THE COMPLETE PRODUCT OF ANNIHILATINGLY UNIQUE DIGRAPHS

Let G be a digraph with n vertices and let A(G) be its adjacency matrix. A monic polynomial f(x) of degree at most n is called an annihilating polynomial of G if f(A(G))=0. G is said to be annihilatingly unique if it possesses a unique annihilating polynomial. Difans and diwheels are two classes of annihilatingly unique digraphs. In this paper, it is shown that the complete product of difan and diwheel is annihilatingly unique.


Introduction
All graphs under consideration in this paper are directed, connected, finite, loopless, and without multiple arcs.By a digraph G = (V ,E), we mean a finite set V (the elements of which are called vertices) together with a set E of ordered pairs of elements of V (these pairs are called arcs).Two vertices are said to be adjacent if they are connected by an arc.Undefined terms and notations can be found in [1,2,4].
A diwalk in a digraph is an alternating sequence of vertices and arcs, v 0 ,x 1 ,v 1 ,...,x k ,v k in which each arc x i is (v i−1 ,v i ).The length of such diwalk is k, the number of occurrences of arcs in it.
A dipath P n is a diwalk of order n in which no vertex is repeated.
A difan F n is a digraph consisting essentially of a dipath P n−1 of n − 1 vertices labelled 1,2,...,n − 1 and an additional vertex n, where there is an arc from n to each of the vertices of P n−1 .Vertex n is called the hub of the difan whereas the arc (n,k) from the hub n to the vertex k of P n−1 is called a spoke.Vertices with labelling i = 1,2,...,n − 1 are called the rim vertices of the difan.
The diwheel W n of order n consists of a directed cycle C n−1 with an additional vertex n joined from it to all the others.Vertex n is called the hub of the diwheel, whereas the arc (n,k) from the hub n to the vertex k of C n−1 is called a spoke.Vertices with labelling i where i = 1,2,...,n − 1 are called the rim vertices of the diwheel.
Let G be a digraph with n vertices.The adjacency matrix A(G) = (a i j ) of G is a square matrix of order n where the (i, j)-entry, a i j , is equal to the number of arcs starting at the vertex i and terminating at the vertex j.
where k is a positive integer and the (i, j)-entry a k i j of A k (G) is the number of different diwalks at length k from the vertex i to vertex j.
The determinant of a square matrix A is denoted by |A| or (det)A.The characteristic polynomial |xI − A(G)| of the adjacency matrix A(G) of G is called the characteristic polynomial of G and is denoted by ψ(x).A monic polynomial f (x) of degree at most n with f (A(G)) = 0 is called an annihilating polynomial of G.The existence of annihilating polynomial of G is guaranteed by its characteristic polynomial.G is said to be annihilatingly unique if it possesses a unique annihilating polynomial.
Theorem 1.1.Let A be an n × n matrix, m(x) its minimum polynomial, and ψ(x) its characteristic polynomial.
(2) If f (x) is any polynomial with f (A) = 0, then m(x) divides f (x); in particular m(x) divides ψ(x).(3) Let {x 1 ,x 2 ,...,x k } be the set of distinct eigenvalues of A, x i having algebraic multiplicity c i .Then where q i satisfies 0 (1.2) A matrix for which the minimum polynomial is equal to the characteristic polynomial is called nonderogatory; otherwise derogatory.The following result follows from Theorem 1.1.

The complete product of digraphs
In this section, we will consider a binary operation on digraphs: the complete product of digraphs. The Let G 1 and G 2 be digraphs with n 1 and n 2 vertices, respectively.The vertices of G 1 ⊗ G 2 are labelled as follows.

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The vertices of G 2 are labelled 1,2,...,n 2 and vertices of G 1 are labelled n 2 + 1,n 2 + 2,...,n 2 + n 1 .Suppose that A(G i ) is the adjacency matrix of G i , i = 1,2.Then the adja- where J (n1) n2 is an n 1 × n 2 matrix with all entries equal to 1 and 0 (n2) n1 is an n 2 × n 1 matrix with all the entries equal to zero.It follows immediately that the characteristic polynomial of G 1 ⊗ G 2 is the product of the characteristic polynomials of G 1 and G 2 .
A question arises: is it true that the complete product of two annihilatingly unique digraphs possesses a unique annihilating polynomial?
The complete product of annihilatingly unique digraphs was first studied by Lam [3].In particular, Lam proved that P q ⊗ P r , P q ⊗ C r , C r ⊗ P q , P q ⊗ W r , W r ⊗ P q are annihilatingly unique.Furthermore, if q and r are relatively prime, then C q ⊗ C r , C q ⊗ W r+1 , W r+1 ⊗ C q , and W q+1 ⊗ W r+1 are annihilatingly unique.
In general, it is not true that the complete product of two annihilatingly unique digraphs possesses a unique annihilating polynomial.For example, C 3 ⊗ W 4 of order 7 is not annihilatingly unique.
Motivated by the above results, we will prove that the complete product of difans and diwheels are annihilatingly unique in the next section.

The complete product of difans and diwheels
In this section, we show that the complete products F q ⊗ W r and W q ⊗ F r of order q + r are annihilatingly unique.The characteristic polynomials for difan of order q and diwheel of order r are x q and x r − x, respectively.Theorem 3.1.F q ⊗ W r and W q ⊗ F r are annihilatingly unique.
Proof.We divide the proof of Theorem 3.1 into two parts and in each part, we will prove that the given complete product of digraphs is annihilatingly unique by showing the equality of its characteristic and minimum polynomials.
Part 1 (F q ⊗ W r is annihilatingly unique).The characteristic polynomial of F q ⊗ W r is given by ψ(x) = x q (x r − x) = x q+1 (x r−1 − 1).From the digraph of F q ⊗ W r , we observe that a (r−1) r,r−1 = 1 and a (q+1) r+q,1 > 1 (this is because there exist at least two diwalks of length q + 1 from vertex r + q to vertex 1.For example, r are two such diwalks of length q + 1 from vertex r + q to vertex 1).This observation implies that A q+1 and A r−1 − I are not equal to 0 and x q+1 and x r−1 − 1 are therefore not the minimum polynomials of ψ(x).
Notice that the only repeated factor of the characteristic polynomial of F q ⊗ W r is x of multiplicity q + 1.We claim that ψ(x) = m(x) = x q+1 (x r−1 − 1).Suppose that m(x) = x q (x r−1 − 1) = x q+r−1 − x q .We will show that m(A) = 0 to obtain a contradiction.From the digraph of F q ⊗ W r , we observe that a (r+q−1) r+q,r = 0 and a (q) r+q,r = 1.This implies that m(A) = A q+r−1 − A q = 0, a contradiction.Therefore, ψ(x) is equal to the minimum polynomial of F q ⊗ W r .Hence, F q ⊗ W r is annihilatingly unique.
Part 2 (W q ⊗ F r is annihilatingly unique).The characteristic polynomial of W q ⊗ F r is given by ψ(x) = x r (x q − x) = x r+1 (x q−1 − 1).Like the case of F q ⊗ W r , from the digraph of W q ⊗ F r , we have a (q−1) r+q,r+q−1 = 1 and a (r+1) r+1,r−1 > 1.This implies that A r+1 and A q−1 − I are not equal to 0 and x r+1 and x q−1 − 1 are therefore not the minimum polynomials of ψ(x).
Notice that the only repeated factor of ψ(x) is x of multiplicity r + 1.We claim that ψ(x) = m(x) = x r+1 (x q−1 − 1).Suppose on the contrary that m(x) = x r (x q−1 − 1).Again, we will show that m(A) = 0 and obtain a contradiction.To this end, we claim that When k = r, the number of diwalks from the vertex r + q to vertex r − 1 can be enumerated in the following three ways.
(i) If the rim vertices of W q are avoided.In this case, there is only one such diwalk, namely r (ii) Move along some rim vertices of W q with the hub vertex r of F r being included in the diwalk.
For each rim vertex of W q , since the hub vertex r of F r is included in the diwalk, the possible number of rim vertices of F r covered in the diwalk is at most r − 2. From the multiplication principle, since W q has q − 1 rim vertices, the number of diwalks from vertex r + q to vertex r − 1 in case (ii) is (q − 1)(r − 2).
(iii) Move along some rim vertices of W q with the hub vertex r of F r being excluded in the diwalk.
In this case, since the hub vertex r of F r is excluded in the diwalk, the possible number of rim vertices of F r contained in the diwalk is r − 1.This implies that the number of diwalks from vertex r + q to vertex r − 1 in case (iii) is (q − 1)(r − 1).
Thus, for the case Likewise, the similar enumeration method for the case when k = r can be used to show that a (k)  r+q,r−1 = 0 + (q − 1)(r − 1) + (q − 1)(r − 1) = 2(q − 1)(r − 1) when k > r.By using the results in the claim, we have This implies that m(A) = A r+q−1 − A r = 0, a contradiction.Therefore ψ(x) is equal to the minimum polynomial of W q ⊗ F r , and W q ⊗ F r is annihilatingly unique and the proof of Part 2 is complete.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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