Spectral integral variations of degree maximal graphs

The concept of the spectral integral variation was introduced by Fan [Fan Yizheng (2002). On spectral integral variations of graphs. Linear and Multilinear Algebra , 50 , 133-142] to study the general graphs with all changed eigenvalues moving up by integers when an edge is added. Here we consider the spectral integral variations of maximal graphs G, and successfully give an equivalent condition for the spectral integral variation of G occurring in two places by adding an edge e. We also characterize whether the graph G + e is maximal so that an explicit interpretation of the above condition is obtained, where G + e denotes the graph obtained from G by adding an edge e.