On the kth Laplacian eigenvalues of trees with perfect matchings

AbstractLet Tn+ be the set of all trees of order n with perfect matchings. In this paper, we prove that for any tree T∈Tn+, its kth largest Laplacian eigenvalue μk(T) satisfies μk(T)=2 when n=2k, and μk(T)⩽⌈n2k⌉+2+(⌈n2k⌉)2+42 when n≠2k. Moreover, this upper bound is sharp when n=0(mod2k)

Graphs with many strong orientations

We establish mild conditions under which a possibly irregular, sparse graph $G$ has "many" strong orientations. Given a graph $G$ on $n$ vertices, orient each edge in either direction with probability $1/2$ independently. We show that if $G$ satisfies a minimum degree condition of $(1+c_1)\log_2{n}$ and has Cheeger constant at least $c_2\frac{\log_2\log_2{n}}{\log_2{n}}$, then the resulting randomly oriented directed graph is strongly connected with high probability. This Cheeger constant bound can be replaced by an analogous spectral condition via the Cheeger inequality. Additionally, we provide an explicit construction to show our minimum degree condition is tight while the Cheeger constant bound is tight up to a $\log_2\log_2{n}$ factor.Comment: 14 pages, 4 figures; revised version includes more background and minor changes that better clarify the expositio

Complexity and heights of tori

We prove detailed asymptotics for the number of spanning trees, called complexity, for a general class of discrete tori as the parameters tend to infinity. The proof uses in particular certain ideas and techniques from an earlier paper. Our asymptotic formula provides a link between the complexity of these graphs and the height of associated real tori, and allows us to deduce some corollaries on the complexity thanks to certain results from analytic number theory. In this way we obtain a conjectural relationship between complexity and regular sphere packings.Comment: 14 page

On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature

Chip-firing may be much faster than you think

A new bound (Theorem \ref{thm:main}) for the duration of the chip-firing game with $N$ chips on a $n$-vertex graph is obtained, by a careful analysis of the pseudo-inverse of the discrete Laplacian matrix of the graph. This new bound is expressed in terms of the entries of the pseudo-inverse. It is shown (Section 5) to be always better than the classic bound due to Bj{\"o}rner, Lov\'{a}sz and Shor. In some cases the improvement is dramatic. For instance: for strongly regular graphs the classic and the new bounds reduce to $O(nN)$ and $O(n+N)$, respectively. For dense regular graphs - $d=(\frac{1}{2}+\epsilon)n$ - the classic and the new bounds reduce to $O(N)$ and $O(n)$, respectively. This is a snapshot of a work in progress, so further results in this vein are in the works