Let A = (aij) be the generic n×n circulant matrix given by aij = xi+j , with subscripts on x interpreted mod n. Define d(n) (resp. p(n)) to be the number of terms in the determinant (resp. permanent) of A. The function p(n) is well-known and has several combinatorial interpretations. The function d(n), on the other hand, has not been studied previously. We show that when n is a prime power, d(n...

The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is identical to the adjacency matrix of a circulant graph is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus...

Abstract In this paper, we consider the k -Fibonacci and k -Lucas sequences {Fk,n}n∈N and {Lk,n}n∈N . Let A = Cr(Fk,0, Fk,1, · · · , Fk,n−1) and B = Cr(Lk,0, Lk,1, · · · , Lk,n−1) be r -circulant matrices. Afterwards, we give upper and lower bounds for the spectral norms of matrices A and B. In addition, we obtain some bounds for the spectral norms of Hadamard and Kronecker products of these ma...

We consider the Youden square formed by deleting one row of a v×v back-circulant Latin square and establish a critical set that contains a number of elements which is equal to v/4 (v even) or (v − 1)/4 (v odd). We show that this critical set is minimal, for v even.

([3]) and tvs(G) ≤ ⌈ 3n δ ⌉ + 1 ([1]). The exact values for some families of graphs are also known, e.g. the value of s(Cin(1, k), given in [2]. We prove that tvs(Cin(1, 2, . . . , k)) = n+2k 2k+1 , while s(Cin(1, 2, . . . , k)) = n+2k−1 2k . In order to do that, we split the graph Cin(1, 2, . . . , k) into segments and label each segment using 0, 1 and 2 in such a way that the weighted degrees...

We show that every weighing matrix of weight n invariant under a finite abelian group G can be generated from a subgroup H of G with |H| ≤ 2n−1. Furthermore, if n is an odd prime power and a proper circulant weighing matrix of weight n and order v exists, then v ≤ 2n−1. We also obtain a lower bound on the weight of group invariant matrices depending on the invariant factors of the underlying

This article identifies the general form of matrix power and trace an integer Special Form 3 × Complex Circulant matrix. The research begins to determine matrix, followed by determining proof is done using mathematical induction. final result this obtain A^n tr(A^n) for n integers in special complex