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 R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. 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 resistance-regular graphs and graphs with few distinct R-eigenvalues. 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 non-adjacent vertices $i,j\in V(G)$.
@article{10_37236_5295,
author = {Jiang Zhou and Zhongyu Wang and Changjiang Bu},
title = {On the resistance matrix of a graph},
journal = {The electronic journal of combinatorics},
year = {2016},
volume = {23},
number = {1},
doi = {10.37236/5295},
zbl = {1333.05191},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5295/}
}
TY - JOUR
AU - Jiang Zhou
AU - Zhongyu Wang
AU - Changjiang Bu
TI - On the resistance matrix of a graph
JO - The electronic journal of combinatorics
PY - 2016
VL - 23
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/5295/
DO - 10.37236/5295
ID - 10_37236_5295
ER -
%0 Journal Article
%A Jiang Zhou
%A Zhongyu Wang
%A Changjiang Bu
%T On the resistance matrix of a graph
%J The electronic journal of combinatorics
%D 2016
%V 23
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/5295/
%R 10.37236/5295
%F 10_37236_5295
Jiang Zhou; Zhongyu Wang; Changjiang Bu. On the resistance matrix of a graph. The electronic journal of combinatorics, Tome 23 (2016) no. 1. doi: 10.37236/5295
