Krystal Guo
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View article: Clique complexes of strongly regular graphs, their eigenvalues, and cohomology groups
Clique complexes of strongly regular graphs, their eigenvalues, and cohomology groups Open
It is known that non-isomorphic strongly regular graphs with the same parameters must be cospectral (have the same eigenvalues). In this paper, we investigate whether the spectra of higher order Laplacians associated with these graphs can …
View article: Cubic graphs with no eigenvalues in the interval (-2,0)
Cubic graphs with no eigenvalues in the interval (-2,0) Open
We give a complete characterisation of the cubic graphs with no eigenvalues in the interval $(-2,0)$. There is one thin infinite family consisting of a single graph on $6n$ vertices for each $n \geqslant 2$, and five ``sporadic'' graphs, n…
View article: Peak state transfer in continuous quantum walks
Peak state transfer in continuous quantum walks Open
We introduce and study peak state transfer, a notion of high state transfer in qubit networks modeled by continuous-time quantum walks. Unlike perfect or pretty good state transfer, peak state transfer does not require fidelity arbitrarily…
View article: On cores of distance-regular graphs
On cores of distance-regular graphs Open
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-Ka…
View article: Spectral approaches for $d$-improper chromatic number
Spectral approaches for $d$-improper chromatic number Open
In this paper, we explore algebraic approaches to $d$-improper and $t$-clustered colourings, where the colouring constraints are relaxed to allow some monochromatic edges. Bilu [J. Comb. Theory Ser. B, 96(4):608-613, 2006] proved a general…
View article: State transfer in discrete-time quantum walks via projected transition matrices
State transfer in discrete-time quantum walks via projected transition matrices Open
In this paper, we analyze state transfer in quantum walks by using combinatorial methods. We generalize perfect state transfer in two-reflection discrete-time quantum walks to a notion that we call 'peak state transfer'; we define peak sta…
View article: Characteristic Polynomials and Hypergraph Generating Functions via Heaps of Pieces
Characteristic Polynomials and Hypergraph Generating Functions via Heaps of Pieces Open
It is a classical result due to Jacobi in algebraic combinatorics that the generating function of closed walks at a vertex $u$ in a graph $G$ is determined by the rational function \[ \frac{ϕ_{G-u}(t)}{ϕ_G(t)} \] where $ϕ_G(t)$ is the char…
View article: Cubic graphs with no eigenvalues in the interval (-1,1)
Cubic graphs with no eigenvalues in the interval (-1,1) Open
We give a complete characterisation of the cubic graphs with no eigenvalues in the open interval $(-1,1)$. There are two infinite families, one due to Guo and Mohar [Linear Algebra Appl. 449:68--75] the other due to Kollár and Sarnak [Comm…
View article: Perfect state transfer in quantum walks on orientable maps
Perfect state transfer in quantum walks on orientable maps Open
A discrete-time quantum walk is the quantum analogue of a Markov chain on a graph. We show that the evolution of a general discrete-time quantum walk that consists of two reflections satisfies a Chebyshev recurrence, under a projection. We…
View article: Selected Open Problems in Continuous-Time Quantum Walks
Selected Open Problems in Continuous-Time Quantum Walks Open
Quantum walks on graphs are fundamental to quantum computing and have led to many interesting open problems in algebraic graph theory. This review article highlights three key classes of open problems in this domain; perfect state transfer…
View article: Selected open problems in continuous-time quantum walks
Selected open problems in continuous-time quantum walks Open
Quantum walks on graphs are fundamental to quantum computing and have led to many interesting open problems in algebraic graph theory. This review article highlights three key classes of open problems in this domain: perfect state transfer…
View article: New eigenvalue bound for the fractional chromatic number
New eigenvalue bound for the fractional chromatic number Open
Given a graph , we let denote the sum of the squares of the positive eigenvalues of the adjacency matrix of , and we similarly define . We prove that and thus strengthen a result of Ando and Lin, who showed the same lower bound for the chr…
View article: Simple eigenvalues of cubic vertex-transitive graphs
Simple eigenvalues of cubic vertex-transitive graphs Open
If ${\mathbf v} \in {\mathbb R}^{V(X)}$ is an eigenvector for eigenvalue $\lambda $ of a graph X and $\alpha $ is an automorphism of X , then $\alpha ({\mathbf v})$ is also an eigenvector for $\lambda $ . Thus, it is rather exceptional for…
View article: Positive and negative square energies of graphs
Positive and negative square energies of graphs Open
The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Let $s^+(G), s^-(G)$ denote the sum of the squares of the positive and negative eigenvalues of $G$, respectively. It was conject…
View article: Positive and Negative Square Energies of Graphs
Positive and Negative Square Energies of Graphs Open
The energy of a graph $G$ is the sum of the absolute values of the eigenvalues of the adjacency matrix of $G$. Let $s^+(G), s^-(G)$ denote the sum of the squares of the positive and negative eigenvalues of $G$, respectively. It was conject…
View article: Perfect state transfer in quantum walks on orientable maps
Perfect state transfer in quantum walks on orientable maps Open
A discrete-time quantum walk is the quantum analogue of a Markov chain on a graph. Zhan [J. Algebraic Combin. 53(4):1187-1213, 2020] proposes a model of discrete-time quantum walk whose transition matrix is given by two reflections, using …
View article: New Eigenvalue Bound for the Fractional Chromatic Number
New Eigenvalue Bound for the Fractional Chromatic Number Open
Given a graph $G$, we let $s^+(G)$ denote the sum of the squares of the positive eigenvalues of the adjacency matrix of $G$, and we similarly define $s^-(G)$. We prove that \[χ_f(G)\ge 1+\max\left\{\frac{s^+(G)}{s^-(G)},\frac{s^-(G)}{s^+(G…
View article: Pseudo-Geometric Strongly Regular Graphs with a Regular Point
Pseudo-Geometric Strongly Regular Graphs with a Regular Point Open
We study pseudo-geometric strongly regular graphs whose second subconstituent with respect to a vertex is a cover of a strongly regular graph or a complete graph. By studying the structure of such graphs, we characterize all graphs contain…
View article: The chromatic index of strongly regular graphs
The chromatic index of strongly regular graphs Open
We determine (partly by computer search) the chromatic index (edge-chromatic number) of many strongly regular graphs (SRGs), including the SRGs of degree k ≤ 18 and their complements, the Latin square graphs and their complements, and the …
View article: Short rainbow cycles in graphs and matroids
Short rainbow cycles in graphs and matroids Open
Let be a simple ‐vertex graph and be a coloring of with colors, where each color class has size at least 2. We prove that contains a rainbow cycle of length at most , which is best possible. Our result settles a special case of a strengthe…
View article: Simple eigenvalues of cubic vertex-transitive graphs
Simple eigenvalues of cubic vertex-transitive graphs Open
If $v$ is an eigenvector for eigenvalue $λ$ of a graph $X$ and $α$ is an automorphism of $X$, then $α(v)$ is also an eigenvector for $λ$. Thus it is rather exceptional for an eigenvalue of a vertex-transitive graph to be simple. We study c…
View article: The Biclique Covering Number of Grids
The Biclique Covering Number of Grids Open
We determine the exact value of the biclique covering number for all grid graphs.
View article: Transversal polynomial of r-fold covers
Transversal polynomial of r-fold covers Open
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…
View article: Diagonal entries of the average mixing matrix
Diagonal entries of the average mixing matrix Open
We study the diagonal entries of the average mixing matrix of continuous quantum walks. The average mixing matrix is a graph invariant; it is the sum of the Schur squares of spectral idempotents of the Hamiltonian. It is non-negative, doub…
View article: The biclique covering number of grids
The biclique covering number of grids Open
We determine the exact value of the biclique covering number for all grid graphs.
View article: A New Perspective on the Average Mixing Matrix
A New Perspective on the Average Mixing Matrix Open
We consider the continuous-time quantum walk defined on the adjacency matrix of a graph. At each instant, the walk defines a mixing matrix which is doubly-stochastic. The average of the mixing matrices contains relevant information about t…