Spectral determinations and eccentricity matrix of graphs

Let G be a connected graph on n vertices. For a vertex u∈G, the eccentricity of u is defined as ε(u)=max⁡{d(u,v)|v∈V(G)}, where d(u,v) denotes the distance between u and v. The eccentricity matrix E(G)=(ϵuv), where ϵuv:={d(u,v)if d(u,v)=min⁡{ε(u),ε(v)},0otherwise, has been firstly introduced in Chemical Graph Theory. In literature, it is also known as the DMAX-matrix. Graphs with the diameter equal to the radius are called self-centered graphs. Two non-isomorphic graphs are said to be M-cospectral with respect to a given matrix M if they have the same M-eigenvalues. In this paper, we show that, when n→∞, the fractions of non-isomorphic cospectral graphs with respect to the adjacency and the eccentricity matrix behave like those only concerning the self-centered graphs with diameter two. Secondly, we prove that a graph G has just two distinct E-eigenvalues if and only if G is an r-antipodal graph. Thirdly, we obtain many pairs of E-cospectral graphs by using strong and lexicographic products. Finally we formulate some problems waiting to be solved in order to build up a spectral theory based on the eccentricity matrix.