Resistance distance and Kirchhoff index of unbalanced blowups of graphs

A balanced blowup graph $G^H$ of G with respect to a fixed graph H is the graph obtained from G by replacing each vertex $v_i\in V\left(G\right)$ with a disjoint copy $H_{v_i}$ of H, and connecting each vertex of $V\left(H_{v_i}\right)$ to each vertex of $V\left(H_{v_j}\right)$ if there is an edge between $v_i$ and $v_j$ in G. In particular, if H is a complete graph (resp. an empty graph), then $G^H$ is called the clique-blowup (resp. independent-blowup) of G. In (Azimi et al. (2021) [1]), A. Azimi et al. presented combinatorial interpretations of Moore-Penrose inverses of incidence matrices of independent-blowups and clique-blowups of trees, and applied these results to obtain formulas for resistance distances between vertices of the corresponding blowup graphs as well as their Kirchhoff index. In this paper, we extend their results to unbalanced blowup graphs of G via combinatorial and electrical network approaches. In addition, our results contain the main results of (Pan et al. (2021) [17]), (Li et al. (2020) [14]) and (Yan et al. (2023) [26]) as special cases.