ON THE SUM OF TOW LARGEST EIGENVALUES OF A GRAPH AND GERNERT'S CONJECTURE

D. Gernet conjectured that the sum of two largest eigenvalues of the adjacency matrix of any simple graph is at most the number of vertices of the graph. This can be proved, in particular, for all also exhibit a nontrivial upper bound for the sum of the two largest eigenvalues and derive a new upper bound for the second largest eigenvalue of the adjacency matrices of the graphs.

Keywords:

Simple graphs , Adjacency matrix , Eigenvalue