  In this paper we study the positive definiteness of meet and join matrices
using a novel approach. When the set $S_n$ is meet closed, we give a sufficient
and necessary condition for the positive definiteness of the matrix $(S_n)_f$.
From this condition we obtain some sufficient conditions for positive
definiteness as corollaries. We also use graph theory and show that by making
some graph theoretic assumptions on the set $S_n$ we are able to reduce the
assumptions on the function $f$ while still preserving the positive
definiteness of the matrix $(S_n)_f$. Dual theorems of these results for join
matrices are also presented. As examples we consider the so-called power GCD
and power LCM matrices as well as MIN and MAX matrices. Finally we give bounds
for the eigenvalues of meet and join matrices in cases when the function $f$
possesses certain monotonic behaviour.
