Rooted rainbow connection number of graphs

Let [Formula: see text] be a connected graph, [Formula: see text] be an edge coloring of [Formula: see text] and [Formula: see text] be a subset of the vertices of [Formula: see text]. [Formula: see text] is called an [Formula: see text]-rooted rainbow coloring if for each vertex [Formula: see text], there is a rainbow path from [Formula: see text] to at least one vertex of [Formula: see text]. The [Formula: see text]-rooted rainbow connection number of [Formula: see text], denoted by [Formula: see text], is the smallest number of colors between all [Formula: see text]-rooted rainbow colorings of [Formula: see text]. When [Formula: see text], this edge coloring is called [Formula: see text]-rooted rainbow coloring, and similarly, we use [Formula: see text] instead of [Formula: see text]. In this paper, we investigate [Formula: see text]-rooted (and [Formula: see text]-rooted) rainbow coloring and [Formula: see text]-rooted (and [Formula: see text]-rooted) rainbow connection number of graphs and their connection with the eccentricity parameters of graphs. Also, we find the upper and lower bounds or exact value of rooted rainbow connection number for some classes of graphs. We will also compare the parameters [Formula: see text]-rooted rainbow connection number and rainbow connection number of some graphs, how close and how far apart they can be from one another.