LAUSR

The Zagreb indices are among the oldest and the most famous topological molecular structure-descriptors. The first Zagreb index is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index is equal to the sum of the products of the degrees of pairs of adjacent vertices of the respective graph. In this paper, we characterize the extremal graphs with maximal, second-maximal, third-maximal, fourth-maximal and minimal, secondminimal, third-minimal Zagreb indices among all Eulerian graphs, and then we give the tight conditions on the Zagreb indices of a graph for the existence of a spanning eulerian subgraph, dominating circuits, spanning circuits, Hamiltonian paths and cycles, respectively.