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

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The first Zagreb index $M_1$ of a graph $G$ is equal to the sum of squaresof degrees of the vertices of $G$. Goubko proved that for trees with $n_1$pendent vertices, $M_1 geq 9,n_1-16$. We show how this result can beextended to hold for any connected graph with cyclomatic number $gamma geq 0$.In addition, graphs with $n$ vertices, $n_1$ pendent vertices, cyclomaticnumber $gamma$, and minimal $M_1$ are characterized. Explicit expressionsfor minimal $M_1$ are given for $gamma=0,1,2$, which directly can be extendedfor $gamma>2$.