On the Resistance Matrix of a Graph

Let G be a connected graph of order n. The resistance matrix of G is defined as RG = (rij(G))n×n, where rij(G) is the resistance distance between two vertices i and j in G. Eigenvalues of RG are called R-eigenvalues of G. If all row sums of RG are equal, then G is called resistance-regular. For any connected graph G, we show that RG 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 RG 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 c1, c2 such that rij(G) = c1 for any adjacent vertices i, j ∈ V (G), and rij(G) = c2 for any non-adjacent vertices i, j ∈ V (G).