Edge-disjoint spanning trees and eigenvalues of regular graphs

Partially answering a question of Paul Seymour, we obtain a sufficient eigenvalue condition for the existence of $k$ edge-disjoint spanning trees in a regular graph, when $k\in \{2,3\}$. More precisely, we show that if the second largest eigenvalue of a $d$-regular graph $G$ is less than $d-\frac{2k-1}{d+1}$, then $G$ contains at least $k$ edge-disjoint spanning trees, when $k\in \{2,3\}$. We construct examples of graphs that show our bounds are essentially best possible. We conjecture that the above statement is true for any $k<d/2$.Comment: 4 figure

Completely Independent Spanning Trees in Some Regular Graphs

Let $k\ge 2$ be an integer and $T_1,\ldots, T_k$ be spanning trees of a graph $G$. If for any pair of vertices $(u,v)$ of $V(G)$, the paths from $u$ to $v$ in each $T_i$, $1\le i\le k$, do not contain common edges and common vertices, except the vertices $u$ and $v$, then $T_1,\ldots, T_k$ are completely independent spanning trees in $G$. For $2k$-regular graphs which are $2k$-connected, such as the Cartesian product of a complete graph of order $2k-1$ and a cycle and some Cartesian products of three cycles (for $k=3$), the maximum number of completely independent spanning trees contained in these graphs is determined and it turns out that this maximum is not always $k$

On the spanning tree packing number of a graph: a survey

AbstractThe spanning tree packing number or STP number of a graph G is the maximum number of edge-disjoint spanning trees contained in G. We use an observation of Paul Catlin to investigate the STP numbers of several families of graphs including quasi-random graphs, regular graphs, complete bipartite graphs, cartesian products and the hypercubes

Abelian sandpiles: an overview and results on certain transitive graphs

We review the Majumdar-Dhar bijection between recurrent states of the Abelian sandpile model and spanning trees. We generalize earlier results of Athreya and Jarai on the infinite volume limit of the stationary distribution of the sandpile model on Z^d, d >= 2, to a large class of graphs. This includes: (i) graphs on which the wired spanning forest is connected and has one end; (ii) transitive graphs with volume growth at least c n^5 on which all bounded harmonic functions are constant. We also extend a result of Maes, Redig and Saada on the stationary distribution of sandpiles on infinite regular trees, to arbitrary exhaustions.Comment: 44 pages. Version 2 incorporates some smaller changes. To appear in Markov Processes and Related Fields in the proceedings of the meeting: Inhomogeneous Random Systems, Stochastic Geometry and Statistical Mechanics, Institut Henri Poincare, Paris, 27 January 201

Thin Trees in Some Families of Graphs

Let =(,) be a graph and let be a spanning tree of . The thinness parameter of denoted by () is the maximum over all cuts of the proportion of the edges of in the cut. Thin trees play an important role in some recent papers on the Asymmetric Traveling Salesman Problem (ATSP). Goddyn conjectured that every graph of sufficiently large edge-connectivity has a spanning tree such that () ≤ . In this thesis, we study the problem of finding thin spanning trees in two families of graphs, namely, (1) distance-regular graphs (DRGs), and (2) planar graphs. For some families of DRGs such as strongly regular graphs, Johnson graphs, Crown graphs, and Hamming graphs, we give a polynomial-time construction of spanning trees of maximum degree ≤ 3 such that () is determined by the parameters of the graph. For planar graphs, we improve the analysis of Merker and Postle ("Bounded Diameter Arboricity", arXiv:1608.05352v1) and show that every 6-edge-connected planar graph has two edge-disjoint spanning trees ,′ such that (),(′) ≤ 14⁄15. For 8-edge-connected planar graphs , we present a simplified version of the techniques of Merker and Postle and show that has two edge-disjoint spanning trees ,′ such that (),(′) ≤ 12⁄13

On the number of spanning trees in random regular graphs

Let $d \geq 3$ be a fixed integer. We give an asympotic formula for the expected number of spanning trees in a uniformly random $d$-regular graph with $n$ vertices. (The asymptotics are as $n\to\infty$, restricted to even $n$ if $d$ is odd.) We also obtain the asymptotic distribution of the number of spanning trees in a uniformly random cubic graph, and conjecture that the corresponding result holds for arbitrary (fixed) $d$. Numerical evidence is presented which supports our conjecture.Comment: 26 pages, 1 figure. To appear in the Electronic Journal of Combinatorics. This version addresses referee's comment

Statistical mechanics on isoradial graphs

Isoradial graphs are a natural generalization of regular graphs which give, for many models of statistical mechanics, the right framework for studying models at criticality. In this survey paper, we first explain how isoradial graphs naturally arise in two approaches used by physicists: transfer matrices and conformal field theory. This leads us to the fact that isoradial graphs provide a natural setting for discrete complex analysis, to which we dedicate one section. Then, we give an overview of explicit results obtained for different models of statistical mechanics defined on such graphs: the critical dimer model when the underlying graph is bipartite, the 2-dimensional critical Ising model, random walk and spanning trees and the q-state Potts model.Comment: 22 page