Solution to a conjecture on resistance distances of block tower graphs

TL;DR

This paper confirms and generalizes a conjecture about the limiting behavior of resistance distances in block tower graphs, providing new insights into their electrical network properties and resistance diametrical pairs.

Contribution

The paper proves the conjecture on resistance distances in block tower graphs and extends it, also identifying all resistance diametrical pairs in these graphs.

Findings

Confirmed the conjecture on resistance distance limits.

Generalized the conjecture to broader cases.

Identified all resistance diametrical pairs in $G_n$.

Abstract

Let $G$ be a connected graph. The resistance distance between two vertices $u$ and $v$ of $G$, denoted by $R_{G}[u,v]$, 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 $D_{r}(G)$, is defined as the maximum resistance distance among all pairs of vertices of $G$. Let $P_n=a_1a_2\ldots a_n$ be the $n$-vertex path graph and $C_{4}=b_{1}b_2b_3b_4b_{1}$ be the 4-cycle. Then the $n$-th block tower graph $G_n$ is defined as the the Cartesian product of $P_n$ and $C_4$, that is, $G_n=P_{n}\square C_4$. Clearly, the vertex set of $G_n$ is $\{(a_i,b_j)|i=1,\ldots,n;j=1,\ldots,4\}$. In [Discrete Appl. Math. 320 (2022) 387--407], Evans and Francis proposed the following conjecture on resistance distances of $G_n$ and $G_{n+1}$: \begin{equation*} \lim_{n \rightarrow \infty}\left(R_{G_{n+1}}[(a_{1},b_1),(a_{n+1},b_3)]-R_{G_{n}}[(a_{1},b_1),(a_{n},b_3)]\right)=\frac{1}{4}. \end{equation*} In this paper, combining algebraic methods and electrical network approaches, we confirm and further generalize this conjecture. In addition, we determine all the resistance diametrical pairs in $G_n$, which enables us to give an equivalent explanation of the conjecture.