The resistance distance between two vertices in a connected graph is defined as the effective resistance between them when each edge of the graph is replaced by a unit resistor. The multiset of all pairwise resistance distances in a graph G constitutes its resistance spectrum, denoted RS(G). A graph G is said to be determined by its resistance spectrum (DRS) if, for any graph H satisfying \({RS}(H) = {RS}(G)\) , we have \(H \cong G\) . While computational studies reveal that the vast majority of small graphs are DRS, providing rigorous mathematical proofs for specific infinite families remains a challenging and active research area. This paper investigates the resistance spectral determination for four significant families of graphs: barbell graphs, chained silicate networks, Dutch windmill graphs, and caterpillar trees. By employing fundamental principles from electrical network theory—including the series and parallel rules, the cut-vertex principle, and Foster’s theorems—we explicitly analyze the resistance structure of these graphs. We then prove that each family is uniquely DRS. Our results contribute new infinite families of graphs to the growing catalog of those known to be DRS, thereby enhancing our understanding of the expressive power of the resistance spectrum in graph characterization.
