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

Abstract

Let G be a graph with vertex set V(G). For $u,v\in V(G)$, we write $v\sim u$ if vertices v and u are adjacent. The Cartesian product of G and H, denoted by $G\Box H$, is the graph with vertex set $V(G)\times V(H)$, where $(x,u)\sim (y,v)$ if and only if $x=y$ and $u\sim v$ in H, or $x\sim y$ in G and $u=v$. The lexicographic product of G and H, denoted by G[H], is the graph with vertex set $V(G)\times V(H)$, where $(x,u)\sim (y,v)$ if and only if $x\sim y$ in G, or $x=y$ and $u\sim v$ 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 $P_n\Box P_m$ 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.