1. Resistance distances in generalized join graphs.
- Author
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Xu, Shaohan and Xu, Kexiang
- Subjects
- *
SPANNING trees , *INDEPENDENT sets - Abstract
Let H be a graph with vertex set V (H) = { v 1 , v 2 , ... , v k }. The generalized join graph H [ G 1 , G 2 , ... , G k ] is obtained from H by replacing each vertex v i with a graph G i and joining each vertex in G i with each vertex in G j provided v i v j ∈ E (H). If every G i is an independent set of n i vertices, then we write H [ G 1 , G 2 , ... , G k ] as H [ n 1 , n 2 , ... , n k ] , which is called the blow-up of H. In this paper we introduce the local complement transformation in electrical networks and obtain an electrically equivalent graph of H [ n 1 , n 2 , ... , n k ]. As their applications, we obtain formulae for resistance distances of some vertex-weighted graphs and give a unified technique to compute resistance distances in H [ G 1 , G 2 , ... , G k ] when every G i for 1 ≤ i ≤ k is a graph consisting of a matching and isolated vertices, which results in closed formulae for resistance distances of H [ G 1 , G 2 , ... , G k ] when H are some given graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2025
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