Explanipedia

Quantum walks on join graphs Open

Steve Kirkland · 2025

Quantum walks on finite and bounded infinite graphs Open

Chris Godsil, Steve Kirkland, Sambit Mohapatra, Hiranmoy Pal · 2025

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 pa…

Laplacian quantum walks on blow-up graphs Open

Hiranmoy Pal, Steve Kirkland · 2025

Laplacian quantum walks on blow-up graphs Open

Steve Kirkland · 2025

This paper is a sequel to the work of Bhattacharjya et al.\ (J. Phys. A-Math. 57.33: 335303, https://doi.org/10.1088/1751-8121/ad6653) on quantum state transfer on blow-up graphs, where instead of the adjacency matrix, we take the Laplacia…

Cut-edge centralities in an undirected graph Open

Dario A. Bini, Guy Latouche, Steve Kirkland, Beatrice Meini · 2024

A centrality measure of the cut-edges of an undirected graph, given in [Altafini et al.~SIMAX 2023] and based on Kemeny's constant, is revisited. A numerically more stable expression is given to compute this measure, and an explicit expres…

Completion problems and sparsity for Kemeny’s constant Open

Steve Kirkland · 2024

For a partially specified stochastic matrix, we consider the problem of completing it so as to minimize Kemeny’s constant. We prove that for any partially specified stochastic matrix for which the problem is well defined, there is a minimi…

Quantum walks on join graphs Open

Steve Kirkland, Hermie Monterde · 2023

The join $X\vee Y$ of two graphs $X$ and $Y$ is the graph obtained by joining each vertex of $X$ to each vertex of $Y$. We explore the behaviour of a continuous quantum walk on a weighted join graph having the adjacency matrix or Laplacian…

Gram mates, sign changes in singular values, and isomorphism Open

Sooyeong Kim, Steve Kirkland · 2023

We study distinct $(0,1)$ matrices $A$ and $B$, called \textit{Gram mates}, such that $AA^T=BB^T$ and $A^TA=B^TB$. We characterize Gram mates where one can be obtained from the other by changing signs of some positive singular values. We c…

Numerical ranges of cyclic shift matrices Open

Mao-Ting Chien, Steve Kirkland, Chi-Kwong Li, Hiroshi Nakazato · 2023

We study the numerical range of an $n\times n$ cyclic shift matrix, which can be viewed as the adjacency matrix of a directed cycle with $n$ weighted arcs. In particular, we consider the change in the numerical range if the weights are rea…

Fiedler vectors with unbalanced sign patterns Open

Sooyeong Kim, Steve Kirkland · 2023

In spectral bisection, a Fielder vector is used for partitioning a graph into two connected subgraphs according to its sign pattern. In this article, we investigate graphs having Fiedler vectors with unbalanced sign patterns such that a pa…

Clusters in Markov Chains via Singular Vectors of Laplacian Matrices Open

Sam Cole, Steve Kirkland · 2021

Suppose that $T$ is a stochastic matrix. We propose an algorithm for identifying clusters in the Markov chain associated with $T$. The algorithm is recursive in nature, and in order to identify clusters, it uses the sign pattern of a left …

On node ranking in graphs Open

Ekaterina Dudkina, Michelangelo Bin, Jane Breen, Emanuele Crisostomi, Pietro Ferraro , et al. · 2021

The ranking of nodes in a network according to their ``importance'' is a classic problem that has attracted the interest of different scientific communities in the last decades. The current COVID-19 pandemic has recently rejuvenated the in…

Preface of the special issue on Numerical ranges and numerical radii Open

Steve Kirkland, Chi-Kwong Li · 2021

The study of the numerical range, numerical radius, and their generalizations has a long and extensive history. Much of the impetus for the high level of activity is the connections and application...

On Kemeny's constant for trees with fixed order and diameter Open

Lorenzo Ciardo, Geir Dahl, Steve Kirkland · 2020

Kemeny's constant $κ(G)$ of a connected graph $G$ is a measure of the expected transit time for the random walk associated with $G$. In the current work, we consider the case when $G$ is a tree, and, in this setting, we provide lower and u…

Complex Hadamard Diagonalisable Graphs Open

Ada Chan, Shaun Fallat, Steve Kirkland, Jephian C.-H. Lin, Shahla Nasserasr , et al. · 2020

In light of recent interest in Hadamard diagonalisable graphs (graphs whose Laplacian matrix is diagonalisable by a Hadamard matrix), we generalise this notion from real to complex Hadamard matrices. We give some basic properties and metho…

Applying circuit theory and landscape linkage maps to reintroduction planning for California Condors Open

Jesse G. Delia, Joseph Brandt, L. Joseph Burnett, Susan M. Haig, Jeff P. Hollenbeck , et al. · 2019

Conservation practitioners are increasingly looking to species translocations as a tool to recover imperiled taxa. Quantitative predictions of where animals are likely to move when released into new areas would allow managers to better add…

Fractional revival of threshold graphs under Laplacian dynamics Open

Steve Kirkland, Xiaohong Zhang · 2019

We consider Laplacian fractional revival between two vertices of a graph $X$. Assume that it occurs at time $\\tau$ between vertices 1 and 2. We prove that for the spectral decomposition $L = \\sum_{r=0}^q \\theta_rE_r$ of the Laplacian ma…

On split graphs with four distinct eigenvalues Open

Felix Goldberg, Steve Kirkland, Anu Varghese, Ambat Vijayakumar · 2019

Switching and partially switching the hypercube while maintaining perfect state transfer Open

Steve Kirkland, Sarah Plosker, Xiaohong Zhang · 2019

A graph is said to exhibit perfect state transfer (PST) if one of its corresponding Hamiltonian matrices, which are based on the vertex-edge structure of the graph, gives rise to PST in a quantum information-theoretic context, namely with …

Number of Source Patches Required for Population Persistence in a Source–Sink Metapopulation with Explicit Movement Open

Julien Arino, Nicolas Bajeux, Steve Kirkland · 2019

The Complexity of Power Graphs Associated With Finite Groups Open

Steve Kirkland, A. R. Moghaddamfar, S. Navid Salehy, S. Nima Salehy, Mahsa Zohouratar · 2018

The power graph P(G) of a finite group G is the graph whose vertex set isG, with two elements in G being adjacent if one of them is a power of theother. The purpose of this paper is twofold: (1) to find the complexity ofa clique-replaced g…

The Complexity of Power Graphs Associated With Finite Groups Open

Steve Kirkland, A. R. Moghaddamfar, S. Navid Salehy, S. Nima Salehy, M. Zohourattar · 2018

The power graph $\mathcal{P}(G)$ of a finite group $G$ is the graph whose vertex set is $G$, and two elements in $G$ are adjacent if one of them is a power of the other. The purpose of this paper is twofold. First, we find the complexity o…

Switching and partially switching the hypercube while maintaining\n perfect state transfer Open

Steve Kirkland, Sarah Plosker, Xiaohong Zhang · 2018

A graph is said to exhibit perfect state transfer (PST) if one of its\ncorresponding Hamiltonian matrices, which are based on the vertex-edge\nstructure of the graph, gives rise to PST in a quantum information-theoretic\ncontext, namely wi…

Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials Open

Steve Kirkland, Darian McLaren, Rajesh Pereira, Sarah Plosker, Xiaohong Zhang · 2017

A simple method for transmitting quantum states within a quantum computer is via a quantum spin chain---that is, a path on $n$ vertices. Unweighted paths are of limited use, and so a natural generalization is to consider weighted paths; th…

The Complexity of Power Graphs Associated With Finite Groups Open

Steve Kirkland, A. R. Moghaddamfar, S. Navid Salehy, S. Nima Salehy, Mahsa Zohouratar · 2017

The power graph P(G) of a finite group G is the graph whose vertex set is G, with two elements in G being adjacent if one of them is a power of the other. The purpose of this paper is twofold: (1) to find the complexity of a clique-replace…

Perfect quantum state transfer using Hadamard diagonalizable graphs Open

Nathaniel Johnston, Steve Kirkland, Sarah Plosker, Rebecca Storey, Xiaohong Zhang · 2017

A Panorama of Mathematics: Pure and Applied Open

Carlos M. da Fonseca, Dinh Van Huynh, Steve Kirkland, Vu Kim Tuan · 2016

A totally positive matrix is a matrix having all its minors positive.The largest amount by which the single entries of such a matrix can be perturbed without losing the property of total positivity is given. Also some completion problems f…

Kemeny's Constant And An Analogue Of Braess' Paradox For Trees Open

Steve Kirkland, Ze Min Zeng · 2016

Given an irreducible stochastic matrix M, Kemenyâs constant K(M) measures the expected time for the corresponding Markov chain to transition from any given initial state to a randomly chosen final state. A combinatorially based expressio…

Preface: Special volume of Electronic Journal of Linear Algebra dedicated to Professor Ravindra B. Bapat Open

Rajendra Bhatia, Steve Kirkland, K. Manjunatha Prasad, Simo Puntanen · 2015

This special volume of the Electronic Journal of Linear Algebra is dedicated to Professor Ravindra B. Bapat on the occasion of his 60th birthday. The volume contains papers related to the International Conference on Linear Algebra & its Ap…

Stationary vectors of stochastic matrices subject to combinatorial constraints Open

Jane Breen, Steve Kirkland · 2015

Given a strongly connected directed graph D, let S_D denote the set of all stochastic matrices whose directed graph is a spanning subgraph of D. We consider the problem of completely describing the set of stationary vectors of irreducible …

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