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

‎The first Zagreb index M_۱ 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_۱‎ ‎pendent vertices‎, ‎M_۱ \geq ۹\,n_۱-۱۶‎. ‎We show how this result can be‎ ‎extended to hold for any connected graph with cyclomatic number \gamma \geq ۰‎. ‎In addition‎, ‎graphs with n vertices‎, ‎n_۱ pendent vertices‎, ‎cyclomatic‎ ‎number \gamma‎, ‎and minimal M_۱ are characterized‎. ‎Explicit expressions‎ ‎for minimal M_۱ are given for \gamma=۰,۱,۲‎, ‎which directly can be extended‎ ‎for \gamma>۲‎.

Keywords:

degree (of vertex) , Zagreb index , First Zagreb index , extremal graphs