Bounds for the positive and negative inertia index of a graph.

Let G be a graph and let A ( G ) be adjacency matrix of G . The positive inertia index (respectively, the negative inertia index) of G , denoted by p ( G ) (respectively, n ( G ) ), is defined to be the number of positive eigenvalues (respectively, negative eigenvalues) of A ( G ) . In this paper, we present the bounds for p ( G ) and n ( G ) as follows: m ( G ) − c ( G ) ≤ p ( G ) ≤ m ( G ) + c ( G ) , m ( G ) − c ( G ) ≤ n ( G ) ≤ m ( G ) + c ( G ) , where m ( G ) and c ( G ) are respectively the matching number and the cyclomatic number of G . Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds respectively. [ABSTRACT FROM AUTHOR]