Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively.