Resistance distance of blowups of trees
A blowup of a graph Γ with respect to a graph Γ′ is the graph ΓΓ′ obtained from Γ by replacing every vertex u of Γ with a disjoint copy Γu′ of Γ′ and attach every vertex of Γu′ to every vertex of Γv′ if there is an edge between u and v in Γ. Accordingly, a P-blowup of a graph Γ is a Γ′-blowup of Γ with Γ′ being a P-graph for a graph-theoretical property P. We present combinatorial interpretations of Moore–Penrose inverses of incidence matrices of independent-blowups and clique-blowups of trees and apply them to obtain formulas for resistance distances between vertices of the corresponding blowup graphs as well as their Kirchhoff indices. Also, we give some results on arbitrary blowups of graphs as well.
Azimi, A.; Bapat, R. B.; and Farrokhi, M., "Resistance distance of blowups of trees" (2021). Journal Articles. 1897.