Bounds and inequalities for the Perron root of a nonnegative matrix. III. Bounds dependent on simple paths and circuits

The paper presents new upper and lower bounds for the Perron root of a nonnegative matrix in terms of the simple circuits of length not exceeding $k$ and the simple paths of length $k$, $1\le k\le n$, in the directed graph of the matrix. For each $k$, $1\le k\le n$, these bounds are intermediate between the circuit bounds and the path-dependent bounds suggested previously, and for $k=1$ and $k=n$ they reduce to the corresponding path-dependent bounds and the circuit bounds, respectively.