Chris D. Godsil

Constructing cospectral graphs

Some new constructions for families of cospectral graphs are derived, and some old ones are considerably generalized. One of our new constructions is sufficiently powerful to produce an estimated 72% of the 51039 graphs on 9 vertices which do not have unique spectrum. In fact, the number of graphs of ordern without unique spectrum is believed to be at leastαn3g−1 for someα>0, wheregn is the number of graphs of ordern andn ≥ 7.

Cycles in graphs

The cycle double cover conjecture asserts that in every bridgeless graph one can find a family C of cycles such that each edge appears in exactly two cycles of C. In a first part of this paper we present the conjecture together with a variety of related problems. In a second part we review four different approaches to the conjecture and present interesting recent results by different authors.

Equiangular lines, mutually unbiased bases, and spin models

We use difference sets to construct interesting sets of lines in complex space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in C^k when k-1 is a prime power. Using semiregular relative difference sets with parameters (k,n,k,@l) we construct sets of n+1 mutually unbiased bases in C^k. We show how to construct these difference sets from commutative semifields and that all known maximal sets of mutually unbiased bases can be obtained in this way, resolving a conjecture about the monomiality of maximal sets. We also relate mutually unbiased bases to spin models.

Feasibility conditions for the existence of walk-regular graphs

Abstract A graph X is walk-regular if the vertex-deleted subgraphs of X all have the same characteristic polynomial. Examples of such graphs are vertex-transitive graphs and distance-regular graphs. We show that the usual feasibility conditions for the existence of a distance-regular graph with a given intersection array can be extended so that they apply to walk-regular graphs. Despite the greater generality, our proofs are more elementary than those usually given for distance-regular graphs. An application to the computation of vertex-transitive graphs is described.

Distance regular covers of the complete graph

Distance regular graphs fall into three families: primitive, antipodal, and bipartite. Each antipodal distance regular graph is a covering graph of a smaller (usually primitive) distance regular graph; the antipodal distance graphs of diameter three are covers of the complete graph, and are the first non-trivial case. Many of the known examples are connected with geometric objects, such as projective planes and general&d quadrangles. We set up a classification scheme, and give new existence conditions and new constructions. A relationship with the theory of equi-isoclinic subspaces of KY”, as studied by Lemmens and Seidel, is investigated.

State transfer on graphs

If X is a graph with adjacency matrix A , then we define H ( t ) to be the operator exp ( i t A ) . We say that we have perfect state transfer in X from the vertex u to the vertex v at time ? if the u v -entry of | H ( ? ) u , v | = 1 . State transfer has been applied to key distribution in commercial cryptosystems, and it seems likely that other applications will be found. We offer a survey of some of the work on perfect state transfer and related questions. The emphasis is almost entirely on the mathematics. Highlights? We survey the interactions between graph theory and perfect state transfer. ? Perfect state transfer is of interest in quantum computing. ? We include some new results and open questions.

A new proof of the Erdős-Ko-Rado theorem for intersecting families of permutations

Let S(n) be the symmetric group on n points. A subset S of S(n) is intersecting if for any pair of permutations @p,@s in S there is a point i@?{1,...,n} such that @p(i)=@s(i). Deza and Frankl [P. Frankl, M. Deza, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A 22 (3) (1977) 352-360] proved that if S@?S(n) is intersecting then |S|@?(n-1)!. Further, Cameron and Ku [P.J. Cameron, C.Y. Ku, Intersecting families of permutations, European J. Combin. 24 (7) (2003) 881-890] showed that the only sets that meet this bound are the cosets of a stabilizer of a point. In this paper we give a very different proof of this same result.

Quantum networks on cubelike graphs

Cubelike graphs are the Cayley graphs of the elementary Abelian group

Asymptotic enumeration of Latin rectangles

{\mathbb{Z}}_{2}^{n}

Number-theoretic nature of communication in quantum spin systems.

(e.g., the hypercube is a cubelike graph). We study perfect state transfer between two particles in quantum networks modeled by a large class of cubelike graphs. This generalizes the results of Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)] and Facer et al. [Phys. Rev. A 92, 187902 (2008)].