concepts not mentioned here we refer, for instance, the readers to [6–8]. Let be a finite simple graph on vertices and edges. We denote the set of vertices of by and the set of edges of by . The complement of , denoted by , is a simple graph on the set of vertices in which two vertices are adjacent if and only if they are not adjacent in . Thus, if and only if . Clearly, , which implies that . We denote the degree of a vertex in a graph by . It is easy to see that for all . We will omit the subscript in the degree and other notation if the graph is clear from the
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