In this paper we specify the class of integral CIRCULANT graphs ICG(n; D), which can be characterized by their order n and the set D of positive divisors of n in such a way that they have the vertex set Zn and the edge set (Formola). This group of graphs is called BMM graphs because of the form of its set of edges. A bipartite G graph is a graph whose vertex set can be divided into two subsets X, and Y such that no two vertices in X and no two vertices in Y are adjacent. The duplicate graph is called complete if each vertex in X is connected to all vertices in Y. This graph is represented by K_ (m, n), if | X | = m and | Y | = n. Multipartite graphs are also defined as bipartite graphs.

In this paper, we investigate the reduced form of CIRCULANT matrices and we show that the problem of computing the q-th roots of a nonsingular CIRCULANT MATRIX A can be reduced to that of computing the q-th roots of two half size matrices B-C and B+C.

Let Crn (Fp) denote the algebra of n´n CIRCULANT matrices over Fp, the finite field of orderp a prime. The order of the unit groups U (Cr3 (Fp)), U (Cr4 (Fp)) and U (Cr5 (Fp)) of algebras of CIRCULANT matrices over Fp are computed.

In this paper, we consider a class of column-weight two quasi-cyclic low-density parity check codes in which the girth can be large enough, as an arbitrary multiple of 8. Then we devote a convolutional form to these codes, such that their generator MATRIX can be obtained by elementary row and column operations on the parity-check MATRIX. Finally, we show that the free distance of the convolutional codes is equal to the minimum distance of their block counterparts.

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Let Crn (F) denote the algebra of n´n CIRCULANT matrices over the field F. In this paper, we study the unit group of Crn (Fpm), where Fpm denotes the Galois field of order pm, p prime.

A graph is called CIRCULANT if it is a Cayley graph on a cyclic group, i.e. its adjacency MATRIX is CIRCULANT. Let D be a set of positive, proper divisors of the integer n > 1. The integral CIRCULANT graph ICGn (D) has the vertex set Z n and the edge set E (ICGn (D)) = {{a, b}; gcd (a – b, n) Î D}. Let n = p1p2….pkm, where p1, p2, …., pk are distinct prime numbers and gcd (p1p2…pk, m) = 1. The open problem posed in paper [A. Ilic, The energy of unitary Cayley graphs, Linear Algebra Appl., 431 (2009) 1881-1889] about calculating the energy of an arbitrary integral CIRCULANT ICGn (D) is completely solved in this paper, where D = {p1, p2, …., pk}.

In this paper, some algebraic and geometrical properties of the rank-k numerical hulls of normal matrices are investigated. A characterization of normal matrices whose rank-1 numerical hulls are equal to their numerical range is given. Moreover, using the extreme points of the numerical range, the higher rank numerical hulls of matrices of the formA1 Å iA2, where A1 and A2 are Hermitian, are investigated. The higher rank numerical hulls of the basic CIRCULANT MATRIX are also studied.

In 2017 Deveci et al. de, ned the Fibonacci-CIRCULANT sequences of the , rst and second kinds as shown, respectively: . . .

Abstract. A recursive-CIRCULANT G (n; d) is defined to be a CIRCULANT graph with n vertices and jumps of powers of d. G (n; d) is vertex-transitive, and has some strong hamiltonian properties. G (n; d) has a recursive structure when n=cdm, 1 £c<d [Theoret. Comput. Sci. 244 (2000) 35-62]. In this paper, we will find the automorphism group of some classes of recursive-CIRCULANT graphs. In particular, we will find that the automorphism group of G (2m; 4) is isomorphic with the group D2.2m, the dihedral group of order 2m+1.