On some metric properties of direct-co-direct product

Direct-co-direct product G⊛H of graphs G and H is a graph on vertex set V(G)×V(H). Two vertices (g,h) and (g′,h′) are adjacent if gg′∈E(G) and hh′∈E(H) or gg′∉E(G) and hh′∉E(H). We show that eccentricity of a vertex of G⊛H for connected non-complete graphs G and H is bounded by five. In addition, we fully describe when the eccentricity is four or five and in all cases one factor must be a star. This is a cornerstone for the distance formula for G⊛H. The disconnected cases of G⊛H are also characterized along the way.