On some metric properties of direct-co-direct product

Direct-co-direct product G O H of graphs G and H is a graph on vertex set V (G ) x V (H) . Two vertices (g, h ) and (g ' , h ' ) are adjacent if gg' E E(G) and hh' E E(H) or gg' E/ E(G) and hh' E/ E(H). We show that eccentricity of a vertex of G O 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 O H. The disconnected cases of G O H are also characterized along the way.& COPY; 2023 Elsevier Inc. All rights reserved.

Automated summary

The direct-co-direct product graph G O H is formed by the vertex set V(G) x V(H). Two vertices (g, h) and (g', h') are adjacent if gg' belongs to E(G) and hh' belongs to E(H), or gg' does not belong to E(G) and hh' does not belong to E(H). We prove that for connected non-complete graphs G and H, the eccentricity of a vertex of G O H is bounded by five. Additionally, we fully describe the cases where the eccentricity is four or five, and in all cases, one factor must be a star. This is crucial for the distance formula of G O H. The disconnected cases of G O H are also characterized.