The exponent and circumdiameter of primitive digraphs

The exponent γ( D) of a primitive digraph D is the smallest m such that for each ordered pair of not necessarily distinct vertices ( u, v), there exists a u ⇝ v walk of length m. If λ( D) is the set of all cycle lengths, then the circumdiameter of D, denoted d c( λ( D)), is the maximum over all orde...

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Bibliographic Details
Published in:	Linear algebra and its applications Vol. 396; pp. 243 - 258
Main Authors:	Dame, L.F., Olesky, D.D., van den Driessche, P.
Format:	Journal Article
Language:	English
Published:	New York, NY Elsevier Inc 01-02-2005 Elsevier Science
Subjects:	Algebra Circumdiameter Exact sciences and technology Exponent Frobenius–Schur index Leslie matrix Linear and multilinear algebra, matrix theory Mathematics Primitive digraph Sciences and techniques of general use 05C20 Frobenius–Schur index Leslie matrix 05C50 Exponent 15A48 Primitive digraph Circumdiameter Upper bound Primitive 15A48 Primitive digraph Frobenius-Schur index Schur complement Digraph
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Summary:	The exponent γ( D) of a primitive digraph D is the smallest m such that for each ordered pair of not necessarily distinct vertices ( u, v), there exists a u ⇝ v walk of length m. If λ( D) is the set of all cycle lengths, then the circumdiameter of D, denoted d c( λ( D)), is the maximum over all ordered pairs of not necessarily distinct vertices ( u, v) of the length of a shortest u ⇝ v walk that intersects cycles of all lengths. It is well known that γ( D) ⩽ ϕ( λ( D)) + d c( λ( D)), in which ϕ( λ( D)) is the Frobenius–Schur index. Several new sufficient conditions and families of digraphs for which equality holds in the above upper bound are given, and some families of digraphs D for which γ( D) = ϕ( λ( D)) + d c( λ( D)) − 1 are defined. Additional sufficient conditions for equality in the above upper bound for γ( D) and a new upper bound for d c( λ( D)) are given for digraphs with large exponent, that is, digraphs on n vertices having γ ( D ) ⩾ ⌊ ( n - 1 ) 2 + 1 2 ⌋ + 2 . The circumdiameter and bounds on the exponent for the digraph of a Leslie matrix are found.
ISSN:	0024-3795 1873-1856
DOI:	10.1016/j.laa.2004.09.004