Some Results on the Resistance-Distance Spectrum

Abstract:

      For vertices $u$ and $v$ in graph $G$, the resistance distance $r_G(u,v)$ between $u$ and $v$ is the effective resistance between them in an electrical network corresponding to $G$ when the resistance between any adjacent vertices is one unit. The resistance-distance eigenvalues of a connected graph $G$ are the eigenvalues of its resistance-distance matrix $R(G)=(r_G(u,v))_{u,v\in V(G)}$. We determine the graph that uniquely minimizes the largest resistance-distance eigenvalue over all connected graphs that are different from the complete graph and the complete graph with one edge deleted and over all connected graphs with fixed number of cut edges, respectively, and we also discuss properties for the smallest resistance-distance eigenvalue.