The resistance distance and Kirchhoff index on quadrilateral graph and pentagonal graph

The quadrilateral graph Q(G) is obtained from G by replacing each edge in G with two parallel paths of length 1 and 3, whereas the pentagonal graph W(G) is obtained from G by replacing each edge in G with two parallel paths of length 1 and 4. In this paper, closed-form formulas of resistance distance and Kirchhoff index for quadrilateral graph and pentagonal graph are obtained whenever G is an arbitrary graph.Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1810.0332

The normalized Laplacians and random walks of the parallel subdivision graphs

The $k$-parallel subdivision graph $S_k(G)$ is generated from $G$ which each edge of $G$ is replaced by $k$ parallel paths of length 2. The $2k$-parallel subdivision graph $S_{2k}(G)$ is constructed from $G$ which each edge of $G$ is replaced by $k$ parallel paths of length 3. In this paper, the normalized Laplacian spectra of $S_k(G)$ and $S_{2k}(G)$ are given. They turn out that the multiplicities of the corresponding eigenvalues are only determined by $k$. As applications, the expected hitting time, the expected commute time and any two-points resistance distance between vertices $i$ and $j$ of $S_k(G)$, the normalized Laplacian spectra of $S_k(G)$ and $S_{2k}(G)$ with $r$ iterations are given. Moreover, the multiplicative degree Kirchhoff index, Kemeny's constant and the number of spanning tress of $S_k(G)$, $S_k^r(G)$, $S_{2k}(G)$ and $S_{2k}^r(G)$ are respectively obtained. Our results have generalized the previous works in Xie et al. and Guo et al. respectively