graphs with fixed number of pendent vertices and minimal first zagreb index

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 $gamma>2$.