A method for constructing graphs with the same resistance spectrum

Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The resistance distance RG(x,y) between two vertices $x,y$ of G is defined to be the effective resistance between the two vertices in the corresponding electrical network in which each edge of G is replaced by a unit resistor. The resistance spectrum RS(G) of a graph G is the multiset of the resistance distances of all pairs of vertices in the graph. This paper presents a method for constructing graphs with the same resistance spectrum. It is obtained that for any positive integer k, there exist at least 2ᵏ graphs with the same resistance spectrum. Furthermore, it is shown that for n ≥ 10, there are at least $2(n-9) p(n-9)$ pairs of graphs of order n with the same resistance spectrum, where $p(n-9)$ is the number of partitions of the integer $n-9$.