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

Abstract

‎The first Zagreb index $M_1$ of a graph $G$ is equal to the sum of squares‎ ‎of 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 be‎ ‎extended to hold for any connected graph with cyclomatic number $gamma geq 0$‎. ‎In addition‎, ‎graphs with $n$ vertices‎, ‎$n_1$ pendent vertices‎, ‎cyclomatic‎ ‎number $gamma$‎, ‎and minimal $M_1$ are characterized‎. ‎Explicit expressions‎ ‎for minimal $M_1$ are given for $gamma=0,1,2$‎, ‎which directly can be extended‎ ‎for $gamma>2$‎.