Kemeny's constant measures how fast a random walker moves around in a graph. Expressions for Kemeny's constant can be quite involved, and for this reason, many lines of research focus on graphs with structure that makes them amenable to more in-depth study (for example, regular graphs, acyclic graphs, and 1-connected graphs). In this article, we study Kemeny's constant for random walks on threshold graphs, which are an interesting family of graphs with properties that make examining Kemeny's constant difficult; that is, they are usually not regular, not acyclic, and not 1-connected. This article is a showcase of various techniques for calculating Kemeny's constant and related random walk parameters for graphs. We establish explicit formulae for $\mathcal{K}(G)$ in terms of the construction code of a threshold graph, and completely determine the ordering of the accessibility indices of vertices in threshold graphs.

A sign pattern $\mathcal{A}$ is a matrix with entries in $\{+,-,0\}$. This article introduces the allow sequence of distinct eigenvalues for an $n\times n$ sign pattern $\mathcal{A}$, defined as $q_{\rm seq}(\mathcal{A})=\langle q_1,\ldots,q_n\rangle$, with $q_k=1$ if there exists a real matrix with exactly $k$ distinct eigenvalues having pattern $\mathcal{A}$, and $q_k=0$ otherwise. For example, $q_{\rm seq}(\mathcal{A})=\langle 0,\ldots,0,1\rangle$ is equivalent to $\mathcal{A}$ requiring all distinct eigenvalues, while $q_{\rm seq}(\mathcal{A})=\langle 1,0,\ldots,0\rangle$ is equivalent to the digraph of $\mathcal{A}$ being acyclic. Relationships between the allow sequence for $\mathcal{A}$ and composite cycles of the digraph of $\mathcal{A}$ are explored to identify zeros in the sequence, while methods based on Jacobian matrices are developed to identify ones in the sequence. When $\mathcal{A}$ is an $n\times n$ irreducible sign pattern, the possible sequences for $q_{\rm seq}(\mathcal{A})$ are completely determined when $n\leq 4$ and when the sequence has at least $n-4$ trailing zeros for $n\geq 5$.

The ranking of nodes in a network according to their centrality or ``importance'' is a classic problem that has attracted the interest of different scientific communities in the last decades. The COVID-19 pandemic has recently rejuvenated the interest in this problem, as the ranking may be used to decide who should be tested, or vaccinated, first, in a population of asymptomatic individuals. In this paper, we review classic methods for node ranking and compare their performance in a benchmark network that considers the community-based structure of society. The outcome of the ranking procedure is then used to decide which individuals should be tested, and possibly quarantined, first. Finally, we also review the extension of these ranking methods to weighted graphs and explore the importance of weights in a contact network by providing a toy model and comparing node rankings for this case in the context of disease spread.

Random walks on graphs are often used to analyse and predict epidemic spreads and to investigate possible control actions to mitigate them. In this study, we first show that models based on random walks with a single stochastic agent (such as Google’s popular PageRank) may provide a poor description of certain features of epidemic spread: most notably, spreading times. Then, we discuss another Markov chain based method that does reflect the correct mean infection times for the disease to spread between individuals in a network, and we determine a procedure that allows one to compute them efficiently via a sampling strategy. Finally, we present a novel centrality measure based on infection times, and we compare its node ranking properties with other centrality measures based on random walks. Our results are provided for a simple SI model for epidemic spreading.

Kemeny's constant for a connected graph $G$ is the expected time for a random walk to reach a randomly-chosen vertex $u$, regardless of the choice of the initial vertex. We extend the definition of Kemeny's constant to non-backtracking random walks and compare it to Kemeny's constant for simple random walks. We explore the relationship between these two parameters for several families of graphs and provide closed-form expressions for regular and biregular graphs. In nearly all cases, the non-backtracking variant yields the smaller Kemeny's constant.

The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games in which both players play optimally. In this paper, we construct a family of graphs having $th_c(G)= \Omega(n^{2/3})$, establish a sublinear upper bound on the cop throttling number, and show that the cop throttling number of chordal graphs is $O(\sqrt{n})$. We also introduce the product cop throttling number $th_c^{\times}(G)$ as a parameter that minimizes the person-hours used by the cops. This parameter extends the notion of speed-up that has been studied in the context of parallel processing and network decontamination. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph $G$, $th_c^{\times}=1+rad(G)$.

In this paper, we determine a formula for Kemeny's constant for a graph with multiple bridges, in terms of quantities that are inherent to the subgraphs obtained upon removal of all bridges and that can be computed independently. With the formula, we consider several optimization problems for Kemeny's constant for graphs with bridges, and we remark on the computational benefit of this formula for the computation of Kemeny's constant. Finally, we discuss some potential applications.

We discuss various models for epidemics on networks that rely on Markov chains. Random walks on graphs are often used to predict epidemic spread and to investigate possible control actions to mitigate them. In this study, we demonstrate that they do not fully reflect the dynamics of epidemics, as they overestimate infection times. Accordingly, we explain how Markov chains may still be used to accurately model the virus spread, and to correctly predict infection times. We also provide an algorithm that efficiently estimates infection times via a sampling strategy. Finally, we present a novel indicator based on infection times, and we compare its node ranking properties with other centrality measures based on random walks.

Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper we show that several NP-hard zero forcing numbers are not characterized by the spectra of several types of associated matrices with a graph. In particular, we consider standard zero forcing, positive semidefinite zero forcing, and skew zero forcing, and provide constructions of infinite families of pairs of cospectral graphs which have different values for these numbers. We explore several methods for obtaining these cospectral graphs including using graph products, graph joins, and graph switching. Among these, we provide a construction involving regular adjacency cospectral graphs; the regularity of this construction also implies cospectrality with respect to several other matrices including the Laplacian, signless Laplacian, and normalized Laplacian. We also provide a construction where pairs of cospectral graphs can have an arbitrarily large difference between their zero forcing numbers.

Given any graph G G , the spread of G G is the maximum difference between any two eigenvalues of the adjacency matrix of G G . In this paper, we resolve a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs. The first states that for all positive integers n n , the n n -vertex graph G G that maximizes spread is the join of a clique and an independent set, with ⌊ 2 n / 3 ⌋ \lfloor 2n/3 \rfloor and ⌈ n / 3 ⌉ \lceil n/3 \rceil vertices, respectively. Using techniques from the theory of graph limits and numerical analysis, we prove this claim for all n n sufficiently large. As an intermediate step, we prove an analogous result for a family of operators in the Hilbert space over L 2 [ 0 , 1 ] \mathscr {L}^2[0,1] . The second conjecture claims that for any fixed m ≤ n 2 / 4 m \leq n^2/4 , if G G maximizes spread over all n n -vertex graphs with m m edges, then G G is bipartite. We prove an asymptotic version of this conjecture. Furthermore, we construct an infinite family of counterexamples, which shows that our asymptotic solution is tight up to lower-order error terms.

This article initiates the study of the minimum number of distinct eigenvalues allowed by a sign pattern. Non-trivial examples of sign patterns that allow matrices with only one eigenvalue include the potentially nilpotent and the spectrally arbitrary sign patterns, in contrast to the inverse eigenvalue problem for graphs. Necessary digraph cycle conditions are developed for a sign pattern to have a matrix realization with exactly one eigenvalue. Certain manipulations of sign patterns provide new n×n patterns that preserve the minimum number of eigenvalues allowed, including some Jacobian methods. The 2×2 and the irreducible 3×3 sign patterns are classified according to the minimum number of eigenvalues allowed by the pattern.

If the Laplacian matrix of a graph has a full set of orthogonal eigenvectors with entries $\pm1$, then the matrix formed by taking the columns as the eigenvectors is a Hadamard matrix and the graph is said to be Hadamard diagonalizable. In this article, we prove that if $n=8k+4$ the only possible Hadamard diagonalizable graphs are $K_n$, $K_{n/2,n/2}$, $2K_{n/2}$, and $nK_1$, and we develop a computational method for determining all graphs diagonalized by a given Hadamard matrix of any order. Using these two tools, we determine and present all Hadamard diagonalizable graphs up to order 36. Note that it is not even known how many Hadamard matrices there are of order 36.

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 interest in this problem, as it is related to the selection of which individuals should be tested in a population of asymptomatic individuals, or which individuals should be vaccinated first. Motivated by the COVID-19 spreading dynamics, in this paper we review the most popular methods for node ranking in undirected unweighted graphs, and compare their performance in a benchmark realistic network, that takes into account the community-based structure of society. Also, we generalize a classic benchmark network originally proposed by Newman for ranking nodes in unweighted graphs, to show how ranks change in the weighted case.

The cop throttling number $th_c(G)$ of a graph $G$ for the game of Cops and Robbers is the minimum of $k + capt_k(G)$, where $k$ is the number of cops and $capt_k(G)$ is the minimum number of rounds needed for $k$ cops to capture the robber on $G$ over all possible games in which both players play optimally. In this paper, we answer in the negative a question from [Breen et al., Throttling for the game of Cops and Robbers on graphs, {\em Discrete Math.}, 341 (2018) 2418--2430.] about whether the cop throttling number of any graph is $O(\sqrt n)$ by constructing a family of graphs having $th_c(G)= \Omega(n^{2/3})$. We establish a sublinear upper bound on the cop throttling number and show that the cop throttling number of chordal graphs is $O(\sqrt{n})$. We also introduce the product cop throttling number $th_c^{\times}(G)$ as a parameter that minimizes the person-hours used by the cops. We establish bounds on the product cop throttling number in terms of the cop throttling number, characterize graphs with low product cop throttling number, and show that for a chordal graph $G$, $th_c^{\times}(G)=1+rad(G)$.

Kemeny's constant is an interesting and useful quantifier describing the global average behavior of a Markov chain. In this article, we examine the sensitivity of Kemeny's constant to perturbations in the transition probabilities. That is, we consider the problem of generating a condition number for Kemeny's constant to give an indication of the size of the change in its value relative to the size of the perturbation. We provide a structured condition number and determine some illuminating upper and lower bounds which connect the conditioning of Kemeny's constant to well-studied condition numbers for the stationary vector of the Markov chain. We also investigate the behavior of this structured condition number for several infinite families of Markov chains.

In a graph theory setting, Kemeny’s constant is a graph parameter which measures a weighted average of the mean first passage times in a random walk on the vertices of the graph. In one sense, Kemeny’s constant is a measure of how well the graph is ‘connected’. An explicit computation for this parameter is given for graphs of order n consisting of two large cliques joined by an arbitrary number of parallel paths of equal length, as well as for two cliques joined by two paths of different length. In each case, Kemeny’s constant is shown to be O(n3), which is the largest possible order of Kemeny’s constant for a graph on n vertices. The approach used is based on interesting techniques in spectral graph theory and includes a generalization of using twin subgraphs to find the spectrum of a graph.

We consider the cop-throttling number of a graph G for the game of Cops and Robbers, which is defined to be the minimum of (k+captk(G)), where k is the number of cops and captk(G) is the minimum number of rounds needed for k cops to capture the robber on G over all possible games. We provide some tools for bounding the cop-throttling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are O(n).

Finite, discrete, time-homogeneous Markov chains are frequently used as a simple mathematical model of real-world dynamical systems. In many such applications, an analysis of clustering behaviour in the states of the system is desirable, and it is well-known that the eigendecomposition of the transition matrix A of the Markov chain can provide such insight. Clustering methods based on the sign pattern in the second eigenvector of A are frequently used when A has dominant eigenvalues that are real. In this paper, we present a method to include an analysis for complex eigenvalues of A which are close to 1. Since a real spectrum is not guaranteed in most applications, this is a valuable result in the area of spectral clustering in Markov chains.

For a Markov chain described by an irreducible stochastic matrix A of order n, the mean first passage time mi,j measures the expected time for the Markov chain to reach state j for the first time given that the system begins in state i, thus quantifying the short-term behaviour of the chain. In this article, a lower bound for the maximum mean first passage time is found in terms of the stationary distribution vector of A, and some matrices for which equality is attained are produced. The main objective of this article is to characterise the directed graphs for which any stochastic matrix A respecting this directed graph attains equality in this lower bound, producing a class of Markov chains with optimal short-term behaviour.