A. L. Dulmage and N. S. Mendelsohn. Gaps in the exponent set of primitive matrices. Illinois J. Math., 8:642--656, 1964. Home/Search Document Not in Database Summary Related Articles Check |
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Exponents of of 2-regular Digraphs - Shen (1999) (Correct)
....that u k v whenever u; v 2 V . The minimum such k is called the exponent of G, denoted exp(G) The local exponent of G at a vertex u 2 V , denoted exp(G : u) is the least integer k such that u k v for each v 2 V . Much work has been done on finding upper bounds for exp(G) see [14] and [3] for example) The diameter bound in Lemma 1 below was proved recently by Shen [9] and Neufeld [6] independently. Neufeld [6] characterized the case of equality in Lemma 1 with the following class of digraphs. Let the family FD consist of the following digraphs G = V; E) The vertex set V = D ....
A. L. Dulmage and N. S. Mendelsohn, Gaps in the exponent set of primitive matrices, Illinois J. Math., 8:642-656 (1964).
Local Exponents of Primitive Digraphs - Shen, Neufeld (1998) (Correct)
.... Gamma 1) 2 1 and showed there is a unique (up to isomorphism) digraph, W n , that attains this bound, where W n = V; E) is defined as follows: V = fu i : 1 i ng and E = f(u i ; u i 1 ) 1 i n Gamma 1g [ f(u n Gamma1 ; u 1 ) u n ; u 1 )g. In 1964, A. L. Dulmage and N. S. Mendelsohn [2] observed that there are gaps in the exponent set ES n = fexp(G) G 2 P n g, where P n is the set of all primitive digraphs of order n. Each gap is a set S of consecutive integers below w n such that no primitive digraph of order n has an exponent in S. In 1981, M. Lewin and Y. Vitek [3] found ....
....s for all 1 k n, where s is the girth of G. J. Shao et al. 6, Theorem 2.1] by applying a theorem contained in B. Liu s doctoral dissertation improved this bound obtaining exp G (k) s(n Gamma 2) k for all 1 k n: In particular, when k = n, this yields the bound of Dulmage and Mendelsohn [2] on exp(G) In Section 2 we provide our own more self contained proof of this result. We define the i th local exponent set ES n (i) fexp G (i) G 2 P n g for each i, where 1 i n. Since exp G (n) exp(G) ES n (n) is the (ordinary) exponent set. It follows from the papers ( 3] 5] 9] ....
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A. L. Dulmage and N. S. Mendelsohn, Gaps in the exponent set of primitive matrices, Illinois J. Math., 8:642-656(1964).
Periodicity and Repetition in Combinatorics on Words - Wang (2004) (Correct)
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A. L. Dulmage and N. S. Mendelsohn. Gaps in the exponent set of primitive matrices. Illinois J. Math., 8:642--656, 1964.
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