On reformulated Zagreb indices

Abstract

The first and second reformulated Zagreb indices are defined respectively in terms of edge-degrees as E M 1 ( G ) = ∑ e ∈ E deg ( e ) 2 and E M 2 ( G ) = ∑ e ∼ f deg ( e ) deg ( f ) , where deg ( e ) denotes the degree of the edge e , and e ∼ f means that the edges e and f are adjacent. We give upper and lower bounds for the first reformulated Zagreb index, and lower bounds for the second reformulated Zagreb index. Then we determine the extremal n -vertex unicyclic graphs with minimum and maximum first and second Zagreb indices, respectively. Furthermore, we introduce another generalization of Zagreb indices.