Primitive Digraphs Research Articles

Upper Bounds of the Generalized Competition Indices of Symmetric Primitive Digraphs with d Loops

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.

The Generalized Competition Indices of Doubly Symmetric Primitive Digraphs with d Loops

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.

Some bounds of the generalized \u03bc-scrambling indices of primitive digraphs with d loops

In 2010, Huang and Liu introduced a useful parameter called the generalized μ-scrambling indices of a primitive digraph. In this paper, we give some bounds for μ-scrambling indices of some primitive digraphs with d loops and the digraphs attained the sharp upper bounds are provided.

Exponents of primitive directed Toeplitz graphs

ABSTRACT Given disjoint non-empty subsets S and T of , a digraph D with the vertex set is called a directed Toeplitz graph provided the arc occurs if and only if or . We investigate strong connectivity and primitivity of directed Toeplitz graphs. We prove that any primitive directed Toeplitz graph with vertices has exponent at least 3 and for each , there is a primitive directed Toeplitz graph of order n which has exponent 3. By Wielandt's result, we know that for a primitive digraph D of order n. We characterize the primitive directed Toeplitz graph, for which the upper bound it attained.

The Exact Formula for the Exponents of the Mixing Digraphs of Register Transformations

A digraph is primitive if some positive degree of it is a complete digraph, i.e. has all possible arcs. The smallest degree of this kind is called the exponent of the digraph. Given a primitive digraph, the elementary local exponent for some vertices $$u$$ and $$v $$ is the least positive integer $$\gamma $$ such that there exists a path from $$u $$ to $$v$$ of any length not less than $$\gamma $$ . For the transformation on the binary $$n $$ -dimensional vector space which is given by a set of $$n $$ coordinate functions, there corresponds some $$n $$ vertex digraph such that a pair $$(u,v) $$ is an arc if the $$v $$ th coordinate component of the transformation depends essentially on the $$u$$ th variable. Such a digraph we call a mixing digraph of the transformation. Under study are the mixing digraphs of $$n$$ -bit shift registers with a nonlinear Boolean feedback function (NFSR), $$n>1 $$ , which are widely used in cryptography. We find the exact formulas for the exponent and elementary local exponents of the $$n $$ -vertex primitive mixing digraph associated to an NFSR. The above results can be applied to evaluating the length of the blank run of the pseudo-random sequences generators based on NFSRs.

The graphic structure of a numerical semigroup

Motivated by the way numerical semigroups arise from primitive digraphs, we initiate the study of the graphic structure of a numerical semigroup. We lay some general groundwork for determining this structure and apply our results to a minimization problem.

Semigroup and Metric Characteristics of Locally Primitive Matrices and Digraphs

The notion of local primitivity for a quadratic 0, 1-matrix of size n × n is extended to any part of the matrix which need not be a rectangular submatrix. A similar generalization is carried out for any set B of pairs of initial and final vertices of the paths in an n-vertex digraph, B ⊆ {(i, j) : 1 ≤ i, j ≤ n}. We establish the relationship between the local B-exponent of a matrix (digraph) and its characteristics such as the cyclic depth and period, the number of nonprimitive matrices, and the number of nonidempotentmatrices in the multiplicative semigroup of all quadratic 0, 1-matrices of order n, etc. We obtain a criterion of B-primitivity and an upper bound for the B-exponent. We also introduce some new metric characteristics for a locally primitive digraph Γ: the k, r-exporadius, the k, r-expocenter, where 1 ≤ k, r ≤ n, and the matex which is defined as the matrix of order n of all local exponents in the digraph Γ. An example of computation of the matex is given for the n-vertex Wielandt digraph. Using the introduced characteristics, we propose an idea for algorithmically constructing realizable s-boxes (elements of round functions of block ciphers) with a relatively wide range of sizes.

О характеристиках локально примитивных орграфов и матриц

О характеристиках локально примитивных орграфов и матриц

Generalized competition index of primitive digraphs

For any positive integers k and m, the k-step m-competition graph Cmk(D) of a digraph D has the same set of vertices as D and there is an edge between vertices x and y if and only if there are distinct m vertices v1, v2, · · ·, vm in D such that there are directed walks of length k from x to vi and from y to vi for all 1 ≤ i ≤ m. The m-competition index of a primitive digraph D is the smallest positive integer k such that Cmk(D) is a complete graph. In this paper, we obtained some sharp upper bounds for the m-competition indices of various classes of primitive digraphs.

The generalized competition indices of primitive minimally strong digraphs

Let D be a primitive digraph of order n and 1≤m≤n. The m-competition index of D is the smallest positive integer k such that every pair of vertices x and y of D have at least m common preys in Dk. In this paper, the upper bound of m-competition indices (1≤m≤n−1) for primitive minimally strong digraphs of order n is obtained. Furthermore, it is shown that for 1≤m≤n−1, there exist “gaps” in the m-competition index set of primitive minimally strong digraphs of order n.

Reset Thresholds of Automata with Two Cycle Lengths

We present several series of synchronizing automata with multiple parameters, generalizing previously known results. Let p and q be two arbitrary co-prime positive integers, [Formula: see text]. We establish reset thresholds of the colorings of primitive digraphs with exactly one simple cycle of length p and one simple cycle of length q. Also, we study reset thresholds of the colorings of primitive digraphs with exactly one simple cycle of length q and two simple cycles of length p. We obtain the reset thresholds of the form [Formula: see text], where [Formula: see text] is a linear function of p and q.

The Scrambling Index of Two-colored Wielandt Digraph

A digraph is primitive provided there is a positive integer k such that for each pair of vertices u and v there exist walks of length k from u to v and from v to u. The scrambling index of a primitive digraph D is the smallest positive integer k such that for each pair of vertices u and v in D there is a vertex w such that there exist walks of length k from u to w and from v to w . A two-colored digraph is a digraph each of whose arc is colored by red or blue. In this paper we generalize the notion of scrambling index of a primitive digraph to that of two-colored digraph. We define the scrambling index of a two-colored digraph D (2) to be the smallest positive integer h+ l over all pairs of nonnegative integers (h, l) such that for each pair of distinct vertices u and v there is a vertex w with the property that there are walks form u to w and from v to w consisting of h red arcs and l blue arcs. For two-colored Wielandt digraph on n 4 vertices we show the scrambling index lies on the interval (n 2 3n + 3, n 2 2n + 2).

The Exponent Set of a Class of Two-Colored Digraphs with One Common Vertex

A two-colored directed digraph D is primitive if and only if there exist nonnegative integers h and k with h+k>0 such that for each pair (i,j) of vertices there is a (h,k)-walk in D from i to j. The exponent of the primitive two-colored digraph D is defined to be the smallest value of h+k over all suchand . With the knowledge of graph theory, a class of two-colored digraphs with two cycles whose uncolored digraph has 3n vertices and consists of one (2n+1)-cycle and one n-cycle is considered.The exponent bound, exponent set and characteristic of extral two-colored digraphs are given.

Primitive digraphs with large exponents and slowly synchronizing automata

We present several infinite series of synchronizing automata for which the minimum length of reset words is close to the square of the number of states. All these automata are tightly related to primitive digraphs with large exponent.

Scrambling index set of primitive digraphs

The scrambling index k(D) of a primitive digraph D is the smallest positive integer k such that for every pair of vertices x and y, there exists a vertex v such that there exist directed walks of length k from x to v and from y to v. In this paper, we study the scrambling index set of primitive digraphs.

DIAMETER OF THE DIRECT PRODUCT OF WIELANDT GRAPH

A digraph D is primitive if there is a positive integer <TEX>$k$</TEX> such that there is a walk of length <TEX>$k$</TEX> between arbitrary two vertices of D. The exponent of a primitive digraph is the least such <TEX>$k$</TEX>. Wielandt graph <TEX>$W_n$</TEX> of order <TEX>$n$</TEX> is known as the digraph whose exponent is <TEX>$n^2-2n+2$</TEX>, which is the maximum of all the exponents of the primitive digraphs of order n. It is known that the diameter of the multiple direct product of a digraph <TEX>$W_n$</TEX> strictly increases according to the multiplicity of the product. And it stops when it attains to the exponent of <TEX>$W_n$</TEX>. In this paper, we find the diameter of the direct product of Wielandt graphs.

The exponent of a digraph and the diameter of its multiple direct product

A new phenomenon pertaining to the diameter of the multiple direct product Dm of a primitive digraph D is found related to exp(D). It is shown that there is a positive integer m, referred to as the critical multiplicity of D, which satisfies the conditiondiam(D)<diam(D2)<⋯<diam(Dm-1)<diam(Dm)=diam(Dm+1)=⋯=exp(D).Further, it is proved that the critical multiplicity m of D satisfies m⩽n-1, where n is the order of D. The extremal cases are classified as follows: for each n, there are two primitive digraphs up to isomorphism having a critical multiplicity of n-1, where one of the digraphs is the Wielandt digraph.

Generalized competition indices of symmetric primitive digraphs

For a primitive digraph D of order n and a positive integer m such that m≤n, the m-competition index of D is defined as the smallest positive integer k such that for every pair of vertices x and y, there exist m distinct vertices v1,v2,…,vm such that there are directed walks of length k from x to vi and from y to vi for 1≤i≤m in D. In this study, we investigate m-competition indices of symmetric primitive digraphs and provide the upper and lower bounds. We also characterize the set of m-competition indices of symmetric primitive digraphs.

The m-competition indices of symmetric primitive digraphs without loops

For positive integers m and n with 1 � mn, 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, we study the generalized competition indices of symmetric primitive digraphs without loops. We determine the generalized competition index set and characterize the digraphs in this class with largest generalized competition index.

On the Set of 2 - Common Consequent of Primitive Digraphs with Exact &amp;lt;i&amp;gt;d&amp;lt;/i&amp;gt; Vertices Having Loop

Let d and n are positive integers, n≥2,1≤d≤ 2.In this paper we obtain that the set of the 2 - common consequent of primitive digraphs of order n with exact d vertices having loop is{1,2,…, n-[]}.