Sharp bounds of the Zagreb indices of k-trees

Abstract

For a graph G, the first Zagreb index M 1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M 2 is equal to the sum of the products of degrees of pairs of adjacent vertices. The Zagreb indices have been the focus of considerable research in computational chemistry dating back to Gutman and Trinajstic in 1972. In 2004, Das and Gutman determined sharp upper and lower bounds for M 1 and M 2 values for trees along with the unique trees that obtain the minimum and maximum M 1 and M 2 values respectively. In this paper, we generalize the results of Das and Gutman to the generalized tree, the k-tree, where the results of Das and Gutman are for k=1. Also by showing that maximal outerplanar graphs are 2-trees, we also extend a result of Hou, Li, Song, and Wei who determined sharp upper and lower bounds for M 1 and M 2 values for maximal outerplanar graphs.