  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:
  $$m(G)-c(G)\leq p(G)\leq m(G)+c(G), \ m(G)-c(G)\leq n(G)\leq m(G)+c(G),$$
  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 or the lower bounds respectively.
