Graphs with fixed number of pendent vertices and minimal first Zagreb index

The first zagreb index m1 of a graph g is equal to the sum of squares of degrees of the vertices of g. goubko proved that for trees with n1 pendent vertices, m1 ≥ 9 n1 − 16. we show how this result can be extended to hold for any connected graph with cyclomatic number γ ≥ 0. in addition, graphs with n vertices, n1 pendent vertices, cyclomatic number γ, and minimal m1 are characterized. explicit expressions for minimal m1 are given for γ = 0, 1, 2, which directly can be extended for γ > 2.