Let DSn(d) denote the set of all doubly symmetric primitive digraphs of order n with d loops, where d is an integer and 1≤d≤n. In this paper, we determine the upper bounds for the m-competition indices(generalized competition indices) of DSn(d), where 1≤m≤n. If n and d satisfy that n is odd and d is odd, or n≤2d−2 and d is even such that d≥2, then the upper bounds for the m-competition indices of DSn(d) can be reached, where 1≤m≤n.
Let D be a primitive digraph of order n. The exponent of a vertex x in V(D) is denoted γD(x), which is the smallest integer q such that for any vertex y, there is a walk of length q from x to y. Let V(D)={v1,v2,⋯,vn}. We order the vertices of V(D) so that γD(v1)≤γD(v2)≤⋯≤γD(vn) is satisfied. Then, for any integer k satisfying 1≤k≤n,γD(vk) is called the kth local exponent of D and is denoted by expD(k). Let DSn(d) represent the set of all doubly symmetric primitive digraphs with n vertices and d loops, where d is an integer such that 1≤d≤n. In this paper, we determine the upper bound for the kth local exponent of DSn(d), where 1≤k≤n. In addition, we find that the upper bound for the kth local exponent of DSn(d) can be reached, where 1≤k≤n.
A digraph (D) is symmetric if (u,v) is an arc of D and if (v,u) is also an arc of D. If a symmetric digraph is primitive and contains d loops, then it is said to be a symmetric primitive digraph with d loops. The m-competition index (generalized competition index) of a digraph is an extension of the exponent and the scrambling index. The m-competition index has been applied to memoryless communication systems in recent years. In this article, we assume that Sn(d) represents the set of all symmetric primitive digraphs of n vertices with d loops, where 1≤d≤n. We study the m-competition indices of Sn(d) and give their upper bounds, where 1≤m≤n. Furthermore, for any integer m satisfying 1≤m≤n, we find that the upper bounds of the m-competition indices of Sn(d) can be reached.
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