the eigenvalues of regular graphs have been well studied. they have strong connections with the expansion constant (alon-milman, tanner), diameter (chung), chromatic and independence number (hoffman) of a graph. in this talk, i will discuss the eigenvalues of irregular graphs. one of the first results of spectral graph theory due to collatz and sinogowitz (1957) states that the spectral radius of a graph is between the average degree and the maximum degree of a graph with equality iff the graph is regular. when the graph is irregular, i will show how can we improve these inequalities. i will conclude with a list of open problems. this is based on joint work with david gregory (queen’s university at kingston, canada) and vlado nikiforov (university of memphis, usa).