Solution to a conjecture on resistance distances of block tower graphs

Let G be a connected graph. The resistance distance between two vertices u and v of G, denoted by RG[u,v], 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 of G, denoted by Dᵣ(G), is defined as the maximum resistance distance among all pairs of vertices of G. Let Pₙ=a₁a₂… aₙ be the n-vertex path graph and C₄=b₁b₂b₃b₄b₁ be the 4-cycle. Then the n-th block tower graph Gₙ is defined as the the Cartesian product of Pₙ and C₄, that is, Gₙ=Pₙ C₄. Clearly, the vertex set of Gₙ is \(aᵢ,bⱼ)|i=1,…,n;j=1,…,4\. In [Discrete Appl. Math. 320 (2022) 387--407], Evans and Francis proposed the following conjecture on resistance distances of Gₙ and Gₙ₊₁: {equation*} limn → ∞(R_{Gₙ₊₁}[(a₁,b_1),(aₙ₊₁,b_3)]-R_{Gₙ}[(a₁,b_1),(aₙ,b_3)])=1/4. {equation*} In this paper, combing algebraic methods and electric network approaches, we confirm and further generalize this conjecture. In addition, we determine all the pair of vertices that reach the resistance diameter of Gₙ, which enables us to give an equivalent explanation of the conjecture.