
Jiang Zhou

Zhongyu Wang

Changjiang Bu
Keywords:
Resistance distance, Resistance matrix, Laplacian matrix, Resistanceregular graph, Reigenvalue
Abstract
Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called Reigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistanceregular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistanceregular graphs and graphs with few distinct Reigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any nonadjacent vertices $i,j\in V(G)$.