1. 2.6 Matrix with Cyclic Structure

2. Index of Imprimitivity, Cyclic Index
Let D be a general digraph
(need not strongly connected)
The index of imprimitivity of D
: = g.c.d of circuit lengths of D
= g.c.d of lengths of closed directed walks of D
The cyclic index of D is the largest positive
integer k such that D is cyclic k-partite

3. Spectral Index
Let
and
The spectral index of A =
: = g.c.d of i at

4. Theorem p.1
, then
Let
(1) The index of imprimitivity of G(A) is the
largest positive integer m such that A
and
have equal corresponding
circuit products.
(2) The cyclic index of G(A) divides the
index of imprimitivity of G(A)

5. Theorem 2.6.1 p.2 (3) The index of imprimitivity of G(A) divides
(4) If
then
the index of imprimitivity of G(A)
(5) If A is irreducible nonnegative, then the
cyclic index, the index of imprimitivity
and the spectral index of A are equal.

10. Theorem p.1
, then
Let
(1) The index of imprimitivity of G(A) is the
largest positive integer m such that A
and
have equal corresponding
circuit products.

11. Combinatorial Spectral Theory of Nonnegative Matrices

12. Spectral Index
Let
and
The spectral index of A =
: = g.c.d of i at