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

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.(c) 2023 Elsevier B.V. All rights reserved.