Deterministic random walks on finite graphs

The rotor‐router model, also known as the Propp machine, is a deterministic process analogous to a random walk on a graph. Instead of distributing tokens to randomly chosen neighbors, the Propp machine deterministically serves the neighbors in a fixed order by associating to each vertex a “rotor‐router” pointing to one of its neighbors. This paper investigates the discrepancy at a single vertex between the number of tokens in the rotor‐router model and the expected number of tokens in a random walk, for finite graphs in general. We show that the discrepancy is bounded by O (mn) at any time for any initial configuration if the corresponding random walk is lazy and reversible, where n and m denote the numbers of nodes and edges, respectively. For a lower bound, we show examples of graphs and initial configurations for which the discrepancy at a single vertex is Ω(m) at any time (> 0). For some special graphs, namely hypercube skeletons and Johnson graphs, we give a polylogarithmic upper bound, in terms of the number of nodes, for the discrepancy. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 46,739–761, 2015     

Deterministic random walks on the integers

Jim Propp’s P-machine, also known as the ‘rotor router model’, is a simple deterministic process that simulates a random walk on a graph. Instead of distributing chips to randomly chosen neighbors, it serves the neighbors in a fixed order.We investigate how well this process simulates a random walk. For the graph being the infinite path, we show that, independent of the starting configuration, at each time and on each vertex, the number of chips on this vertex deviates from the expected number of chips in the random walk model by at most a constant c₁, which is approximately 2.29. For intervals of length L, this improves to a difference of O(logL), for the L₂ average of a contiguous set of intervals even to . All these bounds are tight.     

The geometry of t‐spreads in k‐walk‐regular graphs

A graph is walk‐regular if the number of closed walks of length ? rooted at a given vertex is a constant through all the vertices for all ?. For a walk‐regular graph G with d+1 different eigenvalues and spectrally maximum diameter D=d, we study the geometry of its d‐spreads, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a simplex (or tetrahedron in a three‐dimensional case) and we compute its parameters. Moreover, the results are generalized to the case of k‐walk‐regular graphs, a family which includes both walk‐regular and distance‐regular graphs, and their t‐spreads or vertices at distance t from each other. © 2009 Wiley Periodicals, Inc. J Graph Theory 64:312–322, 2010     

On Trees with Double Domination Number Equal to the 2-Outer-Independent Domination Number Plus One

A vertex of a graph is said to dominate itself and all of its neighbors.A double dominating set of a graph G is a set D of vertices of G,such that every vertex of G is dominated by at least two vertices of D.The double domination number of a graph G is the minimum cardinality of a double dominating set of G.For a graph G =(V,E),a subset D V(G) is a 2-dominating set if every vertex of V(G) \ D has at least two neighbors in D,while it is a 2-outer-independent dominating set of G if additionally the set V(G)\D is independent.The 2-outer-independent domination number of G is the minimum cardinality of a 2-outer-independent dominating set of G.This paper characterizes all trees with the double domination number equal to the 2-outer-independent domination number plus one.     

List Coloring of Random and Pseudo-Random Graphs

choice number of a graph G is the minimum integer k such that for every assignment of a set S(v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S(v). It is shown that the choice number of the random graph G(n, p(n)) is almost surely whenever . A related result for pseudo-random graphs is proved as well. By a special case of this result, the choice number (as well as the chromatic number) of any graph on n vertices with minimum degree at least in which no two distinct vertices have more than common neighbors is at most . Received: October 13, 1997     

Bounds on the cover time

Consider a particle that moves on a connected, undirected graphG withn vertices. At each step the particle goes from the current vertex to one of its neighbors, chosen uniformly at random. Tocover time is the first time when the particle has visited all the vertices in the graph starting from a given vertex. In this paper, we present upper and lower bounds that relate the expected cover time for a graph to the eigenvalues of the Markov chain that describes the random walk above. An interesting consequence is that regular expander graphs have expected cover time (n logn).This research was done while this author was a postdoctoral fellow in the Department of Computer Science, Princeton University, and it was supported in part by DNR grant N00014-87-K-0467.     

Scale Free Properties of Random <Emphasis Type="Italic">k</Emphasis>-Trees

Scale free graphs have attracted attention as their non-uniform structure that can be used as a model for many social networks including the WWW and the Internet. In this paper, we propose a simple random model for generating scale free k-trees. For any fixed integer k, a k-tree consists of a generalized tree parameterized by k, and is one of the basic notions in the area of graph minors. Our model is quite simple and natural; it first picks a maximal clique of size k + 1 uniformly at random, it then picks k vertices in the clique uniformly at random, and adds a new vertex incident to the k vertices. That is, the model only makes uniform random choices twice per vertex. Then (asymptotically) the distribution of vertex degree in the resultant k-tree follows a power law with exponent 2 + 1/k, the k-tree has a large clustering coefficient, and the diameter is small. Moreover, our experimental results indicate that the resultant k-trees have extremely small diameter, proportional to o(log n), where n is the number of vertices in the k-tree, and the o(1) term is a function of k.     

Connectivity threshold of Bluetooth graphs

We study the connectivity properties of random Bluetooth graphs that model certain “ad hoc” wireless networks. The graphs are obtained as “irrigation subgraphs” of the well‐known random geometric graph model. There are two parameters that control the model: the radius r that determines the “visible neighbors” of each vertex and the number of edges c that each vertex is allowed to send to these. The randomness comes from the underlying distribution of vertices in space and from the choices of each vertex. We prove that no connectivity can take place with high probability for a range of parameters r, c and completely characterize the connectivity threshold (in c) for values of r close the critical value for connectivity in the underlying random geometric graph.Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 45–66, 2014     

Competing first passage percolation on random graphs with finite variance degrees

We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate λ₁ (λ₂) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if λ₁ = λ₂, then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable V ∈ (0,1), as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If λ₁ ≠ λ₂, on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.     

On Hamiltonian Powers of Digraphs

 For a strongly connected digraph D, the k-th power D ^k of D is the digraph with the same set of vertices, a vertex x being joined to a vertex y in D ^k if the directed distance from x to y in D is less than or equal to k. It follows from a result of Ghouila-Houri that for every digraph D on n vertices and for every k≥n/2, D ^k is hamiltonian. In the paper we characterize these digraphs D of odd order whose (⌈n/2 ⌉−1)-th power is hamiltonian. Revised: June 13, 1997     

Domination number in graphs with minimum degree two

A set D of vertices of a graph G = （V, E） is called a dominating set if every vertex of V not in D is adjacent to a vertex of D. In 1996, Reed proved that every graph of order n with minimum degree at least 3 has a dominating set of cardinality at most 3n/8. In this paper we generalize Reed＇s result. We show that every graph G of order n with minimum degree at least 2 has a dominating set of cardinality at most （3n ＋IV21）/8, where V2 denotes the set of vertices of degree 2 in G. As an application of the above result, we show that for k ≥ 1, the k-restricted domination number rk （G, γ） ≤ （3n＋5k）/8 for all graphs of order n with minimum degree at least 3.     

The k-Dominating Graph

Given a graph G, the k-dominating graph of G, D _k (G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D _k (G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D _k (G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of D _k (G). In this paper we give conditions that ensure D _k (G) is connected.     

Connected -tuple twin domination in de Bruijn and Kautz digraphs

For a digraph G, a k-tuple twin dominating set D of G for some fixed k≥1 is a set of vertices such that every vertex is adjacent to at least k vertices in D, and also every vertex is adjacent from at least k vertices in D. If the subgraph of G induced by D is strongly connected, then D is called a connected k-tuple twin dominating set of G. In this paper, we give constructions of minimal connected k-tuple twin dominating sets for de Bruijn digraphs and Kautz digraphs.     

On a conjecture of Fink and Jacobson concerning k-domination and k-dependence

A set D of vertices of a graph is k-dependent if every vertex of D is joined to at most k?1 vertices in D. Let β_k(G) be the maximum order of a k-dependent set in G. A set D of vertices of G is k-dominating if every vertex not in D is joined to at least k vertices of D. Let γ_k(G) be the minimum order of a k-dominating set in G. Here we prove the following conjecture of Fink and Jacobson: for any simple graph G and any positive integer k, γ_k(G) ≤ β_k(G).     

Erratum to: On the Extremal Zagreb Indices of Graphs with Cut Edges

For a (molecular) graph, the first Zagreb index M ₁ is equal to the sum of squares of the vertex degrees, and the second Zagreb index M ₂ is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we show that all connected graphs with n vertices and k cut edges, the maximum (resp. minimum) M ₁- and M ₂-value are obtained, respectively, and uniquely, at K _n ^k (resp. P _n ^k ), where K _n ^k is a graph obtained by joining k independent vertices to one vertex of K _n−k and P _n ^k is a graph obtained by connecting a pendent path P _k+1 to one vertex of C _n−k .     

Random Walks on Dihedral Groups

In this paper, we look at the lower bounds of two specific random walks on the dihedral group. The first theorem discusses a random walk generated with equal probabilities by one rotation and one flip. We show that roughly p ² steps are necessary for the walk to become close to uniformly distributed on all of D _2p where p≥3 is an integer. Next we take a random walk on the dihedral group generated by a random k-subset of the dihedral group. The latter theorem shows that it is necessary to take roughly p ^2/(k−1) steps in the typical random walk to become close to uniformly distributed on all of D _2p . We note that there is at least one rotation and one flip in the k-subset, or the random walk generated by this subset has periodicity problems or will not generate all of D _2p .     

An upper bound on the length of a Hamiltonian walk of a maximal planar graph

A Hamiltonian walk of a connected graph is a shortest closed walk that passes through every vertex at least once, and the length of a Hamiltonian walk is the total number of edges traversed by the walk. We show that every maximal planar graph with p(≥ 3) vertices has a Hamiltonian cycle or a Hamiltonian walk of length ≤ 3(p - 3)/2.     

On the Ratio Between 2-Domination and Total Outer-Independent Domination Numbers of Trees

A 2-dominating set of a graph G is a set D of vertices of G such that every vertex of V(G)\D has at least two neighbors in D.A total outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G has a neighbor in D,and the set V(G)\D is independent.The 2-domination(total outer-independent domination,respectively)number of a graph G is the minimum cardinality of a 2-dominating(total outer-independent dominating,respectively)set of G.We investigate the ratio between2-domination and total outer-independent domination numbers of trees.     

Coalescent random walks on graphs

Inspired by coalescent theory in biology, we introduce a stochastic model called “multi-person simple random walks” or “coalescent random walks” on a graph G. There are any finite number of persons distributed randomly at the vertices of G. In each step of this discrete time Markov chain, we randomly pick up a person and move it to a random adjacent vertex. To study this model, we introduce the tensor powers of graphs and the tensor products of Markov processes. Then the coalescent random walk on graph G becomes the simple random walk on a tensor power of G. We give estimates of expected number of steps for these persons to meet all together at a specific vertex. For regular graphs, our estimates are exact.