Spectral radius minus average degree: a better bound
  Collatz and Sinogowitz had proposed to measure the departure of a graph $G$ from regularity by the difference of the (adjacency) spectral radius and the average degree: $\epsilon(G)=\rho(G)-\frac{2m}{n}$. We give here new lower bounds on this quantity, which improve upon the currently known ones. 
Felix Goldberg
math.CO