Proper Eccentric Graphs of Join of Graphs

Abstract

The eccentricity of any vertex [Formula: see text] in a connected graph [Formula: see text] is the length of the largest distance from [Formula: see text] to any other vertex in [Formula: see text]. The eccentric graph of any graph [Formula: see text], denoted by [Formula: see text], is a graph with the same vertex set as [Formula: see text] and two vertices in [Formula: see text] are adjacent if the distance between those vertices is equal to the eccentricity of either of the vertices. A graph [Formula: see text] is properly connected if there is a properly colored path between every pair of vertices in it. In this paper, we define the proper eccentric graph for any properly connected graph [Formula: see text]. We examine the connectivity, proper eccentricity, proper diameter, and several other graph invariants of a proper eccentric graph of the join of any two connected graphs.