Solution to a conjecture on resistance diameter of lexicographic product of paths

Abstract

The resistance distance R[u,v] between two vertices u and v of a graph G is defined as the net effective resistance between them in the electric network constructed from G by replacing each edge with a unit resistor. The resistance diameter Dr(G) of G is the maximum resistance distance among all pairs of vertices of G. Given two path graphs Pn=a1a2…an and Pm=b1b2…bm, let Pn[Pm] be the lexicographic product of Pn and Pm with vertex set {(ai,bj)|i=1,…,n;j=1,…,m}. In [J. Appl. Math. Comput. 68 (2022) 1743–1755], Li et al. proved that for n>10, Dr(Pn[Pm])=R[(a1,b1),(an,bm)]=R[(a1,b1),(an,b1)]=R[(a1,bm),(an,b1)]=R[(a1,bm),(an,bm)]. In addition, they found that the result is not true for n=2. For 3≤n≤10 and enough small m, they checked by computer that the result is still true. Based on their observation, they conjectured that the result is true for 3≤n≤10. In this paper, by combinatorial and electrical network approaches, we confirm the conjecture.