Generalized competition indices of symmetric primitive digraphs

The m-competition indices of symmetric primitive digraphs with loop

Generalized Exponents of Primitive Symmetric Digraphs

A strongly connected digraph D of order n is primitive (aperiodic) provided the greatest common divisor of its directed cycle lengths equals 1. For such a digraph there is a minimum integer t, called the exponent of D, such that given any ordered pair of vertices x and y there is a directed walk from x to y of length t. The exponent of D is the largest of n ‘generalized exponents’ that may be a...

Ela the M-competition Indices of Symmetric Primitive Digraphs without Loops

Abstract. For positive integers m and n with 1 ≤ m ≤ n, the m-competition index (generalized competition index) of a primitive digraph D of order n is the smallest positive integer k such that for every pair of vertices x and y in D, there exist m distinct vertices v1, v2, . . . , vm such that there exist walks of length k from x to vi and from y to vi for each i = 1, . . . , m. In this paper, ...

Well primitive digraphs

A primitive digraph D is said to be well primitive if the local exponents of D are all equal. In this paper we consider well primitive digraphs of two special types: digraphs that contain loops, and symmetric digraphs with shortest odd cycle of length r. We show that the upper bound of the exponent of the well primitive digraph is n− 1 in both these classes of digraphs, and we characterize the ...

Local Exponents of Primitive Digraphs

A digraph G = (V; E) is primitive if, for some positive integer k, there is a u ! v walk of length k for every pair u; v of vertices of 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 there is a u ! v walk of length k for each v 2 V. Let V = f1; 2; ; ng. Following Brualdi and Liu, ...