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.