On the resistance diameter of the Cartesian and lexicographic product of paths

Let G be a graph with vertex set V(G). For u, v is an element of V(G), we write v similar to u if vertices v and u are adjacent. The Cartesian product of G and H, denoted by G square H, is the graph with vertex set V(G) x V(H), where (x, u) similar to (y, v) if and only if x = y and u similar to v in H, or x similar to y in G and u = v. The lexicographic product of G and H, denoted by G[H], is the graph with vertex set V(G) xV(H), where (x, u) similar to (y, v) if and only if x similar to y in G, or x = y and u similar to v in H. The resistance diameter of graph G refers to the maximum resistance distance among all pairs of vertices in G. Let P-n be the path of n vertices. In this paper, the resistance diameters of P-n square P-m and P-n[P-m] are studied. Meanwhile, the maximal resistance distance, which is among some pairs of vertices in the lexicographic product of connected graph and orderable graph, is given.

Automated summary

The paper investigates the resistance diameters of P-n square P-m and P-n[P-m]. Additionally, it provides the maximal resistance distance among some pairs of vertices in the lexicographic product of connected graph and orderable graph.