A method for constructing graphs with the same resistance spectrum

Abstract

Let G be a simple graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. The resistance distance $R_G(x,y)$ 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 $\mathrm{RS}\left(G\right)$ of a graph G is the multiset of the resistance distances between all pairs of vertices in the graph. This paper presents a novel method for constructing graphs with the same resistance spectrum. It is obtained that for any positive integer k, there exist at least $2^k$ graphs with the same resistance spectrum. Furthermore, it is shown that for $n\ge 10$, there are at least $2\left(\right(n-10\left)p\right(n-9)+q(n-9\left)\right)$ pairs of graphs of order n with the same resistance spectrum, where $p(n-9)$ and $q(n-9)$ are the numbers of partitions of the integer $n-9$ and simple graphs of order $n-9$, respectively.