On spectral irregularity of graphs

The spectral radius \(\rho (G)\) of a graph G is the largest eigenvalue of the adjacency matrix of G. For a graph G with maximum degree \(\Delta (G)\), it is known that \(\rho (G)\le \Delta (G)\) with equality when G is connected if and only if G is regular. So the quantity \(\beta (G)=\Delta (G)-\rho (G)\) is a spectral measure of irregularity of G. In this paper, we identify the trees of order \(n\ge 12\) with the first 15 largest \(\beta \)-values, the unicyclic graphs of order \(n\ge 17\) with the first 16 largest \(\beta \)-values, as well as the bicyclic graphs of order \(n\ge 30\) with the first 11 largest \(\beta \)-values. We also determine the graphs with the largest \(\beta \)-values among all connected graphs with given order and clique number.