Maximal first Zagreb index of trees with given Roman domination number

Abstract

The first Zagreb index of graphs is defined to be the sum of squares of degrees of all the vertices of graphs. It drew a great deal of attention in the past half-century. In this paper, we study the relationship between the first Zagreb index and Roman domination number of graphs. More precisely, we characterize the trees with the maximum first Zagreb index among trees with given Roman domination number.

Keywords

• the first zagreb index
• domination number
• roman domination number