Sharp lower bounds for the Zagreb indices of unicyclic graphs

The first Zagreb index M-1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M-2 is equal to the sum of the products of the degrees of pairs of adjacent vertices of the respective graph. In this paper we present the lower bound on M-1 and M-2 among all unicyclic graphs of given order, maximum degree, and cycle length, and characterize graphs for which the bound is attained. Moreover, we obtain some relations between the Zagreb indices for unicyclic graphs.