Let S be a subset of the cyclic group Z n . The cyclic Haar graph H (Z n ,  S ) is the bipartite graph with color classes Z n + and Z n − , and edges { x + ,  y − }, where x ,  y ∈ Z n and y −  x ∈  S . In this paper we give sufficient and necessary conditions for the isomorphism of two connected cyclic Haar graphs of valency 4 .

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Isomorphic tetravalent cyclic Haar graphs

https://faculty.nps.edu/pstanica/research/2015DM_CirculantMRS.pdf

Circulant matrices and affine equivalence of monomial rotation symmetric Boolean functions

Let $S$ be a subset of the cyclic group $\Z_n$. The cyclic Haar graph $H(\Z_n,S)$ is the bipartite graph with color classes $\Z_n^+$ and $\Z_n^-,$ and edges $\{x^+,y^-\},$ where $x,y \in \Z_n$ and $y - x \in S$. In this paper we give sufficient and necessary conditions for the isomorphism of two connected cyclic Haar graphs of valency 4.

Rotation symmetric Boolean functions have been extensively studied for about 15 years because of their applications in cryptography and coding theory. Until recently little was known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009. For the much more complicated case of cubic rotation symmetric functions generated by a single monomial, the affine equivalence classes under permutations which preserve rotation symmetry were determined in 2011. It was conjectured then that the cubic equivalence classes are the same if all nonsingular affine transformations, not just permutations, are allowed. This conjecture is probably difficult, but here we take a step towards it by proving that the cubic affine equivalence classes found in 2011 are the same if all permutations, not just those preserving rotation symmetry, are allowed. The needed new idea uses the theory of circulant matrices.

Permutation equivalence of cubic rotation symmetric Boolean functions

A configuration of points and lines is cyclic if it has an automorphism which permutes its points in a full cycle. A closed formula is derived for the number of non-isomorphic connected cyclic configurations of type (v_3), i.e., which have v points and lines, and each point/line is incident with exactly 3 lines/points. In addition, a Bays-Lambossy type theorem is proved for cyclic configurations if the number of points is a product of two primes or a prime power.

The number of cyclic configurations of type $(v_3)$ and the isomorphism problem

We consider n-by-n circulant matrices having entries 0 and 1 Such matrices can be identified with sets of residues mod n, corresponding to the columns in which the top row contains an entry 1 Let A and B be two such matrices, and suppose that the corresponding residue sets S_A and S_B have size at most 3 We prove that the following are equivalent: (1) there are integers u,v mod n, with u a unit, such that S_A = uS_B + v; (2) there are permutation matrices P,Q such that A=PBQ Our proof relies on some new results about vanishing sums of roots of unity We give examples showing this result is not always true for denser circulants, as well as results showing it continues to hold in some situations We also explain how our problem relates to the Adam problem on isomorphisms of circulant directed graphs

Doug Wiedemann

Papers

Equivalence of sparse circulants: the bipartite Ádám problem