arXiv subjects

Chris Godsil

Publications and source records attributed to Chris Godsil.

At least 19 recordsLinked to original sources

arXiv2026

Compact graphs and quantum automorphisms

Compact graphs are graphs for which the fractional automorphism polytope has no genuinely fractional vertices. This paper proposes a quantum analogue of this idea by evaluating the fundamental magic unitary of the quantum automorphism group on states, which we show to produce a closed convex set of doubly stochastic matrices sitting between the classical automorphism polytope and the full fractional automorphism polytope. Our main result is that the natural quantum analogue of compactness is classical, that is, a quantum compact graph is classically compact. We also relate this set to the quantum orbital algebra and obtain a hierarchy of classical and quantum compactness pseudo notions. The framework recovers familiar consequences of compactness through commutants and suggests quantum analogues of generous transitivity and distance-transitivity. We also isolate examples and open problems indicating where quantum symmetries may strictly refine the classical compactness theory.

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arXiv2026

Total Conformal Rigidity in Graphs

We introduce and study a generalization of conformal rigidity for graphs. A graph is $k$-conformally rigid if the uniform edge weights simultaneously maximize the sum of the $k$ smallest nontrivial Laplacian eigenvalues and minimize the sum of the $k$ largest, over all normalized non-negative weight assignments. A graph that is $k$-conformally rigid for every $k$ is called totally conformally rigid. Our main result is a complete characterization: a graph is totally conformally rigid if and only if it is edge-rigid, meaning every canonical spectral embedding onto a Laplacian eigenspace is edge-isometric. We further show this is equivalent to all edges of the graph being pairwise Laplacian-cospectral, that is, the removal of any single edge yields a graph with the same Laplacian characteristic polynomial. Using semidefinite programming duality, we establish this equivalence and derive a polynomial-time algorithm for deciding edge-rigidity using only integer arithmetic. We provide a combinatorial characterization of edge-rigidity in terms of Laplacian walks and connect it to the walk-regularity of signed line graphs. We show that a graph is edge-rigid if and only if it is either $1$-walk-regular or $1$-walk-biregular, and we finally show an equivalence based on monotone gauges and gauge duality. As an application, we derive two non-trivial combinatorial consequences of total conformal rigidity, relating it to the number of spanning trees and the Kirchhoff index of the graph.

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arXiv2025

Quantum walks on finite and bounded infinite graphs

A weighted graph $G$ with countable vertex set is bounded if there is an upper bound on the maximum of the sum of absolute values of all edge weights incident to a vertex in $G$. In this paper, we prove a fundamental result on equitable partitions of bounded weighted graphs with twin subgraphs and use this fact to construct finite and bounded infinite graphs with pair and plus state transfer with the adjacency matrix as a Hamiltonian. We show that for each $k \ge 3$, (i) there are infinitely many connected unweighted graphs with maximum degree $k$ admitting pair state transfer at $\tau\in\{\frac{\pi}{\sqrt{2}},\frac{\pi}{2}\}$, and (ii) there are infinitely many signed graphs with exactly one negative edge weight and whose underlying unweighted graphs have maximum degree $k$ admitting plus state transfer at $\tau\in\{\frac{\pi}{\sqrt{2}},\frac{\pi}{2}\}$. Parallel results are proven for perfect state transfer between a plus state and a pair state, and for the existence of sedentary pair and plus states. We further prove that almost all connected unweighted finite planar graphs admit pair state transfer at $\tau\in\{\frac{\pi}{\sqrt{2}},\frac{\pi}{2}\}$, and almost all connected unweighted finite planar graphs can be assigned a single negative edge weight resulting in plus state transfer, or perfect state transfer between a plus state and a pair state, at $\tau\in\{\frac{\pi}{\sqrt{2}},\frac{\pi}{2}\}$. Analogous results are shown to hold for unweighted finite trees. Using blow-up graphs, Cayley graphs and graphs with tails, we construct new infinite families of (finite and infinite) unweighted graphs and signed graphs admitting pair or plus state transfer.

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arXiv2025

On cores of distance-regular graphs

We look at the question of which distance-regular graphs are core-complete, meaning they are isomorphic to their own core or have a complete core. We build on Roberson's homomorphism matrix approach by which method he proved the Cameron-Kazanidis conjecture that strongly regular graphs are core-complete. We develop the theory of the homomorphism matrix for distance-regular graphs of diameter $d$. We derive necessary conditions on the cosines of a distance-regular graph for it to admit an endomorphism into a subgraph of smaller diameter $e<d$. As a consequence of these conditions, we show that if $X$ is a primitive distance-regular graph where the subgraph induced by the set of vertices furthest away from a vertex $v$ is connected, any retraction of $X$ onto a diameter-$d$ subgraph must be an automorphism, which recovers Roberson's result for strongly regular graphs as a special case for diameter $2$. We illustrate the application of our necessary conditions through computational results. We find that no antipodal, non-bipartite distance-regular graphs of diameter 3, with degree at most $50$ admits an endomorphism to a diameter 2 subgraph. We also give many examples of intersection arrays of primitive distance-regular graphs of diameter $3$ which are core-complete. Our methods include standard tools from the theory of association schemes, particularly the spectral idempotents. Keywords: algebraic graph theory, distance-regular graphs, association schemes, graph homomorphisms

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arXiv2025

Perfect state transfer between real pure states

Pure states correspond to one-dimensional subspaces of $\mathbb{C}^n$ represented by unit vectors. In this paper, we develop the theory of perfect state transfer (PST) between real pure states with emphasis on the adjacency and Laplacian matrices as Hamiltonians of a graph representing a quantum spin network. We characterize PST between real pure states based on the spectral information of a graph and prove three fundamental results: (i) every periodic real pure state $\mathbf{x}$ admits perfect state transfer with another real pure state $\mathbf{y}$, (ii) every connected graph admits perfect state transfer between real pure states, and (iii) for any pair of real pure states $\mathbf{x}$ and $\mathbf{y}$ and for any time $\tau$, there exists a real symmetric matrix $M$ such that $\mathbf{x}$ and $\mathbf{y}$ admits perfect state transfer relative to $M$ at time $\tau$. We also determine all real pure states that admit PST in complete graphs, complete bipartite graphs, paths, and cycles. This leads to a complete characterization of pair and plus state transfer in paths and complete bipartite graphs. We give constructions of graphs that admit PST between real pure states. Finally, using results on the spread of graphs, we prove that amongst all $n$-vertex simple unweighted graphs, the least minimum PST time between real pure states relative to the Laplacian is attained by any join graph, while the it is attained by the join of an empty graph and a complete graph of appropriate sizes relative to the adjacency matrix.

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arXiv2024

Degree-Similar Graphs

The degree matrix of a graph is the diagonal matrix with diagonal entries equal to the degrees of the vertices of $X$. If $X_1$ and $X_2$ are graphs with respective adjacency matrices $A_1$ and $A_2$ and degree matrices $D_1$ and $D_2$, we say that $X_1$ and $X_2$ are degree similar if there is an invertible real matrix $M$ such that $M^{-1}A_1M=A_2$ and $M^{-1}D_1M=D_2$. If graphs $X_1$ and $X_2$ are degree similar, then their adjacency matrices, Laplacian matrices, unsigned Laplacian matrices and normalized Laplacian matrices are similar. We first show that the converse is not true. Then, we provide a number of constructions of degree-similar graphs. Finally, we show that the matrices $A_1-\mu D_1$ and $A_2-\mu D_2$ are similar over the field of rational functions $\mathbb{Q}(\mu)$ if and only if the Smith normal forms of the matrices $tI-(A_1-\mu D_1)$ and $tI-(A_2-\mu D_2)$ are equal.

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arXiv2024

Oriented or signed Cayley graphs with all eigenvalues integer multiples of $\sqrtΔ$

Let $G$ be a finite abelian group. Bridges and Mena characterized the Cayley graphs of $G$ that have only integer eigenvalues. Here we consider the $(0,1,-1)$ adjacency matrix of an oriented Cayley graph or of a signed Cayley graph $X$ on $G$. We give a characterization of when all the eigenvalues of $X$ are integer multiples of $\sqrtΔ$ for some square-free integer $Δ$. These are exactly the oriented or signed Cayley graphs on which the continuous quantum walks are periodic, a necessary condition for walks on such graphs to admit perfect state transfer. This also has applications in the study of uniform mixing on oriented Cayley graphs, as the occurrence of local uniform mixing at vertex $a$ in an oriented graph $X$ implies periodicity of the walk at $a$. We give examples of oriented Cayley graphs which admit uniform mixing or multiple state transfer.

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arXiv2024

Quantum independence and chromatic numbers

We construct a new graph on 120 vertices whose quantum and classical independence numbers are different. At the same time, we construct an infinite family of graphs whose quantum chromatic numbers are smaller than the classical chromatic numbers. Furthermore, we discover the relation to Kochen-Specker sets that characterizes quantum cocliques that are strictly bigger than classical ones. Finally, we prove that for graphs with independence number is two, quantum and classical independence numbers coincide.

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arXiv2023

A Note on Eigenvalues of Cayley Graphs

A graph is called integral if all its eigenvalues are integers. A Cayley graph is called normal if its connection set is a union of conjugacy classes. We show that a non-empty integral normal Cayley graph for a group of odd order has an odd eigenvalue.

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arXiv2023

Cospectral graphs obtained by edge deletion

Let $M\circ N$ denote the Schur product of two matrices $M$ and $N$. A graph $X$ with adjacency matrix $A$ is walk regular if $A^k\circ I$ is a constant times $I$ for each $k\ge0$, and $X$ is 1-walk-regular if it is walk regular and $A^k\circ A$ is a constant times $A$ for each $k\ge0$. Assume $X$ is 1-walk regular. Here we show that by deleting an edge in $X$, or deleting edges of a graph inside a clique of $X$, we obtain families of graphs that are not necessarily isomorphic, but are cospectral with respect to four types of matrices: the adjacency matrix, Laplacian matrix, unsigned Laplacian matrix, and normalized Laplacian matrix. Furthermore, we show that removing edges of Laplacian cospectral graphs in cliques of a 1-walk regular graph results in Laplacian cospectral graphs; removing edges of unsigned Laplacian cospectral graphs whose complements are also cospectral with respect to the unsigned Laplacian in cliques of a 1-walk regular graph results in unsigned Laplacian cospectral graphs.

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arXiv2023

Strongly cospectral vertices in normal Cayley graphs

We prove an upper bound on the number of pairwise strongly cospectral vertices in a normal Cayley graph, in terms of the multiplicities of its eigenvalues. We use this to determine an explicit bound in Cayley graphs of $\mathbb{Z}_2^d$ and $\mathbb{Z}_4^d$. We also provide some infinite families of Cayley graphs of $\mathbb{Z}_2^d$ with a set of four pairwise strongly cospectral vertices and show that such graphs exist in every dimension.

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arXiv2023

State Transfer in Complex Quantum Walks

Given a graph with Hermitian adjacency matrix $H$, perfect state transfer occurs from vertex $a$ to vertex $b$ if the $(b,a)$-entry of the unitary matrix $\exp(-iHt)$ has unit magnitude for some time $t$. This phenomenon is relevant for information transmission in quantum spin networks and is known to be monogamous under real symmetric matrices. We prove the following results: 1. For oriented graphs (whose nonzero weights are $\pm i$), the oriented $3$-cycle and the oriented edge are the only graphs where perfect state transfer occurs between every pair of vertices. This settles a conjecture of Cameron et al. On the other hand, we construct an infinite family of oriented graphs with perfect state transfer between any pair of vertices on a subset of size four. 2. There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity whenever the adjacency matrix has algebraic entries (as shown by Godsil). 3. There are infinite families with non-monogamous pretty good state transfer in rooted graph products. In particular, we generalize known results on double stars (due to Fan and Godsil) and on paths with loops (due to Kempton, Lippner and Yau). The latter extends the experimental observation of quantum transport (made by Zimbor\'{a}s et al.) and shows non-monogamous pretty good state transfer can occur amongst distant vertices.

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arXiv2022

Transversal polynomial of r-fold covers

We explore the interplay between algebraic combinatorics and algorithmic problems in graph theory by defining a polynomial with connections to correspondence colouring (also known as DP-colouring), a recent generalization of list-colouring, and the Unique Games Conjecture. Like the chromatic polynomial of a graph, we are able to evaluate this polynomial at a point, despite the complexity of computing this polynomial. We construct a cover of a graph $X$ by blowing up each vertex to a set of $r$ vertices and joining each pair of sets corresponding to adjacent vertices by a matching with $r$ edges. To each cover $Y$ of $X$ we associate a polynomial $ξ(Y,t)$, called the transversal polynomial. The coefficient $t^k$ of $ξ(Y,t)$ is the number of $k$-edge induced subgraphs of $Y$ whose vertex set is a transversal of the set system given by the blown-up vertices. We show that $ξ(Y,t)$ satisfies a contraction-deletion formula, and that if $n=|V_X|$ and the cover has index $r$, then $ξ(Y,-(r-1)) \equiv 0 \mod r^n$.

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arXiv2022

On state transfer in Cayley graphs for abelian groups

In this paper, we characterize perfect state transfer in Cayley graphs for abelian groups that have a cyclic Sylow-2-subgroup. This generalizes a result of Bašić from 2013 where he provides a similar characterization for Cayley graphs of cyclic groups.

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arXiv2022

Of Shadows and Gaps in Spatial Search

Spatial search occurs in a connected graph if a continuous-time quantum walk on the adjacency matrix of the graph, suitably scaled, plus a rank-one perturbation induced by any vertex will unitarily map the principal eigenvector of the graph to the characteristic vector of the vertex. This phenomenon is a natural continuous-time analogue of Grover search. The spatial search is said to be optimal if it occurs with constant fidelity and in time inversely proportional to the shadow of the target vertex on the principal eigenvector. Extending a result of Chakraborty et al. (Physical Review A, 102:032214, 2020), we prove a simpler characterization of optimal spatial search. Based on this characterization, we observe that some families of distance-regular graphs, such as Hamming and Grassmann graphs, have optimal spatial search. We also show a matching lower bound on time for spatial search with constant fidelity, which extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary proofs employ standard tools, such as Weyl inequalities and Cauchy determinant formula.

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arXiv2022

Irrational quantum walks

The adjacency matrix of a graph G is the Hamiltonian for a continuous-time quantum walk on the vertices of G. Although the entries of the adjacency matrix are integers, its eigenvalues are generally irrational and, because of this, the behaviour of the walk is typically not periodic. In consequence we can usually only compute numerical approximations to parameters of the walk. In this paper, we develop theory to exactly study any quantum walk generated by an integral Hamiltonian. As a result, we provide exact methods to compute the average of the mixing matrices, and to decide whether pretty good (or almost) perfect state transfer occurs in a given graph. We also use our methods to study geometric properties of beautiful curves arising from entries of the quantum walk matrix, and discuss possible applications of these results.

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arXiv2022

Hamiltonians of Bipartite Walks

In this paper, we introduce a discrete quantum walk model called bipartite walks. Bipartite walks include many known discrete quantum walk models, like arc-reversal walks, vertex-face walks. For the transition matrix of a quantum walk, there is a Hamiltonian associated with it. We will study the Hamiltonians of the bipartite walks. Let $S$ be a skew-symmetric matrix. We are mainly interested in the Hamiltonians of the form $iS$. We show that the Hamiltonian can be written as $iS$ if and only if the adjacency matrix of the bipartite graph is invertible. We show that arc-reversal walks and vertex-face walks are special cases of bipartite walks. Via the Hamiltonians, phenomena of bipartite walks lead to phenomena of continuous walks. We show in detail how we use bipartite walks on paths to construct universal perfect state transfer in continuous walks.

math.CO↗

arXiv2022

Fractional revival on non-cospectral vertices

Perfect state transfer and fractional revival can be used to move information between pairs of vertices in a quantum network. While perfect state transfer has received a lot of attention, fractional revival is newer and less studied. One problem is to determine the differences between perfect state transfer and fractional revival. If perfect state transfer occurs between two vertices in a graph, the vertices must be cospectral. Further if there is perfect state transfer between vertices $a$ and $b$ in a graph, there cannot be perfect state transfer from $a$ to any other vertex. No examples of unweighted graphs with fractional revival between non-cospectral vertices were known; here we give an infinite family of such graphs. No examples of unweighted graphs where the pairs involved in fractional revival overlapped were known; we give examples of such graphs as well.

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