Expected cover times of random walks on symmetric graphs

We give very simple proofs for an (N–1)H _N–1 lower bound and anN ² upper bound for the expected cover time of symmetric graphs.     

Further results on the expected hitting time,the cover cost and the related invariants of graphs

A close relation between hitting times of the simple random walk on a graph, the Kirchhoff index, the resistance-centrality, and related invariants of unicyclic graphs is displayed. Combining graph transformations and some other techniques, sharp upper and lower bounds on the cover cost (resp. reverse cover cost) of a vertex in an $n$-vertex unicyclic graph are determined. All the corresponding extremal graphs are identified.     

On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.     

Random walks on highly symmetric graphs

We consider uniform random walks on finite graphs withn nodes. When the hitting times are symmetric, the expected covering time is at least 1/2n logn-O(n log logn) uniformly over all such graphs. We also obtain bounds for the covering times in terms of the eigenvalues of the transition matrix of the Markov chain. For distance-regular graphs, a general lower bound of (n-1) logn is obtained. For hypercubes and binomial coefficient graphs, the limit law of the covering time is obtained as well.     

Random walks on colored graphs

We initiate a study of random walks on undirected graphs with colored edges. In our model, a sequence of colors is specified before the walk begins, and it dictates the color of edge to be followed at each step. We give tight upper and lower bounds on the expected cover time of a random walk on an undirected graph with colored edges. We show that, in general, graphs with two colors have exponential expected cover time, and graphs with three or more colors have doubly-exponential expected cover time. We also give polynomial bounds on the expected cover time in a number of interesting special cases. We described applications of our results to understanding the dominant eigenvectors of products and weighted averages of stochastic matrices, and to problems on time-inhomogeneous Markov chains. © 1994 John Wiley & Sons, Inc.     

On the number of maximal bipartite subgraphs of a graph

We show new lower and upper bounds on the maximum number of maximal induced bipartite subgraphs of graphs with n vertices. We present an infinite family of graphs having 105^n/10 ≈ 1.5926ⁿ; such subgraphs show an upper bound of O(12^n/4) = O(1.8613ⁿ) and give an algorithm that finds all maximal induced bipartite subgraphs in time within a polynomial factor of this bound. This algorithm is used in the construction of algorithms for checking k‐colorability of a graph. © 2004 Wiley Periodicals, Inc. J Graph Theory 48: 127–132, 2005     

Non-bipartiteness of Graphs and the Upper Bounds of Dirichlet Forms

The sum of the lower bound and the upper one of the spectrum of our discrete Laplacian is less than or equal to 2. The equality holds if a graph is bipartite while the converse does not hold for general infinite graphs. In this paper, we give an estimate of the upper bounds of Dirichlet forms and using this estimate together with an h-transform, we show that the sum is strictly less than 2 for a certain class of infinite graphs.Dedicated to Professor Yoichiro Takahashi on his sixtieth birthday.     

Some small ball probabilities for Gaussian processes under nonuniform norms

We establish lower and upper bounds for the small ball probability of a centered Gaussian process(X(t)) _t[0,1] ^N under Hölder-type norms as well as upper bounds for some more general functionals. This extends recently established results for the uniform norm. In addition, our proof of the lower bound is considerably simpler. In the special caseN=1 we establish precise estimates under a wider class of norms including in particular the Besov norms.     

图的代数连通度的界(英文)

本文首先给出了简单图的度序列的平方和的上界,利用这些结果,求出了简单图的代数连通度的几个上下界并确定了它们的临界图。另外,文章也给出了加权图的代数连通度的一个下界。     

First Hitting Time Analysis of the Independence Metropolis Sampler

In this paper, we study a special case of the Metropolis algorithm, the Independence Metropolis Sampler (IMS), in the finite state space case. The IMS is often used in designing components of more complex Markov Chain Monte Carlo algorithms. We present new results related to the first hitting time of individual states for the IMS. These results are expressed mostly in terms of the eigenvalues of the transition kernel. We derive a simple form formula for the mean first hitting time and we show tight lower and upper bounds on the mean first hitting time with the upper bound being the product of two factors: a “local” factor corresponding to the target state and a “global” factor, common to all the states, which is expressed in terms of the total variation distance between the target and the proposal probabilities. We also briefly discuss properties of the distribution of the first hitting time for the IMS and analyze its variance. We conclude by showing how some non-independence Metropolis–Hastings algorithms can perform better than the IMS and deriving general lower and upper bounds for the mean first hitting times of a Metropolis–Hastings algorithm.     

Energy of generalized line graphs

The energy of a graph is equal to the sum of the absolute values of its eigenvalues. Line graphs play an important role in the study of graph theory. Generalized line graphs extend the ideas of both line graphs and cocktail party graphs. In this paper, we establish relations between the energy of the generalized line graph of a graph G and the Laplacian and signless Laplacian energies of G. We give upper and lower bounds for the energy of generalized line graphs. Finally, we present upper and lower bounds for some special graphs.     

Covering times of random walks on bounded degree trees and other graphs

The motivating problem for this paper is to find the expected covering time of a random walk on a balanced binary tree withn vertices. Previous upper bounds for general graphs ofO(|V| |E|)⁽¹⁾ andO(|V| |E|/d _min)⁽²⁾ imply an upper bound ofO(n ²). We show an upper bound on general graphs ofO( |E| log |V|), which implies an upper bound ofO(n log² n). The previous lower bound was (|V| log |V|) for trees.⁽²⁾ In our main result, we show a lower bound of (|V| (log _{d max} |V|)²) for trees, which yields a lower bound of (n log² n). We also extend our techniques to show an upper bound for general graphs ofO(max{E T_i} log |V|).     

Several sharp upper bounds for the largest laplacian eigenvalue of a graph

Let K be the quasi-Laplacian matrix of a graph G and B be the adjacency matrix of the line graph of G,respectively.In this paper,we first present two sharp upper bounds for the largest Laplacian eigenvalue of G by applying the non-negative matrix theory to the similar matrix D~(-1/2) KD~(1/2) and U~(-1/2)BU~(1/2),respectively,where D is the degree diagonal matrix of G and U=diag(d_u,d_v,:uv∈E(G)). And then we give another type of the upper bound in terms of the degree of the vertex and the edge number of G.Moreover,we determine all extremal graphs which achieve these upper bounds.Finally, some examples are given to illustrate that our results are better than the earlier and recent ones in some sense.     

A linear bound on the Euler number of threefolds of Calabi–Yau and of general type

Let X⊂ℙ ^N be either a threefold of Calabi–Yau or of general type (embedded with r K _X ). In this article we give lower and upper bounds, linear on the degree of X and N, for the Euler number of X. As a corollary we obtain the boundedness of the region described by the Chern ratios of threefolds with ample canonical bundle and a new upper bound for the number of nodes of a complete intersection threefold. Received: 26 April 2000 / Revised version: 20 November 2000     

The stable set problem and the lift-and-project ranks of graphs

We study the lift-and-project procedures for solving combinatorial optimization problems, as described by Lovász and Schrijver, in the context of the stable set problem on graphs. We investigate how the procedures' performances change as we apply fundamental graph operations. We show that the odd subdivision of an edge and the subdivision of a star operations (as well as their common generalization, the stretching of a vertex operation) cannot decrease the N₀-, N-, or N₊-rank of the graph. We also provide graph classes (which contain the complete graphs) where these operations do not increase the N₀- or the N-rank. Hence we obtain the ranks for these graphs, and we also present some graph-minor like characterizations for them. Despite these properties we give examples showing that in general most of these operations can increase these ranks. Finally, we provide improved bounds for N₊-ranks of graphs in terms of the number of nodes in the graph and prove that the subdivision of an edge or cloning a vertex can increase the N₊-rank of a graph.Research of these authors was supported in part by a PREA from Ontario, Canada and research grants from NSERC.Mathematics Subject Classification (2000): 0C10, 90C22, 90C27, 47D20     

New bounds for the chromatic number of graphs

In this article we first give an upper bound for the chromatic number of a graph in terms of its degrees. This bound generalizes and modifies the bound given in 11 . Next, we obtain an upper bound of the order of magnitude for the coloring number of a graph with small K_2,t (as subgraph), where n is the order of the graph. Finally, we give some bounds for chromatic number in terms of girth and book size. These bounds improve the best known bound, in terms of order and girth, for the chromatic number of a graph when its girth is an even integer. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:110–122, 2008     

Bounds for algebraic gossip on graphs

We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading: Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(Δn) where Δ is the maximum degree of the graph. This leads to a tight bound of for bounded degree graphs and an upper bound of O(n²) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of . Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 45, 185–217, 2014     

Isoperimetric numbers of graph bundles

The main aim of this paper is to give some upper and lower bounds for the isoperimetric numbers of graph coverings or graph bundles, with exact values in some special cases. In addition, we show that the isoperimetric number of any covering graph is not greater than that of the base graph. Mohar's theorem for the isoperimetric number of the cartesian product of a graph and a complete graph can be extended to a more general case: The isoperimetric numberi(G × K _2n) of the cartesian product of any graphG and a complete graphK _2n on even vertices is the minimum of the isoperimetric numberi(G) andn, and it is also a sharp lower bound of the isoperimetric numbers of all graph bundles over the graphG with fiberK _2n. Furthermore, ifn 2i(G) then the isoperimetric number of any graph bundle overG with fibreK _n is equal to the isoperimetric numberi(G) ofG. Partially supported by The Ministry of Education, Korea.     

Phase transitions of the Moran process and algorithmic consequences

The Moran process is a random process that models the spread of genetic mutations through graphs. On connected graphs, the process eventually reaches “fixation,” where all vertices are mutants, or “extinction,” where none are. Our main result is an almost‐tight upper bound on expected absorption time. For all ?>0, we show that the expected absorption time on an n‐vertex graph is o(n^3+?). Specifically, it is at most , and there is a family of graphs where it is Ω(n³). In proving this, we establish a phase transition in the probability of fixation, depending on the mutants' fitness r. We show that no similar phase transition occurs for digraphs, where it is already known that the expected absorption time can be exponential. Finally, we give an improved fully polynomial randomized approximation scheme (FPRAS) for approximating the probability of fixation. On degree‐bounded graphs where some basic properties are given, its running time is independent of the number of vertices.     

Multiple cover time

Motivated by applications in Markov estimation and distributed computing, we define the blanket time of an undirected graph G to be the expected time for a random walk to hit every vertex of G within a constant factor of the number of times predicted by the stationary distribution. Thus the blanket time is, essentially, the number of steps required of a random walk in order that the observed distribution reflect the stationary distribution. We provide substantial evidence for the following conjecture: that the blanket time of a graph never exceeds the cover time by more than a constant factor. In other words, at the cost of a multiplicative constant one can hit every vertex often instead of merely once. We prove the conjecture in the case where the cover time and maximum hitting time differ by a logarithmic factor. This case includes almost all graphs, as well as most “natural” graphs: the hypercube, k-dimensional lattices for k ≥ 2, balanced k-ary trees, and expanders. We further prove the conjecture for perhaps the most natural graphs not falling in the above case: paths and cycles. Finally, we prove the conjecture in the case of independent stochastic processes. © 1996 John Wiley & Sons, Inc. Random Struct. Alg., 9 , 403–411 (1996)