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Chris Godsil

Publications and source records attributed to Chris Godsil.

At least 73 records · Page 4Linked to original sources
arXiv2012

Control by quantum dynamics on graphs

We address the study of controllability of a closed quantum system whose dynamical Lie algebra is generated by adjacency matrices of graphs. We characterize a large family of graphs that renders a system controllable. The key property is a novel graph-theoretic feature consisting of a particularly disordered cycle structure. Disregarding efficiency of control functions, but choosing subfamilies of sparse graphs, the results translate into continuous-time quantum walks for universal computation.

quant-ph↗
arXiv2011

Average mixing of continuous quantum walks

If $X$ is a graph with adjacency matrix $A$, then we define $H(t)$ to be the operator $\exp(itA)$. The Schur (or entrywise) product $H(t)\circ H(-t)$ is a doubly stochastic matrix and, because of work related to quantum computing, we are concerned the \textsl{average mixing matrix}. This can be defined as the limit of $C^{-1} \int_0^C H(t)\circ H(-t)\dt$ as $C\to\infty$. We establish some of the basic properties of this matrix, showing that it is positive semidefinite and that its entries are always rational. We find that for paths and cycles this matrix takes on a surprisingly simple form, thus for the path it is a linear combination of $I$, $J$ (the all-ones matrix), and a permutation matrix.

math.CO↗
arXiv2011

Quantum Walks on Regular Graphs and Eigenvalues

We study the transition matrix of a quantum walk on strongly regular graphs. It is proposed by Emms, Hancock, Severini and Wilson in 2006, that the spectrum of $S^+(U^3)$, a matrix based on the amplitudes of walks in the quantum walk, distinguishes strongly regular graphs. We find the eigenvalues of $S^+(U)$ and $S^+(U^2)$ for regular graphs.

math.CO↗
arXiv2011

Perfect state transfer in cubelike graphs

Suppose $C$ is a subset of non-zero vectors from the vector space $\mathbb{Z}_2^d$. The cubelike graph $X(C)$ has $\mathbb{Z}_2^d$ as its vertex set, and two elements of $\mathbb{Z}_2^d$ are adjacent if their difference is in $C$. If $M$ is the $d\times |C|$ matrix with the elements of $C$ as its columns, we call the row space of $M$ the code of $X$. We use this code to study perfect state transfer on cubelike graphs. Bernasconi et al have shown that perfect state transfer occurs on $X(C)$ at time $π/2$ if and only if the sum of the elements of $C$ is not zero. Here we consider what happens when this sum is zero. We prove that if perfect state transfer occurs on a cubelike graph, then it must take place at time $τ=π/2D$, where $D$ is the greatest common divisor of the weights of the code words. We show that perfect state transfer occurs at time $π/4$ if and only if D=2 and the code is self-orthogonal.

math.CO↗
arXiv2011

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 several 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.

quant-ph↗
arXiv2011

When can perfect state transfer occur?

Let $X$ be a graph on $n$ vertices with with adjacency matrix $A$ and let $H(t)$ denote the matrix-valued function $\exp(iAt)$. If $u$ and $v$ are distinct vertices in $X$, we say perfect state transfer from $u$ to $v$ occurs if there is a time $τ$ such that $|H(τ)_{u,v}| = 1$. Our chief problem is to characterize the cases where perfect state transfer occurs. We show that if perfect state transfer does occur in a graph, then the spectral radius is an integer or a quadratic irrational; using this we prove that there are only finitely many graphs with perfect state transfer and with maximum valency at most 4K4. We also show that if perfect state transfer from $u$ to $v$ occurs, then the graphs $X\setminus u$ and $X\setminus v$ are cospectral and any automorphism of $X$ that fixes $u$ must fix $v$ (and conversely).

math.CO↗
arXiv2010

Connectivity and Minimal Distance Spectral Radius of Graphs

In this paper, we study how the distance spectral radius behaves when the graph is perturbed by grafting edges. As applications, we also determine the graph with $k$ cut vertices (respectively, $k$ cut edges) with the minimal distance spectral radius.

math.SP↗
arXiv2010

Generalized Hamming schemes

We introduce a class of association schemes that generalizes the Hamming scheme. We derive generating functions for their eigenvalues, and use these to obtain a version of MacWilliams theorem.

math.CO↗
arXiv2010

Periodic Graphs

Let $X$ be a graph on $n$ vertices with with adjacency matrix $A$ and let $H(t)$ denote the matrix-valued function $\exp(iAt)$. If $u$ and $v$ are distinct vertices in $X$, we say perfect state transfer}from $u$ to $v$ occurs if there is a time $τ$ such that $|H(τ)_{u,v}|=1$. If $u\in V(X)$ and there is a time $\sg$ such that $|H(\sg)_{u,u}|=1$, we say $X$ is periodic at $u$ with period $\sg$. We show that if perfect state transfer from $u$ to $v$ occurs at time $τ$, then $X$ is periodic at both $u$ and $v$ with period $2τ$. We extend previous work by showing that a regular graph with at least four distinct eigenvalues is periodic with respect to some vertex if and only if its eigenvalues are integers. We show that, for a class of graphs $X$ including all vertex-transitive graphs, if perfect state transfer occurs at time $τ$, then $H(τ)$ is a scalar multiple of a permutation matrix of order two with no fixed points. Using certain Hadamard matrices, we construct a new infinite family of graphs on which perfect state transfer occurs.

math.CO↗
arXiv2010

Controllable Subsets in Graphs

Let $X$ be a graph on $v$ vertices with adjacency matrix $A$, and let let $S$ be a subset of its vertices with characteristic vector $z$. We say that the pair $(X,S)$ is controllable if the vectors $A^rz$ for $r=1,\ldots,v-1$ span $\mathbb{R}^v$. Our concern is chiefly with the cases where $S=V(X)$, or $S$ is a single vertex. In this paper we develop the basic theory of controllable pairs. We will see that if $(X,S)$ is controllable then the only automorphism of $X$ that fixes $S$ as a set is the identity. If $(X,S)$ is controllable for some subset $S$ then the eigenvalues of $A$ are all simple.

math.CO↗
arXiv2009

Distinguishing Primitive Permutation Groups

Let $G$ be a permutation group acting on a set $V$. A partition $π$ of $V$ is distinguishing if the only element of $G$ that fixes each cell of $π$ is the identity. The distinguishing number of $G$ is the minimum number of cells in a distinguishing partition. We prove that if $G$ is a primitive permutation group and $|V|\ge336$, its distinguishing number is two.

math.CO↗
arXiv2008

Quantum Networks on Cubelike Graphs

Cubelike graphs are the Cayley graphs of the elementary abelian group (Z_2)^n (e.g., the hypercube is a cubelike graph). We give conditions for perfect state transfer between two particles in quantum networks modeled by a large class of cubelike graphs. This generalizes results of Christandl et al. [Phys. Rev. Lett. 92, 187902 (2004)] and Facer et al. [Phys. Rev. A 92, 187902 (2008)].

quant-ph↗
arXiv2008

Cores of Geometric Graphs

Cameron and Kazanidis have recently shown that rank-3 graphs are either cores or have complete cores, and they asked whether this holds for all strongly regular graphs. We prove that this is true for the point graphs and line graphs of generalized quadrangles and that when the number of points is sufficiently large, it is also true for the block graphs of Steiner systems and orthogonal arrays.

math.CO↗
arXiv2007

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 π, σin S there is a point i in {1,...,n} such that π(i)=σ(i). Deza and Frankl \cite{MR0439648} proved that if S a subset of S(n) is intersecting then |S| \leq (n-1)!. Further, Cameron and Ku \cite{MR2009400} show 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.

math.CO↗
arXiv2007

Type-II Matrices and Combinatorial Structures

Type-II matrices are a class of matrices used by Jones in his work on spin models. In this paper we show that type-II matrices arise naturally in connection with some interesting combinatorial and geometric structures.

math.CO↗
arXiv2007

Two characterizations of crooked functions

We give two characterizations of crooked functions: one based on the minimum distance of a Preparata-like code, and the other based on the distance-regularity of a crooked graph.

math.CO↗
arXiv2005

Colouring Lines in Projective Space

Let $V$ be a vector space of dimension $v$ over a field of order $q$. The $q$-Kneser graph has the $k$-dimensional subspaces of $V$ as its vertices, where two subspaces $α$ and $β$ are adjacent if and only if $α\capβ$ is the zero subspace. This paper is motivated by the problem of determining the chromatic numbers of these graphs. This problem is trivial when $k=1$ (and the graphs are complete) or when $v<2k$ (and the graphs are empty). We establish some basic theory in the general case. Then specializing to the case $k=2$, we show that the chromatic number is $q^2+q$ when $v=4$ and $(q^{v-1}-1)/(q-1)$ when $v > 4$. In both cases we characterise the minimal colourings.

math.CO↗