Primitive Digraphs Research Articles

Bounds on the generalized μ-scrambling indices of primitive digraphs

Huang and Liu [Generalized scrambling indices of primitive a digraph, Linear Algebra Appl. 433 (2010), pp. 1798–1808] gave the definitions of the generalized μ-scrambling indices and the generalized scrambling indices of a primitive digraph and also provided some exact lower and upper bounds for the generalized scrambling indices of various classes of primitive digraphs. In this paper, we give some exact lower and upper bounds for the generalized μ-scrambling indices of various classes of primitive digraphs.

On generalized competition index of a primitive tournament

For positive integers k and m and a digraph D , the k -step m -competition graph C m k ( D ) of D has the same set of vertices as D and an edge between vertices x and y if and only if there exist m distinct vertices v 1 , v 2 , … , v m in D such that there exist directed walks of length k from x to v i and from y to v i for 1 ≤ i ≤ m . The m -competition index of a primitive digraph D is the smallest positive integer k such that C m k ( D ) is a complete graph. In this paper, we study the m -competition indices of primitive tournaments and provide an upper bound for the m -competition index of a primitive tournament.

A bound of generalized competition index of a primitive digraph

For a positive integer m , where 1 ≤ m ≤ n , the m -competition index (generalized competition index) of a primitive digraph D is the smallest positive integer k such that for every pair of vertices x and y , there exist m distinct vertices v 1 , v 2 , … , v m such that there exist directed walks of length k from x to v i and from y to v i for 1 ≤ i ≤ m . The m -competition index is a generalization of the scrambling index and the exponent of a primitive digraph. In this paper, we study the upper bound of the m -competition index of a primitive digraph using its order and girth.

COMPETITION INDICES OF STRONGLY CONNECTED DIGRAPHS

Cho and Kim (4) and Kim (6) introduced the concept of the competition index of a digraph. Cho and Kim (4) and Akelbek and Kirk- land (1) also studied the upper bound of competition indices of primitive digraphs. In this paper, we study the upper bound of competition in- dices of strongly connected digraphs. We also study the relation between competition index and ordinary index for a symmetric strongly connected digraph.

A bound on the generalized competition index of a primitive matrix using Boolean rank

For a positive integer m where 1 ⩽ m ⩽ n , the m -competition index (generalized competition index) of a primitive digraph is the smallest positive integer k such that for every pair of vertices x and y , there exist m distinct vertices v 1 , v 2 , … , v m such that there are directed walks of length k from x to v i and from y to v i for 1 ⩽ i ⩽ m . The m -competition index is a generalization of the scrambling index and the exponent of a primitive digraph. In this study, we determine an upper bound on the m -competition index of a primitive digraph using Boolean rank and give examples of primitive Boolean matrices that attain the bound.

GENERALIZED $r$-EXPONENTS OF PRIMITIVE DIGRAPHS

In this paper, as natural extensions of the generalized exponents of primitive digraphs introduced by R.A. Brualdi and B. Liu, we introduce the generalized $r$-exponents of primitive digraphs, where $1\leq r\leq n$. Moreover, some properties and sharp upper bounds of the generalized $r$-exponents of primitive digraphs are obtained, respectively.

Generalized scrambling indices of a primitive digraph

From the background of a memoryless communication system, we introduce the definition of the generalized scrambling indices for a primitive digraph, which are natural extensions of the scrambling index and m-competition index. Moreover, we provide some exact lower and upper bounds for the generalized scrambling indices of various classes of primitive digraphs, respectively.

The scrambling index of primitive digraphs

In 2009, Akelbek and Kirkland introduced a useful parameter called the scrambling index of a primitive digraph D , which is the smallest positive integer k such that for every pair of vertices u and v , there is a vertex w such that we can get to w from u and v in D by directed walks of length k . In this paper, we obtain some new upper bounds for the scrambling index of primitive digraphs. Moreover, the maximum index problem, the extremal matrix problem and the index set problem for the scrambling index of various classes of primitive digraphs (e.g. primitive digraphs with d loops, minimally strong digraphs, nearly decomposable digraphs, micro-symmetric digraphs, etc.) are settled, respectively.

On a conjecture about the kth lower multiexponent

In this paper, the conjecture about the kth lower multiexponent f(n,k) proposed by R.A. Brualdi and B. Liu is proved to be true for the following cases: (1) k=n−i, where i=2,3,4,5; (2) small n, where n≤8; (3) the class of primitive micro-symmetric digraphs of order n.

Generalized competition index of a primitive digraph

For positive integers k and m , and a digraph D , the k -step m -competition graph C m k ( D ) of D has the same set of vertices as D and an edge between vertices x and y if and only if there are distinct m vertices v 1 , v 2 , … , v m in D such that there are directed walks of length k from x to v i and from y to v i for 1 ⩽ i ⩽ m . In this paper, we present the definition of m -competition index for a primitive digraph. The m -competition index of a primitive digraph D is the smallest positive integer k such that C m k ( D ) is a complete graph. We study m -competition indices of primitive digraphs and provide an upper bound for the m -competition index of a primitive digraph.

On the exponents of two-colored digraphs with two cycles†

We consider the special primitive two-colored digraphs whose uncolored digraph has n+s vertices and consists of one n-cycle and one n−t-cycle. We give some primitivity conditions and an upper bound on the exponent. Further, for the case s=0, we give a tight upper bound on the exponent and the characterization of extremal two-colored digraphs. †Research supported by NNSF of China (No. 10571163) and NSF of Shanxi (No. 20041010).

On k-primitivity of two classes of digraphs

In this paper, we study two classes of primitive digraphs, and show that there are k -colorings that are k -primitive.

Sign patterns that require or allow power-positivity

A matrix A is power-positive if some positive integer power of A is entrywise positive. A sign pattern A is shown to require power-positivity if and only if either A or −A is nonnegative and has a primitive digraph, or equivalently, either A or −A requires eventual positivity. A sign pattern A is shown to be potentially power-positive if and only if A or −A is potentially eventually positive.

Primitive digraphs with the largest scrambling index

The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). In [M. Akelbek, S. Kirkland, Coefficients of ergodicity and the scrambling index, preprint] we gave the upper bound on k(D) in terms of the order and the girth of a primitive digraph D. In this paper, we characterize all the primitive digraphs such that the scrambling index is equal to the upper bound.

Coefficients of ergodicity and the scrambling index

For a primitive stochastic matrix S, upper bounds on the second largest modulus of an eigenvalue of S are very important, because they determine the asymptotic rate of convergence of the sequence of powers of the corresponding matrix. In this paper, we introduce the definition of the scrambling index for a primitive digraph. The scrambling index of a primitive digraph D is the smallest positive integer k such that for every pair of vertices u and v, there is a vertex w such that we can get to w from u and v in D by directed walks of length k; it is denoted by k(D). We investigate the scrambling index for primitive digraphs, and give an upper bound on the scrambling index of a primitive digraph in terms of the order and the girth of the digraph. By doing so we provide an attainable upper bound on the second largest modulus of eigenvalues of a primitive matrix that make use of the scrambling index.

The k-point exponent set of primitive digraphs with girth 2

Let D = (V,E) be a primitive digraph. The vertex exponent of D at a vertex v ∈ V, denoted by expD(v), is the least integer p such that there is a v → u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that expD(v1) ≤ expD(v2) ≤ ... ≤ expD(vn). Then expD(vk) is called the k-point exponent of D and is denoted by expD(k), 1 ≤ k ≤ n. In this paper we define e(n, k):=max{expD(k)|D ∈ PD(n, 2)} and E(n, k):= {expD(k)|D ∈ PD(n, 2)}, where PD(n, 2) is the set of all primitive digraphs of order n with girth 2. We completely determine e(n, k) and E(n, k) for all n, k with n ≥ 3 and 1 ≤ k ≤ n.

Primitive digraphs with smallest large exponent

A primitive digraph D on n vertices has large exponent if its exponent, γ ( D ) , satisfies α n ⩽ γ ( D ) ⩽ w n , where α n = ⌊ w n / 2 ⌋ + 2 and w n = ( n - 1 ) 2 + 1 . It is shown that the minimum number of arcs in a primitive digraph D on n ⩾ 5 vertices with exponent equal to α n is either n + 1 or n + 2 . Explicit constructions are given for fixed n even and odd, for a primitive digraph on n vertices with exponent α n and n + 2 arcs. These constructions extend to digraphs with some exponents between α n and w n . A necessary and sufficient condition is presented for the existence of a primitive digraph on n vertices with exponent α n and n + 1 arcs. Together with some number theoretic results, this gives an algorithm that determines for fixed n whether the minimum number of arcs is n + 1 or n + 2 .

The [formula omitted]th local exponent of doubly symmetric primitive matrices

Let D = ( V , E ) be a primitive digraph. The local exponent of D at a vertex u ∈ V , denoted by γ D ( u ) , is the least integer p such that there is an u → v walk of length p for each v ∈ V . Let V = { v 1 , v 2 , … , v n } . Following Brualdi and Liu, we order the vertices of V so that γ D ( v 1 ) ≤ γ D ( v 2 ) ≤ ⋯ ≤ γ D ( v n ) . Then γ D ( v k ) is called the k th local exponent of D and is denoted by exp D ( k ) , 1 ≤ k ≤ n . In this work we define exp ( n , k ) = max { exp G ( k ) | G = G ( A ) with A ∈ DSP ( n ) } , where DSP ( n ) is the set of all n × n doubly symmetric primitive matrices and G ( A ) is the associated graph of matrix A . For n ≥ 3 , we determine that exp ( n , k ) = n − 1 for all k with 1 ≤ k ≤ n .

Exponents of a class of two-colored digraphs

A two-colored digraph D is primitive if there exist nonnegative integers h and k with h+k>0 such that for each pair (i, j) of vertices there exists an (h, k)-walk in D from i to j. The exponent of the primitive two-colored digraph D is the minimum value of h+k taken over all such h and k. In this article, we consider special primitive two-colored digraphs whose uncolored digraph has n+s vertices and consist of one n-cycle and one (n − 2)-cycle. We give the bounds on the exponents, and the characterizations of the extremal two-colored digraphs.

BOUND FOR 2-EXPONENTS OF PRIMITIVE EXTREMAL MINISTRONG DIGRAPHS

We consider 2-colored digraphs of the primitive min- istrong digraphs having given exponents. In this paper we give bounds for 2-exponents of primitive extremal ministrong digraphs.