Kemeny's constant for a connected graph G is the expected time for a random walk to reach a rando... more 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.
We develop a general spectral framework to analyze quantum fractional revival in quantum spin net... more We develop a general spectral framework to analyze quantum fractional revival in quantum spin networks. In particular, we introduce generalizations of the notions of cospectral and strongly cospectral vertices to arbitrary subsets of vertices, and give various examples. This work resolves two open questions of Chan et. al. ["Quantum Fractional Revival on graphs". Discrete
2022 IEEE International Parallel and Distributed Processing Symposium (IPDPS)
In recent years, graph theoretic considerations have become increasingly important in the design ... more In recent years, graph theoretic considerations have become increasingly important in the design of HPC interconnection topologies. One approach is to seek optimal or near-optimal families of graphs with respect to a particular graph theoretic property, such as diameter. In this work, we consider topologies which optimize the spectral gap. We study a novel HPC topology, SpectralFly, designed around the Ramanujan graph construction of Lubotzky, Phillips, and Sarnak (LPS). We show combinatorial properties, such as diameter, bisection bandwidth, average path length, and resilience to link failure, of SpectralFly topologies are better than, or comparable to, similarly constrained DragonFly, SlimFly, and BundleFly topologies. Additionally, we simulate the performance of SpectralFly on a representative sample of microbenchmarks using the Structure Simulation Toolkit Macroscale Element Library simulator and study cost-minimizing layouts, demonstrating considerable benefit of the SpectralFly topology.
We initiate the study of pretty good quantum fractional revival in graphs, a generalization of pr... more We initiate the study of pretty good quantum fractional revival in graphs, a generalization of pretty good quantum state transfer in graphs. We give a complete characterization of pretty good fractional revival in a graph in terms of the eigenvalues and eigenvectors of the adjacency matrix of a graph. This characterization follows from a lemma due to Kronecker on Diophantine approximation, and is similar to the spectral characterization of pretty good state transfer in graphs. Using this, we give complete characterizations of when pretty good fractional revival can occur in paths and in cycles.
We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number... more We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for simple graphs and a conjecture for weighted graphs.
We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how... more We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. Furthermore, we find an expression for the eigenvalues of the non-backtracking matrix in terms of eigenvalues of the adjacency matrix, and use this to upper-bound the spectral radius of the non-backtracking matrix, and to give a lower bound on the spectrum. We also investigate properties of a graph that can be determined by the spectrum. Specifically, we prove that the number of components, the number of degree 1 vertices, and whether or not the graph is bipartite are all determined by the spectrum of the non-backtracking matrix.
We initiate the study of approximate quantum fractional revival in graphs, a generalization of pr... more We initiate the study of approximate quantum fractional revival in graphs, a generalization of pretty good quantum state transfer in graphs. We give a complete characterization of approximate fractional revival in a graph in terms of the eigenvalues and eigenvectors of the adjacency matrix of a graph. This characterization follows from a lemma due to Kronecker on Diophantine approximation, and is similar to the spectral characterization of pretty good state transfer in graphs. Using this, we give a complete characterizations of when approximate fractional revival can occur in paths and in cycles.
This thesis has two primary areas of focus. First we study connection graphs, which are weighted ... more This thesis has two primary areas of focus. First we study connection graphs, which are weighted graphs in which each edge is associated with a d-dimensional rotation matrix for some fixed dimension d, in addition to a scalar weight. Second, we study non-backtracking random walks on graphs, which are random walks with the additional constraint that they cannot return to the immediately previous state at any given step. Our work in connection graphs is centered on the notion of consistency, that is, the product of rotations moving from one vertex to another is independent of the path taken, and a generalization called epsilon-consistency. We present higher dimensional versions of the combinatorial Laplacian matrix and normalized Laplacian matrix from spectral graph theory, and give results characterizing the consistency of a connection graph in terms of the spectra of these matrices. We generalize several tools from classical spectral graph theory, such as PageRank and effective resi...
We present two conjectures related to strong embeddings of a graph into a surface. The first conj... more We present two conjectures related to strong embeddings of a graph into a surface. The first conjecture relates equivalence of integer quadratic forms given by the Laplacians of graphs, 2-isomorphism of 2connected graphs, and strong embeddings of graphs. We prove various special cases of this conjecture, and give evidence for it. The second conjecture, motivated by ideas from physics and number theory, gives a lower bound on the number of strong embeddings of a graph. If true, this conjecture would imply the well-known Strong Embedding Conjecture.
We prove a homology vanishing theorem for graphs with positive Bakry-Émery curvature, analogous t... more We prove a homology vanishing theorem for graphs with positive Bakry-Émery curvature, analogous to a result of Bochner on manifolds [2]. Specifically, we prove that if a graph has positive curvature at every vertex, then the first path homology group is trivial. We prove this by first showing that a graph with positive curvature can have no non-trivial infinite cover preserving 3-cycles and 4-cycles, and then by giving a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. We also relate cycle spaces of graphs to gain graphs with abelian gain group, and relate these to coverings of graphs.
We obtain a general formula for the resistance distance (or effective resistance) between any pai... more We obtain a general formula for the resistance distance (or effective resistance) between any pair of nodes in a general family of graphs which we call flower graphs. Flower graphs are obtained from identifying nodes of multiple copies of a given base graph in a cyclic way. We apply our general formula to two specific families of flower graphs, where the base graph is either a complete graph or a cycle. We also obtain bounds on the Kirchhoff index and Kemeny's constant of general flower graphs using our formula for resistance. For flower graphs whose base graph is a complete graph or a cycle, we obtain exact, closed form expressions for the Kirchhoff index and Kemeny's constant.
The Minimum Rank, Inverse Inertia, and Inverse Eigenvalue Problems for Graphs Mark C. Kempton Dep... more The Minimum Rank, Inverse Inertia, and Inverse Eigenvalue Problems for Graphs Mark C. Kempton Department of Mathematics Master of Science For a graph G we define S(G) to be the set of all real symmetric n× n matrices whose offdiagonal zero/nonzero pattern is described by G. We show how to compute the minimum rank of all matrices in S(G) for a class of graphs called outerplanar graphs. In addition, we obtain results on the possible eigenvalues and possible inertias of matrices in S(G) for certain classes of graph G. We also obtain results concerning the relationship between two graph parameters, the zero forcing number and the path cover number, related to the minimum rank problem.
We prove a homology vanishing theorem for graphs with positive Bakry-\'Emery curvature, analo... more We prove a homology vanishing theorem for graphs with positive Bakry-\'Emery curvature, analogous to a classic result of Bochner on manifolds \cite{Bochner}. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau \cite{Grigoryan2}. %\Hm{added the fundamental group curvature relation} We moreover prove that the fundamental group is finite for graphs with positive Bakry-\'Emery curvature, analogous to a classic result of Myers on manifolds \cite{Myers1941}. The proofs draw on several separate areas of graph theory. We study graph coverings, gain graphs, and cycle spaces of graphs, in addition to the Bakry-\'Emery curvature and the path homology. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a...
The cyclic edge-connectivity of a graph G is the least k such that there exists a set of k edges ... more The cyclic edge-connectivity of a graph G is the least k such that there exists a set of k edges whose removal disconnects G into components where every component contains a cycle. We show the cyclic edge-connectivity is defined for graphs with minimum degree at least 3 and girth at least 4, as long as G is not K t,3. From the proof of this result it follows that, other than K 3,3 , the cyclic edge-connectivity is bounded above by (d − 2)g for d-regular graphs of girth g ≥ 4. We then provide a spectral condition for when this upper bound on cyclic edge-connectivity is tight.
Kemeny's constant of a simple connected graph G is the expected length of a random walk from i to... more Kemeny's constant of a simple connected graph G is the expected length of a random walk from i to any given vertex j = i. We provide a simple method for computing Kemeny's constant for 1-separable via effective resistance methods from electrical network theory. Using this formula, we furnish a simple proof that the path graph on n vertices maximizes Kemeny's constant for the class of undirected trees on n vertices. Applying this method again, we simplify existing expressions for the Kemeny's constant of barbell graphs and demonstrate which barbell maximizes Kemeny's constant. This 1-separation identity further allows us to create sufficient conditions for the existence of Braess edges in 1-separable graphs. We generalize the notion of the Braess edge to Braess sets, collections of non-edges in a graph such that their addition to the base graph increases the Kemeny constant. We characterize Braess sets in graphs with any number of twin pendant vertices, generalizing work of Kirkland et. al. [9] and Ciardo [7].
In this paper we study quantum state transfer (also called quantum tunneling) on graphs when ther... more In this paper we study quantum state transfer (also called quantum tunneling) on graphs when there is a potential function on the vertex set. We present two main results. First, we show that for paths of length greater than three, there is no potential on the vertices of the path for which perfect state transfer between the endpoints can occur. In particular, this answers a question raised by Godsil in Section 20 of [1]. Second, we show that if a graph has two vertices that share a common neighborhood, then there is a potential on the vertex set for which perfect state transfer will occur between those two vertices. This gives numerous examples where perfect state transfer does not occur without the potential, but adding a potential makes perfect state transfer possible. In addition, we investigate perfect state transfer on graph products, which gives further examples where perfect state transfer can occur.
Pólya's random walk theorem states that a random walk on a d-dimensional grid is recurrent for d ... more Pólya's random walk theorem states that a random walk on a d-dimensional grid is recurrent for d = 1, 2 and transient for d ≥ 3. We prove a version of Pólya's random walk theorem for non-backtracking random walks. Namely, we prove that a non-backtracking random walk on a d-dimensional grid is recurrent for d = 2 and transient for d = 1, d ≥ 3. Along the way, we prove several useful general facts about non-backtracking random walks on graphs. In addition, our proof includes an exact enumeration of the number of closed non-backtracking random walks on an infinite 2-dimensional grid. This enumeration suggests an interesting combinatorial link between non-backtracking random walks on grids, and trinomial coefficients.
A spanning 2-forest separating vertices u and v of an undirected connected graph is a spanning fo... more A spanning 2-forest separating vertices u and v of an undirected connected graph is a spanning forest with 2 components such that u and v are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices u and v in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating u and v divided by the number of spanning trees in the graph. There are also well-known matrix theoretic methods for calculating resistance distance, but the way in which the structure of the underlying graph determines resistance distance via these methods is not well understood. For any connected graph G with a 2-separator separating vertices u and v, we show that the number of spanning trees and spanning 2-forests separating u and v can be expressed in terms of these same quantities for the smaller separated graphs, which makes computation significantly more tractable. An important special case is the preservation of the number of spanning 2-forests if u and v are in the same smaller graph. In this paper we demonstrate that this method of calculating resistance distance is more suitable for certain structured families of graphs than the more standard methods. We apply our results to count the number of spanning 2-forests and calculate the resistance distance in a family of Sierpinski triangles and in the family of linear 2-trees with a single bend.
Kemeny's constant for a connected graph G is the expected time for a random walk to reach a rando... more 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.
We develop a general spectral framework to analyze quantum fractional revival in quantum spin net... more We develop a general spectral framework to analyze quantum fractional revival in quantum spin networks. In particular, we introduce generalizations of the notions of cospectral and strongly cospectral vertices to arbitrary subsets of vertices, and give various examples. This work resolves two open questions of Chan et. al. ["Quantum Fractional Revival on graphs". Discrete
2022 IEEE International Parallel and Distributed Processing Symposium (IPDPS)
In recent years, graph theoretic considerations have become increasingly important in the design ... more In recent years, graph theoretic considerations have become increasingly important in the design of HPC interconnection topologies. One approach is to seek optimal or near-optimal families of graphs with respect to a particular graph theoretic property, such as diameter. In this work, we consider topologies which optimize the spectral gap. We study a novel HPC topology, SpectralFly, designed around the Ramanujan graph construction of Lubotzky, Phillips, and Sarnak (LPS). We show combinatorial properties, such as diameter, bisection bandwidth, average path length, and resilience to link failure, of SpectralFly topologies are better than, or comparable to, similarly constrained DragonFly, SlimFly, and BundleFly topologies. Additionally, we simulate the performance of SpectralFly on a representative sample of microbenchmarks using the Structure Simulation Toolkit Macroscale Element Library simulator and study cost-minimizing layouts, demonstrating considerable benefit of the SpectralFly topology.
We initiate the study of pretty good quantum fractional revival in graphs, a generalization of pr... more We initiate the study of pretty good quantum fractional revival in graphs, a generalization of pretty good quantum state transfer in graphs. We give a complete characterization of pretty good fractional revival in a graph in terms of the eigenvalues and eigenvectors of the adjacency matrix of a graph. This characterization follows from a lemma due to Kronecker on Diophantine approximation, and is similar to the spectral characterization of pretty good state transfer in graphs. Using this, we give complete characterizations of when pretty good fractional revival can occur in paths and in cycles.
We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number... more We conjecture a new lower bound on the algebraic connectivity of a graph that involves the number of vertices of high eccentricity in a graph. We prove that this lower bound implies a strengthening of the Laplacian Spread Conjecture. We discuss further conjectures, also strengthening the Laplacian Spread Conjecture, that include a conjecture for simple graphs and a conjecture for weighted graphs.
We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how... more We investigate the spectrum of the non-backtracking matrix of a graph. In particular, we show how to obtain eigenvectors of the non-backtracking matrix in terms of eigenvectors of a smaller matrix. Furthermore, we find an expression for the eigenvalues of the non-backtracking matrix in terms of eigenvalues of the adjacency matrix, and use this to upper-bound the spectral radius of the non-backtracking matrix, and to give a lower bound on the spectrum. We also investigate properties of a graph that can be determined by the spectrum. Specifically, we prove that the number of components, the number of degree 1 vertices, and whether or not the graph is bipartite are all determined by the spectrum of the non-backtracking matrix.
We initiate the study of approximate quantum fractional revival in graphs, a generalization of pr... more We initiate the study of approximate quantum fractional revival in graphs, a generalization of pretty good quantum state transfer in graphs. We give a complete characterization of approximate fractional revival in a graph in terms of the eigenvalues and eigenvectors of the adjacency matrix of a graph. This characterization follows from a lemma due to Kronecker on Diophantine approximation, and is similar to the spectral characterization of pretty good state transfer in graphs. Using this, we give a complete characterizations of when approximate fractional revival can occur in paths and in cycles.
This thesis has two primary areas of focus. First we study connection graphs, which are weighted ... more This thesis has two primary areas of focus. First we study connection graphs, which are weighted graphs in which each edge is associated with a d-dimensional rotation matrix for some fixed dimension d, in addition to a scalar weight. Second, we study non-backtracking random walks on graphs, which are random walks with the additional constraint that they cannot return to the immediately previous state at any given step. Our work in connection graphs is centered on the notion of consistency, that is, the product of rotations moving from one vertex to another is independent of the path taken, and a generalization called epsilon-consistency. We present higher dimensional versions of the combinatorial Laplacian matrix and normalized Laplacian matrix from spectral graph theory, and give results characterizing the consistency of a connection graph in terms of the spectra of these matrices. We generalize several tools from classical spectral graph theory, such as PageRank and effective resi...
We present two conjectures related to strong embeddings of a graph into a surface. The first conj... more We present two conjectures related to strong embeddings of a graph into a surface. The first conjecture relates equivalence of integer quadratic forms given by the Laplacians of graphs, 2-isomorphism of 2connected graphs, and strong embeddings of graphs. We prove various special cases of this conjecture, and give evidence for it. The second conjecture, motivated by ideas from physics and number theory, gives a lower bound on the number of strong embeddings of a graph. If true, this conjecture would imply the well-known Strong Embedding Conjecture.
We prove a homology vanishing theorem for graphs with positive Bakry-Émery curvature, analogous t... more We prove a homology vanishing theorem for graphs with positive Bakry-Émery curvature, analogous to a result of Bochner on manifolds [2]. Specifically, we prove that if a graph has positive curvature at every vertex, then the first path homology group is trivial. We prove this by first showing that a graph with positive curvature can have no non-trivial infinite cover preserving 3-cycles and 4-cycles, and then by giving a combinatorial interpretation of the first path homology in terms of the cycle space of a graph. We also relate cycle spaces of graphs to gain graphs with abelian gain group, and relate these to coverings of graphs.
We obtain a general formula for the resistance distance (or effective resistance) between any pai... more We obtain a general formula for the resistance distance (or effective resistance) between any pair of nodes in a general family of graphs which we call flower graphs. Flower graphs are obtained from identifying nodes of multiple copies of a given base graph in a cyclic way. We apply our general formula to two specific families of flower graphs, where the base graph is either a complete graph or a cycle. We also obtain bounds on the Kirchhoff index and Kemeny's constant of general flower graphs using our formula for resistance. For flower graphs whose base graph is a complete graph or a cycle, we obtain exact, closed form expressions for the Kirchhoff index and Kemeny's constant.
The Minimum Rank, Inverse Inertia, and Inverse Eigenvalue Problems for Graphs Mark C. Kempton Dep... more The Minimum Rank, Inverse Inertia, and Inverse Eigenvalue Problems for Graphs Mark C. Kempton Department of Mathematics Master of Science For a graph G we define S(G) to be the set of all real symmetric n× n matrices whose offdiagonal zero/nonzero pattern is described by G. We show how to compute the minimum rank of all matrices in S(G) for a class of graphs called outerplanar graphs. In addition, we obtain results on the possible eigenvalues and possible inertias of matrices in S(G) for certain classes of graph G. We also obtain results concerning the relationship between two graph parameters, the zero forcing number and the path cover number, related to the minimum rank problem.
We prove a homology vanishing theorem for graphs with positive Bakry-\'Emery curvature, analo... more We prove a homology vanishing theorem for graphs with positive Bakry-\'Emery curvature, analogous to a classic result of Bochner on manifolds \cite{Bochner}. Specifically, we prove that if a graph has positive curvature at every vertex, then its first homology group is trivial, where the notion of homology that we use for graphs is the path homology developed by Grigor'yan, Lin, Muranov, and Yau \cite{Grigoryan2}. %\Hm{added the fundamental group curvature relation} We moreover prove that the fundamental group is finite for graphs with positive Bakry-\'Emery curvature, analogous to a classic result of Myers on manifolds \cite{Myers1941}. The proofs draw on several separate areas of graph theory. We study graph coverings, gain graphs, and cycle spaces of graphs, in addition to the Bakry-\'Emery curvature and the path homology. The main results follow as a consequence of several different relationships developed among these different areas. Specifically, we show that a...
The cyclic edge-connectivity of a graph G is the least k such that there exists a set of k edges ... more The cyclic edge-connectivity of a graph G is the least k such that there exists a set of k edges whose removal disconnects G into components where every component contains a cycle. We show the cyclic edge-connectivity is defined for graphs with minimum degree at least 3 and girth at least 4, as long as G is not K t,3. From the proof of this result it follows that, other than K 3,3 , the cyclic edge-connectivity is bounded above by (d − 2)g for d-regular graphs of girth g ≥ 4. We then provide a spectral condition for when this upper bound on cyclic edge-connectivity is tight.
Kemeny's constant of a simple connected graph G is the expected length of a random walk from i to... more Kemeny's constant of a simple connected graph G is the expected length of a random walk from i to any given vertex j = i. We provide a simple method for computing Kemeny's constant for 1-separable via effective resistance methods from electrical network theory. Using this formula, we furnish a simple proof that the path graph on n vertices maximizes Kemeny's constant for the class of undirected trees on n vertices. Applying this method again, we simplify existing expressions for the Kemeny's constant of barbell graphs and demonstrate which barbell maximizes Kemeny's constant. This 1-separation identity further allows us to create sufficient conditions for the existence of Braess edges in 1-separable graphs. We generalize the notion of the Braess edge to Braess sets, collections of non-edges in a graph such that their addition to the base graph increases the Kemeny constant. We characterize Braess sets in graphs with any number of twin pendant vertices, generalizing work of Kirkland et. al. [9] and Ciardo [7].
In this paper we study quantum state transfer (also called quantum tunneling) on graphs when ther... more In this paper we study quantum state transfer (also called quantum tunneling) on graphs when there is a potential function on the vertex set. We present two main results. First, we show that for paths of length greater than three, there is no potential on the vertices of the path for which perfect state transfer between the endpoints can occur. In particular, this answers a question raised by Godsil in Section 20 of [1]. Second, we show that if a graph has two vertices that share a common neighborhood, then there is a potential on the vertex set for which perfect state transfer will occur between those two vertices. This gives numerous examples where perfect state transfer does not occur without the potential, but adding a potential makes perfect state transfer possible. In addition, we investigate perfect state transfer on graph products, which gives further examples where perfect state transfer can occur.
Pólya's random walk theorem states that a random walk on a d-dimensional grid is recurrent for d ... more Pólya's random walk theorem states that a random walk on a d-dimensional grid is recurrent for d = 1, 2 and transient for d ≥ 3. We prove a version of Pólya's random walk theorem for non-backtracking random walks. Namely, we prove that a non-backtracking random walk on a d-dimensional grid is recurrent for d = 2 and transient for d = 1, d ≥ 3. Along the way, we prove several useful general facts about non-backtracking random walks on graphs. In addition, our proof includes an exact enumeration of the number of closed non-backtracking random walks on an infinite 2-dimensional grid. This enumeration suggests an interesting combinatorial link between non-backtracking random walks on grids, and trinomial coefficients.
A spanning 2-forest separating vertices u and v of an undirected connected graph is a spanning fo... more A spanning 2-forest separating vertices u and v of an undirected connected graph is a spanning forest with 2 components such that u and v are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices u and v in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating u and v divided by the number of spanning trees in the graph. There are also well-known matrix theoretic methods for calculating resistance distance, but the way in which the structure of the underlying graph determines resistance distance via these methods is not well understood. For any connected graph G with a 2-separator separating vertices u and v, we show that the number of spanning trees and spanning 2-forests separating u and v can be expressed in terms of these same quantities for the smaller separated graphs, which makes computation significantly more tractable. An important special case is the preservation of the number of spanning 2-forests if u and v are in the same smaller graph. In this paper we demonstrate that this method of calculating resistance distance is more suitable for certain structured families of graphs than the more standard methods. We apply our results to count the number of spanning 2-forests and calculate the resistance distance in a family of Sierpinski triangles and in the family of linear 2-trees with a single bend.
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Papers by Mark Kempton