Spectral rational variation in two places for adjacency matrix is impossible

Let G = (V, E) be a simple graph and {λ1(G), … , λn(G)} be its adjacency spectrum. It is easy to see that if an edge is added between two isolated vertices, then one zero eigenvalue increases by 1, and another zero eigenvalue decreases by 1. Let G+ be a connected graph obtained from G by adding an edge e ∉ E(G). In this paper, it will be proved that the spectrum of G+ is different from that of G only in two places with one eigenvalue increases by m and another eigenvalue decreases by m, where m > 0 is a rational number, if and only if G is an empty graph with order 2. It will also be proved that one cannot construct a new adjacency integral connected graph with order n ⩾ 3 from a known one by adding an edge.