A method for constructing graphs with the same resistance spectrum

Abstract

Let $G=(V(G),E(G))$ be a graph with vertex set $V(G)$ and edge set $E(G)$. 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 $\mathrm{RS}(G)$ 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 $2^k$ graphs with the same resistance spectrum. Furthermore, it is shown that for $n \geq 10$, 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 $n-9$.

Figures (12)

• Figure 1: $13$ pairs of nonisomorphic graphs with the same resistance spectrum WeissteinRSEquivalent.
• Figure 2: $G (S, A, \mathcal{H},T)$.
• Figure 3: Two graphs $T_1$ and $T_2$ holding the relation $\mathcal{U}$.
• Figure 4: $G_{2,7,8}^{I}$
• Figure 5: $Q_{1, 1, 1, 1}$ and $Q_{1, 1, 1, 2}$

Theorems & Definitions (18)

• Definition 2.1
• Example 2.2
• Proposition 2.3
• Definition 2.4
• Example 2.5
• Lemma 2.6
• Proposition 2.7
• Example 2.8
• Proposition 2.9
• Theorem 2.10