Densities of Minor-Closed Graph Families

We define the limiting density of a minor-closed family of simple graphs F to be the smallest number k such that every n-vertex graph in F has at most kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting densities. This set of numbers is countable, well-ordered, and closed; its order type is at least {\omega}^{\omega}. It is the closure of the set of densities of density-minimal graphs, graphs for which no minor has a greater ratio of edges to vertices. By analyzing density-minimal graphs of low densities, we find all limiting densities up to the first two cluster points of the set of limiting densities, 1 and 3/2. For multigraphs, the only possible limiting densities are the integers and the superparticular ratios i/(i+1).Comment: 19 pages, 4 figure

The genus of curve, pants and flip graphs

This article is about the graph genus of certain well studied graphs in surface theory: the curve, pants and flip graphs. We study both the genus of these graphs and the genus of their quotients by the mapping class group. The full graphs, except for in some low complexity cases, all have infinite genus. The curve graph once quotiented by the mapping class group has the genus of a complete graph so its genus is well known by a theorem of Ringel and Youngs. For the other two graphs we are able to identify the precise growth rate of the graph genus in terms of the genus of the underlying surface. The lower bounds are shown using probabilistic methods.Comment: 26 pages, 9 figure

Edge reconstruction of the Ihara zeta function

We show that if a graph $G$ has average degree $\bar d \geq 4$, then the Ihara zeta function of $G$ is edge-reconstructible. We prove some general spectral properties of the edge adjacency operator $T$: it is symmetric for an indefinite form and has a "large" semi-simple part (but it can fail to be semi-simple in general). We prove that this implies that if $\bar d>4$, one can reconstruct the number of non-backtracking (closed or not) walks through a given edge, the Perron-Frobenius eigenvector of $T$ (modulo a natural symmetry), as well as the closed walks that pass through a given edge in both directions at least once. The appendix by Daniel MacDonald established the analogue for multigraphs of some basic results in reconstruction theory of simple graphs that are used in the main text.Comment: 19 pages, 2 pictures, in version 2 some minor changes and now including an appendix by Daniel McDonal